Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results

Philipp Zimmermann Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain [email protected]

Abstract.

The main purpose of this article is to establish new uniqueness results for Calderón type inverse problems related to damped nonlocal wave equations. To achieve this goal we extend the theory of very weak solutions to our setting, which allows to deduce an optimal Runge approximation theorem. With this result at our disposal, we can prove simultaneous determination results in the linear and semilinear regime.

Keywords. Fractional Laplacian, wave equations, nonlinear PDEs, inverse problems, Runge approximation, very weak solutions.

Mathematics Subject Classification (2020): Primary 35R30; secondary 26A33, 42B37

1. Introduction

In recent years, inverse problems for nonlocal partial differential equations (PDEs) of elliptic, parabolic and hyperbolic type have been studied. This line of research was initiated by Ghosh, Salo and Uhlman [GSU20], in which they have considered the (partial data) Calderón problem related to the fractional Schrödinger equation

(1.1)

\begin{cases}((-\Delta)^{s}+q)u=0&\text{ in }\Omega,\\ u=\varphi&\text{ in }\Omega_{e},\end{cases}

where $\Omega\subset{\mathbb{R}}^{n}$ is a bounded domain, $\Omega_{e}={\mathbb{R}}^{n}\setminus\overline{\Omega}$ , $0<s<1$ , $q$ is a suitable potential and $(-\Delta)^{s}$ is the fractional Laplacian which is the operator with Fourier symbol $|\xi|^{2s}$ . In this problem one asks whether the knowledge of the (partial) Dirichlet to Neumann (DN) map

(1.2)

\Lambda_{q}\varphi=(-\Delta)^{s}u_{\varphi}|_{W_{2}},\quad\varphi\in C_{c}^{\infty}(W_{1}),

where $W_{1},W_{2}\subset\Omega_{e}$ are given measurement sets (i.e. nonempty open sets) and $u_{\varphi}$ denotes the unique solution to (1.1), uniquely determines the potential $q$ . The overall strategy to establish unique determination results for the above Calderón problem is as follows (see [GSU20, RS20, RZ23]):

(i)

Integral identity: Assume that the potentials $q_{j}$ are suitably regular, then one can write

(1.3)

\langle(\Lambda_{q_{1}}-\Lambda_{q_{2}})\varphi_{1},\varphi_{2}\rangle=\int_{\Omega}(q_{1}-q_{2})(u_{\varphi_{1}}-\varphi_{1}),(u_{\varphi_{2}}-\varphi_{2})\,dx,

when the right hand side is interpreted accordingly.

(ii)
Establish one of the following Runge approximation theorems:
1. (I)
  
  $\mathcal{R}_{W}=\{u_{f}|_{\Omega}\,;\,f\in C_{c}^{\infty}(W)\}$ is dense in $L^{2}(\Omega)$ (see [GSU20] for $q\in L^{\infty}(\Omega)$ ).
2. (II)
  
  $\mathscr{R}_{W}=\{u_{f}-f\,;\,f\in C_{c}^{\infty}(W)\}$ is dense in $\widetilde{H}^{s}(\Omega)$ (see [RS20] for Sobolev multipliers $q$ or [RZ23] for local, bounded bilinear forms).
(iii)

If the potentials $q_{j}$ for $j=1,2$ have suitable continuity properties, then $\Lambda_{q_{1}}=\Lambda_{q_{2}}$ together with ii ensure that there holds $q_{1}=q_{2}$ in $\Omega$ .

In iiII, the space $\widetilde{H}^{s}(\Omega)$ is the closure of $C_{c}^{\infty}(\Omega)$ in the energy space

H^{s}({\mathbb{R}}^{n})=\{u\in\mathscr{S}^{\prime}({\mathbb{R}}^{n})\,;\,\|u\|_{H^{s}({\mathbb{R}}^{n})}\vcentcolon=\|\langle D\rangle^{s}u\|_{L^{2}({\mathbb{R}}^{n})}<\infty\},

where $\langle D\rangle^{s}$ is the Bessel potential operator. Observe the similarity of the above strategy to the one of [SU87] for showing unique determination for the classical Calderón problem, where instead of the Runge approximation theorem suitable geometric optics solutions are used. Moreover, the Runge approximation ii relies on a Hahn–Banach argument and the unique continuation property (UCP) of the fractional Laplacian $(-\Delta)^{s}$ . For more results on Calderón problems for elliptic nonlocal PDEs, we refer the interested reader to [GLX17, CLR20, CLL19, LL22, LL23, LZ23, KLZ24, KLW22, LRZ22, LTZ24b, LLU23, CGRU23, LLU23, RZ23, RZ24, CRTZ22, LZ24, FGKU21, Fei21, FKU24, LNZ24] and the references therein.

1.1. Mathematical model and main results

Recently, the above approach for solving elliptic nonlocal inverse problems has also been adapted to deduce uniqueness results for the Calderón problem of nonlocal hyperbolic equations. Let us next describe some of these results in more detail and for this purpose consider the problem

(1.4)

\begin{cases}\partial_{t}^{2}u+\lambda(-\Delta)^{s}\partial_{t}u+(-\Delta)^{s}u+f(u)=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega,\end{cases}

where $\lambda\in{\mathbb{R}}$ and $f\colon\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a possibly nonlinear function. If the problem (1.4) is well-posed in the energy class $H^{s}({\mathbb{R}}^{n})$ , then for any two given measurement sets $W_{1},W_{2}\subset\Omega_{e}$ we may introduce the DN map $\Lambda^{\lambda}_{f}$ via

\Lambda^{\lambda}_{f}\varphi=\left.(\lambda(-\Delta)^{s}\partial_{t}u_{\varphi}+(-\Delta)^{s}u_{\varphi})\right|_{(W_{2})_{T}},

whenever $\varphi$ is supported in $(W_{1})_{T}$ and $u_{\varphi}$ is the solution of (1.4). The Calderón problem for (1.4) reads as follows:

Question 1.

Does the DN map $\Lambda^{\lambda}_{f}$ uniquely determine the function $f$ ?

A suitable class of nonlinearities are the so-called weak nonlinearities, which are defined next.

Definition 1.1.

We call a Carathéodory function $f\colon\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ weak nonlinearity, if it satisfies the following conditions:

(i)

$f$ has partial derivative $\partial_{\tau}f$ , which is a Carathéodory function, and there exists $a\in L^{p}(\Omega)$ such that

(1.5)

\left|\partial_{\tau}f(x,\tau)\right|\lesssim a(x)+|\tau|^{r}

for all $\tau\in{\mathbb{R}}$ and a.e. $x\in\Omega$ . Here the exponents $p$ and $r$ satisfy the restrictions

(1.6)

\begin{cases}n/s\leq p\leq\infty,&\,\text{if }\,2s<n,\\ 2<p\leq\infty,&\,\text{if }\,2s=n,\\ 2\leq p\leq\infty,&\,\text{if }\,2s\geq n.\end{cases}

and

(1.7)

\begin{cases}0\leq r<\infty,&\,\text{if }\,2s\geq n,\\ 0\leq r\leq\frac{2s}{n-2s},&\,\text{if }\,2s<n,\end{cases}

respectively. Moreover, $f$ fulfills the integrability condition $f(\cdot,0)\in L^{2}(\Omega)$ .

(ii)

The function $F\colon\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ defined via

$F(x,\tau)=\int_{0}^{\tau}f(x,\rho)\,d\rho$

satisfies $F(x,\tau)\geq 0$ for all $\tau\in{\mathbb{R}}$ and $x\in\Omega$ .

Let us note that $f(x,\tau)=q(x)|\tau|^{r}\tau$ with $r$ satisfying (1.7) and $0\leq q\in L^{\infty}(\Omega)$ is a weak nonlinearity. Using the above notions, we can now discuss some of the existing results.

(a)

The article [KLW22] gives a positive answer for $\lambda=0,f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(\Omega)$ . Their proof relied on the observation that the related nonlocal wave equation (1.4) satisfies an $L^{2}(\Omega_{T})$ Runge approximation theorem.
(b)

The work [Zim24] deals on the one hand with the linear case $\lambda=1$ , $f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(0,T;L^{p}(\Omega))$ , where $p$ satisfies the restrictions (1.6), and $q$ is weakly continuous in $t$ and on the other hand with the nonlinear case $\lambda=1$ and $f$ is a $r+1$ homogeneous, weak nonlinearity. The uniqueness proofs use substantially that due to presence of the viscosity term $(-\Delta)^{s}\partial_{t}$ solutions $u$ to (1.4) satisfy $\partial_{t}u\in L^{2}(0,T;H^{s}({\mathbb{R}}^{n}))$ and as a consequence the linearized equations have the Runge approximation property in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ .
(c)

In [LTZ24c], uniqueness is proved in the case $\lambda=0$ and $f$ satisfies the same properties as in b, but with the additional restriction $r\leq 1$ . This article only uses an $L^{2}(\Omega_{T})$ Runge approximation result for the linearized nonlocal wave equation.
(d)

By establishing a theory for very weak solutions of linear nonlocal wave equations with $\lambda=0$ , the authors of [LTZ24a] could deduce an optimal $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ Runge approximation theorem for these equations. This allowed to extend the results in c to the cases $r>1$ and additionally showed that one can recover any linear perturbation $q\in L^{p}(\Omega)$ with $p$ satisfying the restrictions (1.6). Furthermore, by this improved Runge approximation theorem the authors could also treat the case of serially or asymptotically polyhomogeneous nonlinearities (see [LTZ24a, Theorem 1.5]).

In this context, let us also mention the recent article [FY24] which deals with the Calderón problem for a third order semilinear, nonlocal, viscous wave equation.

The goal of this paper us to present an extension of the models described in b and c, which we discuss next. Let $0<s<1$ and suppose that we have given coefficients $(\gamma,q)\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ , where $0<s<\alpha\leq 1$ and $1\leq p\leq\infty$ satisfies the restrictions in (1.6). Then we define the following damped, nonlocal wave operator

(1.8)

L_{\gamma}\vcentcolon=\partial_{t}^{2}+\gamma\partial_{t}+(-\Delta)^{s}

and consider the problem

(1.9)

\begin{cases}L_{\gamma}u+f(u)=F&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

where $f$ is a weak nonlinearity or $f(u)=q(x)u$ . In fact, this is a possibly nonlinear generalization of the model (G2) with $s_{1}=0$ in [Zim24, Section 1.1]. By [LTZ24c, Proposition 3.7] (see Section 2.1 for the linear case), we know that the problem (1.9) is well-posed, whenever the source $F$ , exterior condition $\varphi$ and initial conditions $u_{0},u_{1}$ are sufficiently regular. Thus, we can introduce the related (partial) DN map via

(1.10)

\Lambda_{\gamma,f}\varphi\vcentcolon=\left.(-\Delta)^{s}u_{\varphi}\right|_{(W_{2})_{T}},

where $W_{1},W_{2}\subset\Omega_{e}$ are some measurement sets, $\varphi$ is supported in $(W_{1})_{T}$ and $u_{\varphi}$ is the unique solution of (1.9) with $u_{0}=u_{1}=0$ . Then we ask the following question:

Question 2.

Does the partial DN map $\Lambda_{\gamma,f}$ uniquely determine the damping coefficient $\gamma$ and the function $f$ ?

In this work we establish the following affirmative answers to this question, whereas the first result discusses the linear case and the second one the semilinear perturbations.

Theorem 1.2 (Uniqueness for linear perturbations).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that for $j=1,2$ we have given coefficients $(\gamma_{j},q_{j})\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ and let $\Lambda_{\gamma_{j},q_{j}}$ be the DN map associated to the problem

(1.11)

\begin{cases}(L_{\gamma_{j}}+q_{j})u=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega\end{cases}

for $j=1,2$ . If $W_{1},W_{2}\subset\Omega_{e}$ are two measurement sets such that

(1.12)

\left.\Lambda_{\gamma_{1},q_{1}}\varphi\right|_{(W_{2})_{T}}=\left.\Lambda_{\gamma_{2},q_{2}}\varphi\right|_{(W_{2})_{T}}

for all $\varphi\in C_{c}^{\infty}((W_{1})_{T})$ , then there holds

(1.13)

\gamma_{1}=\gamma_{2}\text{ and }q_{1}=q_{2}\text{ in }\Omega.

Theorem 1.3 (Uniqueness for semilinear perturbations).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ and $0<s<\alpha\leq 1$ . Assume that for $j=1,2$ we have given coefficients $\gamma_{j}\in C^{0,\alpha}({\mathbb{R}}^{n})$ and $r+1$ homogeneous, weak nonlinearities $f_{j}$ , where $r>0$ satisfies (1.7). Let $\Lambda_{\gamma_{j},f_{j}}$ be the DN map associated to the problem

(1.14)

\begin{cases}L_{\gamma_{j}}u+f_{j}(u)=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega\end{cases}

for $j=1,2$ . If $W_{1},W_{2}\subset\Omega_{e}$ are two measurement sets such that

(1.15)

\left.\Lambda_{\gamma_{1},f_{1}}\varphi\right|_{(W_{2})_{T}}=\left.\Lambda_{\gamma_{2},f_{2}}\varphi\right|_{(W_{2})_{T}}

for all $\varphi\in C_{c}^{\infty}((W_{1})_{T})$ , then there holds

(1.16)

\gamma_{1}=\gamma_{2}\text{ in }\Omega\text{ and }f_{1}=f_{2}\text{ in }\Omega\times{\mathbb{R}}.

Remark 1.4.

For simplicity we restrict our attention to homogeneous nonlinearities $f$ , but the unique determination remains valid in some polyhomogeneous cases as described in [LTZ24a] for $\gamma=0$ .

1.2. Organization of the article

The rest of this article is structured as follows. In Section 2, we establish the existence of unique weak and very weak solutions to damped nonlocal wave equations. In Section 3 we then move on to the inverse problem part of this work. First, in Section 3.1 we establish the optimal Runge approximation theorem. Afterwards, in Section 3.2 we prove a suitable integral identity that allows us to recover simultaneously the damping coefficient $\gamma$ and potential $q$ in Section 3.3. Finally, Section 3.4 contains the proof of the simultaneous determination of the damping coefficient $\gamma$ and the homogeneous nonlinearity $f$ .

2. Weak and very weak solutions to damped, nonlocal wave equations

The main purpose of this section is to show existence of unique weak and very weak solutions to damped, nonlocal wave equations (DNWEQ)

(2.1)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

where $L_{\gamma}$ is given by (1.8) and only the case $\varphi=0$ is considered for very weak solutions.

2.1. Weak solutions

This section deals with the well-posedness of (2.1) for regular sources, exterior conditions and initial data. We also prove well-posedness for the case when instead of initial values the values at $t=T$ are specified, which will be needed for the development of the theory of very weak solutions.

Theorem 2.1 (Weak solutions to homogeneous DNWEQ).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$ . Then for any $F\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ ¹¹1Here and below we set $\widetilde{L}^{2}(\Omega)\vcentcolon=\widetilde{H}^{0}(\Omega)$ . and initial conditions $(u_{0},u_{1})\in\widetilde{H}^{s}(\Omega)\times\widetilde{L}^{2}(\Omega)$ , there exists a unique weak solution $u\in C([0,T];\widetilde{H}^{s}(\Omega))\cap C^{1}([0,T];\widetilde{L}^{2}(\Omega))$ of

(2.2)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

which means that $(u(0),\partial_{t}u(0))=(u_{0},u_{1})$ in $\widetilde{H}^{s}(\Omega)\times\widetilde{L}^{2}(\Omega)$ and there holds

(2.3)

\begin{split}&\frac{d}{dt}\langle\partial_{t}u,v\rangle_{L^{2}(\Omega)}+\langle\gamma\partial_{t}u,v\rangle_{L^{2}(\Omega)}+\langle(-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^{2}({\mathbb{R}}^{n})}+\langle qu,v\rangle_{L^{2}(\Omega)}\\ &=\langle F,v\rangle_{L^{2}(\Omega)}\end{split}

for all $v\in\widetilde{H}^{s}(\Omega)$ in the sense of distributions on $(0,T)$ . Moreover, the unique solution $u$ obeys the energy identity

(2.4)

\begin{split}&\|\partial_{t}u(t)\|_{L^{2}(\Omega)}^{2}+\|(-\Delta)^{s/2}u(t)\|_{L^{2}({\mathbb{R}}^{n})}^{2}+2\int_{0}^{t}\langle\gamma\partial_{t}u(\tau)+qu(\tau),\partial_{t}u(\tau)\rangle_{L^{2}(\Omega)}\,d\tau\\ &=\|(-\Delta)^{s/2}u_{0}\|_{L^{2}({\mathbb{R}}^{n})}^{2}+\|u_{1}\|_{L^{2}(\Omega)}^{2}+2\int_{0}^{t}\langle F(\tau),\partial_{t}u(\tau)\rangle_{L^{2}(\Omega)}\,d\tau\end{split}

which implies

(2.5)

\begin{split}&\|\partial_{t}u(t)\|_{L^{2}(\Omega)}+\|(-\Delta)^{s/2}u(t)\|_{L^{2}({\mathbb{R}}^{n})}\\ &\leq C(\|u_{1}\|_{L^{2}(\Omega)}+\|(-\Delta)^{s/2}u_{0}\|_{L^{2}({\mathbb{R}}^{n})}+\|F\|_{L^{2}(0,t;L^{2}(\Omega))})\end{split}

for all $0\leq t\leq T$ and some $C>0$ only depending on $\|q\|_{L^{p}(\Omega)}$ , $\|\gamma\|_{L^{{\infty}(\Omega)}}$ and $T>0$ .

Proof.

Throughout the proof, we endow $\widetilde{H}^{s}(\Omega)$ with the equivalent norm $\|u\|_{\widetilde{H}^{s}(\Omega)}=\|(-\Delta)^{s/2}u\|_{L^{2}({\mathbb{R}}^{n})}$ (see [LTZ24c, Lemma 2.3]) and we introduce the following continuous sesquilinear forms

(2.6)

a_{0}(u,v)=\langle(-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^{2}({\mathbb{R}}^{n})},\quad a_{1}(u,v)=\langle qu,v\rangle_{L^{2}(\Omega)}

for $u,v\in\widetilde{H}^{s}(\Omega)$ and

(2.7)

b(u,v)=\langle\gamma u,v\rangle_{L^{2}(\Omega)}

for $u,v\in\widetilde{L}^{2}(\Omega)$ . Next, recall that by [LTZ24c, eq. (3.7)] one has

(2.8)

\|qu\|_{L^{2}(\Omega)}\leq C\|q\|_{L^{p}(\Omega)}\|u\|_{\widetilde{H}^{s}(\Omega)}

for all $u\in\widetilde{H}^{s}(\Omega)$ . It is not hard to see that we can invoke the existence and uniqueness results [DL92, Chapter XVIII, §5, Theorem 3 & 4] (see [DL92, p. 571]), which ensure the existence of a unique, real-valued solution $u\in C([0,T];\widetilde{H}^{s}(\Omega))\cap C^{1}([0,T];\widetilde{L}^{2}(\Omega))$ to (2.2). Furthermore, by [DL92, Chapter XVIII, §5, Lemma 7] the solution $u$ satisfies the following energy identity

(2.9)

\begin{split}&\|\partial_{t}u(t)\|_{L^{2}(\Omega)}^{2}+\|(-\Delta)^{s/2}u(t)\|_{L^{2}({\mathbb{R}}^{n})}^{2}+2\int_{0}^{t}\langle\gamma\partial_{t}u(\tau)+qu(\tau),\partial_{t}u(\tau)\rangle_{L^{2}(\Omega)}\,d\tau\\ &=\|(-\Delta)^{s/2}u_{0}\|_{L^{2}({\mathbb{R}}^{n})}^{2}+\|u_{1}\|_{L^{2}(\Omega)}^{2}+2\int_{0}^{t}\langle F(\tau),\partial_{t}u(\tau)\rangle_{L^{2}(\Omega)}\,d\tau\end{split}

for $0\leq t\leq T$ . Hence, we have shown the identity (2.4). Let us define $\Psi\in C([0,T])$ by

\Psi(t)\vcentcolon=\|\partial_{t}u(t)\|_{L^{2}(\Omega)}^{2}+\|(-\Delta)^{s/2}u(t)\|_{L^{2}({\mathbb{R}}^{n})}^{2}

for $0\leq t\leq T$ . Using (2.8), (2.9) and $\gamma\in L^{\infty}(\Omega)$ , we get

\begin{split}&\Psi(t)\leq\Psi(0)+\int_{0}^{t}\|F(\tau)\|_{L^{2}(\Omega)}^{2}\,d\tau+C\int_{0}^{t}(1+\|\gamma\|_{L^{\infty}(\Omega)}+\|q\|^{2}_{L^{p}(\Omega)})\Psi(\tau)\,d\tau\end{split}

and via Gronwall’s inequality we deduce the energy estimate

(2.10)

\begin{split}&\Psi(t)\leq C(\Psi(0)+\|F\|_{L^{2}(0,t;L^{2}(\Omega))}^{2})\end{split}

for all $0\leq t\leq T$ and some $C>0$ only depending on $\|q\|_{L^{p}(\Omega)}$ , $\|\gamma\|_{L^{\infty}(\Omega)}$ and $T>0$ . This establishes the estimate (2.5). ∎

As a consequence we have the following result:

Proposition 2.2 (Weak solutions to inhomogeneous DNWEQ).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$ . Then for any $F\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ , exterior condition $\varphi\in C^{2}([0,T];H^{2s}({\mathbb{R}}^{n}))$ and initial conditions $(u_{0},u_{1})\in H^{s}({\mathbb{R}}^{n})\times L^{2}({\mathbb{R}}^{n})$ satisfying the compatibility conditions $u_{0}-\varphi(0)\in\widetilde{H}^{s}(\Omega)$ and $u_{1}-\partial_{t}\varphi(0)\in\widetilde{L}^{2}(\Omega)$ , there exists a unique weak solution $u\in C([0,T];H^{s}({\mathbb{R}}^{n}))\cap C^{1}([0,T];L^{2}({\mathbb{R}}^{n}))$ of

(2.11)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

which means that $u$ satisfies (2.3), the $(u(0),\partial_{t}u(0))=(u_{0},u_{1})$ in $H^{s}({\mathbb{R}}^{n})\times L^{2}({\mathbb{R}}^{n})$ and $u=\varphi$ in $(\Omega_{e})_{T}$ means that $u(t)=\varphi(t)$ a.e. in $\Omega_{e}$ for any $0<t<T$ . Furthermore, the following energy estimate holds

(2.12)

\begin{split}&\|\partial_{t}u(t)\|_{L^{2}({\mathbb{R}}^{n})}+\|(-\Delta)^{s/2}u(t)\|_{L^{2}({\mathbb{R}}^{n})}\\ &\leq C(\|u_{1}\|_{L^{2}({\mathbb{R}}^{n})}+\|(-\Delta)^{s/2}u_{0}\|_{L^{2}({\mathbb{R}}^{n})}+\|\varphi\|_{C^{2}([0,t];H^{2s}({\mathbb{R}}^{n}))}+\|F\|_{L^{2}(0,t;L^{2}(\Omega))})\end{split}

for any $0\leq t\leq T$ .

Proof.

Observe, under the current regularity assumptions and compatibility conditions, that $u$ solves (2.11) if and only if $w\vcentcolon=u-\varphi$ solves

(2.13)

\begin{cases}(L_{\gamma}+q)w=F-(L_{\gamma}+q)\varphi&\text{ in }\Omega_{T},\\ w=0&\text{ on }(\Omega_{e})_{T},\\ w(0)=u_{0}-\varphi(0),\,\partial_{t}w(0)=u_{1}-\partial_{t}\varphi(0)&\text{ on }\Omega.\end{cases}

The only fact to keep in mind is that if $u\in C([0,T];H^{s}({\mathbb{R}}^{n}))$ , then the condition $u(t)=\varphi(t)$ a.e. in $\Omega_{e}$ is equivalent to $u(t)-\varphi(t)\in\widetilde{H}^{s}(\Omega)$ as $\Omega\subset{\mathbb{R}}^{n}$ is a bounded Lipschitz domain. So, the assertions of Propsition 2.2 follow immediately from Theorem 2.1. ∎

Next, let us define for any $g\in L^{1}_{loc}(V_{T})$ , $V\subset{\mathbb{R}}^{n}$ open, its time-reversal

(2.14)

g^{\star}(x,t)=g(x,T-t).

Then, we have the following lemma.

Lemma 2.3.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$ . Let $F\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ , $\varphi\in C^{2}([0,T];H^{2s}({\mathbb{R}}^{n}))$ and $(u_{0},u_{1})\in H^{s}({\mathbb{R}}^{n})\times L^{2}({\mathbb{R}}^{n})$ satisfying the compatibility conditions $u_{0}-\varphi(0)\in\widetilde{H}^{s}(\Omega)$ and $u_{1}-\partial_{t}\varphi(0)\in\widetilde{L}^{2}(\Omega)$ . Then $u$ solves

(2.15)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

if and only if $u^{\star}$ solves

(2.16)

\begin{cases}(L_{-\gamma}+q)v=F^{\star}&\text{ in }\Omega_{T},\\ v=\varphi^{\star}&\text{ on }(\Omega_{e})_{T},\\ v(T)=u_{0},\,\partial_{t}v(T)=u_{1}&\text{ on }\Omega.\end{cases}

In particular, for any $F\in L^{2}(\Omega_{T})$ , $\varphi\in C^{2}([0,T];H^{2s}({\mathbb{R}}^{n}))$ and $(u_{0},u_{1})\in H^{s}({\mathbb{R}}^{n})\times L^{2}({\mathbb{R}}^{n})$ satisfying the compatibility conditions $u_{0}-\varphi(T)\in\widetilde{H}^{s}(\Omega)$ and $u_{1}-\partial_{t}\varphi(T)\in\widetilde{L}^{2}(\Omega)$ , there exists a unique solution $u^{\star}$ of

(2.17)

\begin{cases}(L_{-\gamma}+q)v=F&\text{ in }\Omega_{T},\\ v=\varphi&\text{ on }(\Omega_{e})_{T},\\ v(T)=u_{0},\,\partial_{t}v(T)=u_{1}&\text{ on }\Omega.\end{cases}

Proof.

First, note that by the proof of Proposition 2.2, we can assume without loss of generality that $\varphi=0$ . Secondly, one easily sees that $\partial_{t}u^{\star}=-(\partial_{t}u)^{\star}$ and thus a change of variables in (2.3) gives the asserted equivalence. The unique solvability of (2.17) follows from the equivalence and Theorem 2.1. ∎

2.2. Very weak solutions

Let us start by making some simple observations. Suppose that $u$ and $v$ are smooth solutions of the problems

(2.18)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

and

(2.19)

\begin{cases}(L_{-\gamma}+q)v=G&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(T)=0,\,\partial_{t}v(T)=0&\text{ on }\Omega,\end{cases}

respectively. If we multiply the PDE (2.18) by $v$ and integrate over $\Omega_{T}$ , then we get

(2.20)

\begin{split}&\int_{\Omega_{T}}Fv\,dxdt=\int_{\Omega_{T}}[(L_{\gamma}+q)u]v\,dxdt\\ &=\int_{\Omega}u_{0}\partial_{t}v(0)\,dx-\int_{\Omega}u_{1}v(0)\,dx-\int_{\Omega}\gamma u_{0}v(0)\,dx+\int_{\Omega_{T}}u(L_{-\gamma}+q)v\,dxdt\\ &=\int_{\Omega}u_{0}\partial_{t}v(0)\,dx-\int_{\Omega}u_{1}v(0)\,dx-\int_{\Omega}\gamma u_{0}v(0)\,dx+\int_{\Omega_{T}}Gu\,dxdt.\end{split}

Notice that if $G\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ , $v\in L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ and $(v(0),\partial_{t}v(0))\in\widetilde{H}^{s}(\Omega)\times\widetilde{L}^{2}(\Omega)$ , then one can make sense of the first integral and the last line in (2.20), even in the case $F\in L^{2}(0,T;H^{-s}(\Omega))$ , $(u_{0},u_{1})\in\widetilde{L}^{2}(\Omega)\times H^{-s}(\Omega)$ and $u\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ . Here, $H^{-s}(\Omega)\subset\mathscr{D}^{\prime}(\Omega)$ is defined by

H^{-s}(\Omega)=\{u|_{\Omega}\,;\,u\in H^{-s}({\mathbb{R}}^{n})\}

and it can be identified with the dual space of $\widetilde{H}^{s}(\Omega)$ , when $\Omega$ is Lipschitz. The previous computation motivates the following definition.

Definition 2.4 (Very weak solutions).

(2.21)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

whenever there holds²²2Here and below we sometimes write $\langle\cdot,\cdot\rangle$ to denote the duality pairing between $H^{-s}(\Omega)\times\widetilde{H}^{s}(\Omega)$ .

(2.22)

\int_{0}^{T}\langle G,u\rangle_{L^{2}(\Omega)}\,dt=\int_{0}^{T}\langle F,v\rangle\,dt+\langle u_{1},v(0)\rangle-\langle u_{0},\partial_{t}v(0)\rangle_{L^{2}(\Omega)}+\langle\gamma u_{0},v(0)\rangle

for all $G\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ , where $v\in C([0,T];\widetilde{H}^{s}(\Omega))\cap C^{1}([0,T];\widetilde{L}^{2}(\Omega))$ is the unique weak solution of the adjoint equation

(2.23)

\begin{cases}(L_{-\gamma}+q)v=G&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(T)=0,\,\partial_{t}v(T)=0&\text{ on }\Omega,\end{cases}

(see Theorem 2.1).

Next, let us recall the following well-posedness result of very weak solutions.

Theorem 2.5 (Very weak solutions for $\gamma=q=0$ ,[LTZ24a, Theorem 3.6]).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ and $0<s<1$ . Then for any given source $F\in L^{2}(0,T;H^{-s}(\Omega))$ and initial conditions $(u_{0},u_{1})\in\widetilde{L}^{2}(\Omega)\times H^{-s}(\Omega)$ , there exists a unique solution to

(2.24)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})u=F&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega\end{cases}

and it satisfies the following energy estimate

(2.25)

\|u(t)\|_{L^{2}(\Omega)}+\|\partial_{t}u(t)\|_{H^{-s}(\Omega)}\leq C(\|u_{0}\|_{L^{2}(\Omega)}+\|u_{1}\|_{H^{-s}(\Omega)}+\|F\|_{L^{2}(0,t;H^{-s}(\Omega))})

for all $0\leq t\leq T$ .

Hence, we have a well-defined solution map.

Proposition 2.6 (Solution map).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<1$ and let $X_{s}\vcentcolon=\widetilde{L}^{2}(\Omega)\times H^{-s}(\Omega)$ be endowed with the usual product norm

\|(u,w)\|_{X_{s}}\vcentcolon=(\|u\|_{L^{2}(\Omega)}^{2}+\|w\|_{H^{-s}(\Omega)}^{2})^{1/2}.

Then the solution map $S\colon L^{2}(0,T;H^{-s}(\Omega))\to C([0,T];X_{s})$ defined by

(2.26)

S(F)\vcentcolon=(u,\partial_{t}u),

where $u\in C([0,T];\widetilde{L}^{2}(\Omega))\cap C^{1}([0,T];H^{-s}(\Omega))$ is the unique solution of (2.24) with $(u_{0},u_{1})=0$ . Moreover, the solution map is continuous and satisfies the estimate

(2.27)

\|S(F)(t)\|_{X_{s}}\leq C\|F\|_{L^{2}(0,t;H^{-s}(\Omega))}

for any $0\leq t\leq T$ .

Proof.

First of all note that the solution map $S$ is well-defined by Theorem 2.5. The estimate (2.27) follows from (2.25), which together with the linearity of $S$ gives the continuity of $S$ . Observe that the linearity of $S$ is a direct consequence of the unique solvability of (2.24) and the fact that the PDE is linear. ∎

Theorem 2.7.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<1$ and suppose that $\mathcal{F}\colon C([0,T];X_{s})\to L^{2}(0,T;H^{-s}(\Omega))$ satisfies the Lipschitz estimate

(2.28)

\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\|_{H^{-s}(\Omega)}\leq C\|U(t)-V(t)\|_{X_{s}}

for a.e. $0\leq t\leq T$ and $U,V\in C([0,T];X_{s})$ . Then for all $(u_{0},u_{1})\in X_{s}$ , there exists a unique solution $u$ of

(2.29)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})u=\mathcal{F}(u,\partial_{t}u)&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

that is the formula (2.22) holds with $F$ replaced by $\mathcal{F}(u,\partial_{t}u)$ in which we test against every weak solution $v$ of the adjoint equation

(2.30)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})v=G&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(T)=0,\,\partial_{t}v(T)=0&\text{ on }\Omega\end{cases}

with $G\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ .

Proof of Theorem 2.7.

Let $u_{h}\in C([0,T];\widetilde{L}^{2}(\Omega))\cap C^{1}([0,T];H^{-s}(\Omega))$ be the unique solution to

(2.31)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})u=0&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega\end{cases}

and let us set $U_{h}\vcentcolon=(u_{h},\partial_{t}u_{h})\in C([0,T];X_{s})$ . Furthermore, we define the operator $\mathcal{T}\colon C([0,T];X_{s})\to C([0,T];X_{s})$ as

(2.32)

\mathcal{T}(U)\vcentcolon=U_{h}+S(\mathcal{F}(U)),

which is well-defined by (2.6) and the properties of $\mathcal{F}$ . Next, we show that $\mathcal{T}$ has a unique fixed point $U=(U_{1},U_{2})$ .

Step 1. Existence. Let $U,V\in C([0,T];X_{s})$ , then by linearity of $S$ , (2.27) and (2.28) we get

\begin{split}\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{X_{s}}&=\|S(\mathcal{F}(U))(t)-S(\mathcal{F}(V))(t)\|_{X_{s}}\\ &=\|S(\mathcal{F}(U)-\mathcal{F}(V))(t)\|_{X_{s}}\\ &\leq C\|\mathcal{F}(U)-\mathcal{F}(V)\|_{L^{2}(0,t;H^{-s}(\Omega))}\\ &\leq C\|U-V\|_{L^{2}(0,t;X_{s})}.\end{split}

Next, let us define the following norm on $X_{s}$

(2.33)

\|U\|_{\theta}\vcentcolon=\sup_{0\leq t\leq T}\left(e^{-\theta t}\|U(t)\|_{X_{s}}\right)

for $\theta>0$ , which will be fixed in a moment. Then we have the estimate

\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{X_{s}}\leq C\left(\int_{0}^{t}e^{2\theta\tau}\,d\tau\right)^{1/2}\|U-V\|_{\theta}\leq\frac{C}{(2\theta)^{1/2}}e^{\theta t}\|U-V\|_{\theta}

and hence there holds

\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{\theta}\leq\frac{C}{(2\theta)^{1/2}}\|U-V\|_{\theta}.

Therefore, we deduce that $\mathcal{T}$ is a strict contraction from the complete metric space $(C([0,T];X_{s}),\|\cdot\|_{\theta})$ to itself, when $\theta>0$ is chosen such that $C/(2\theta)^{1/2}<1$ . Now, we may invoke Banach’s fixed point theorem to obtain a unique fixed point $U=(u,w)$ of $\mathcal{T}$ . Next, observe that the definition of the solution map $S$ and $U=\mathcal{T}(U)=U_{h}+S(\mathcal{F}(U))$ imply

u=u_{h}+u_{n}\text{ and }w=\partial_{t}u,

where $u_{n}$ solves

(2.34)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})v=\mathcal{F}(U)&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(0)=0,\,\partial_{t}v(0)=0&\text{ on }\Omega.\end{cases}

Going back to the definition of very weak solutions, we see this implies that $u$ solves (2.29).

Step 2. Uniqueness. Suppose $\tilde{u}\in C([0,T];\widetilde{L}^{2}(\Omega))\cap C^{1}([0,T];H^{-s}(\Omega))$ is any other solution to (2.29), then $\bar{u}\vcentcolon=u-\widetilde{u}$ solves

(2.35)

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})v=\mathcal{F}(u,\partial_{t}u)-\mathcal{F}(\widetilde{u},\partial_{t}\widetilde{u})&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(0)=0,\,\partial_{t}v(0)=0&\text{ on }\Omega.\end{cases}

Thus, applying the energy estimate (2.25) together with the Lipschitz assumption on $\mathcal{F}$ , we see that

\begin{split}\|U(t)-\widetilde{U}(t)\|^{2}_{X_{s}}&\leq C\int_{0}^{t}\|\mathcal{F}(U)(\tau)-\mathcal{F}(\widetilde{U})(\tau)\|_{H^{-s}(\Omega)}^{2}\,d\tau\\ &\leq C\int_{0}^{t}\|U(t)-\widetilde{U}(t)\|^{2}_{X_{s}}\,d\tau,\end{split}

where $U=(u,\partial_{t}u)$ and $\widetilde{U}=(\widetilde{u},\partial_{t}\widetilde{u})$ . So, Gronwall’s inequality shows that $u=\widetilde{u}$ . This establishes the uniqueness assertion and we can conclude the proof. ∎

As an application of Theorem 2.7, we can show the unique solvability of (2.1) for rough source and initial data.

Theorem 2.8 (Very weak solutions to DNWEQ).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ . Then for any $F\in L^{2}(0,T;H^{-s}(\Omega))$ and $(u_{0},u_{1})\in\widetilde{L}^{2}(\Omega)\times H^{-s}(\Omega)$ , there exists a unique solution of

(2.36)

\begin{cases}(L_{\gamma}+q)u=F&\text{ in }\Omega_{T},\\ u=0&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega.\end{cases}

Proof.

Let us define the mapping $\mathcal{F}\colon C([0,T];X_{s})\to L^{2}(0,T;H^{-s}(\Omega))$ by

\mathcal{F}(U)(t)\vcentcolon=F-\gamma w(t)-qu(t),

where $U=(u,w)\in C([0,T];X_{s})$ . On the one hand, using the estimate (2.8) we see that for any $u\in\widetilde{L}^{2}(\Omega)$ one has $qu\in H^{-s}(\Omega)$ and there holds

(2.37)

\begin{split}\|qu\|_{H^{-s}(\Omega)}&=\sup_{\|v\|_{\widetilde{H}^{s}(\Omega)}\leq 1}|\langle u,qv\rangle_{L^{2}(\Omega)}|\\ &\leq C\|q\|_{L^{p}(\Omega)}\|u\|_{L^{2}(\Omega)}.\end{split}

On the other hand, by applying [CRTZ24, Lemma 3.1] and $\partial\Omega\in C^{0}$ we deduce that for any $v\in\widetilde{H}^{s}(\Omega)$ one has $\gamma v\in\widetilde{H}^{s}(\Omega)$ and it obeys the estimate

(2.38)

\|\gamma v\|_{\widetilde{H}^{s}(\Omega)}\leq C\|\gamma\|_{C^{0,\alpha}({\mathbb{R}}^{n})}\|v\|_{\widetilde{H}^{s}(\Omega)}.

Thus, we can again infer from a duality argument that $H^{-s}(\Omega)\ni w\mapsto\gamma w\in H^{-s}(\Omega)$ is a continuous map satisfying

(2.39)

\begin{split}\|\gamma w\|_{H^{-s}(\Omega)}&=\sup_{\|v\|_{\widetilde{H}^{s}(\Omega)}\leq 1}|\langle\gamma w,v\rangle|\\ &=\sup_{\|v\|_{\widetilde{H}^{s}(\Omega)}\leq 1}|\langle w,\gamma v\rangle|\\ &\leq C\|\gamma\|_{C^{0,\alpha}({\mathbb{R}}^{n})}\|w\|_{H^{-s}(\Omega)}.\end{split}

From the estimates (2.37) and (2.39), we easily deduce that $\mathcal{F}$ is well-defined and satisfies the Lipschitz estimate

(2.40)

\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\|_{H^{-s}(\Omega)}\leq C(\|\gamma\|_{C^{0,\alpha}({\mathbb{R}}^{n})}+\|q\|_{L^{p}(\Omega)})\|U(t)-V(t)\|_{X_{s}}

for all $U,V\in C([0,T];X_{s})$ . Thus, we can apply Theorem 2.7 to get the existence of a unique solution to (2.36) in the sense that for any $G\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ and corresponding solution $v$ of (2.30), there holds

(2.41)

\int_{0}^{T}\langle G,u\rangle_{L^{2}(\Omega)}\,dt=\int_{0}^{T}\langle(F-\gamma\partial_{t}u-qu),v\rangle\,dt+\langle u_{1},v(0)\rangle-\langle u_{0},\partial_{t}v(0)\rangle_{L^{2}(\Omega)}.

It remains to verify that $u$ is indeed a solution of (2.36) in the sense of Definition 2.4. For this purpose let $G\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ and suppose that $v$ is the unique solution to (2.23). Hence, $v$ solves

\begin{cases}(\partial_{t}^{2}+(-\Delta)^{s})v=\widetilde{G}&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(T)=0,\,\partial_{t}v(T)=0&\text{ on }\Omega\end{cases}

with $\widetilde{G}=G+\gamma\partial_{t}v-qv\in L^{2}(0,T;\widetilde{L}^{2}(\Omega))$ (see (2.8)). Next, we claim that there holds

(2.42)

\int_{0}^{T}\langle\gamma\partial_{t}u,v\rangle\,dt=-\int_{0}^{T}\langle\gamma\partial_{t}v,u\rangle_{L^{2}(\Omega)}\,dt-\langle\gamma u_{0},v(0)\rangle.

For this purpose, let us consider for $\varepsilon>0$ the unique solution $v_{\varepsilon}\in H^{1}(0,T;\widetilde{H}^{s}(\Omega))$ with $\partial_{t}^{2}v_{\varepsilon}\in L^{2}(0,T;H^{-s}(\Omega))$ to the following parabolically regularized problem

(2.43)

\begin{cases}(\partial_{t}^{2}-\varepsilon(-\Delta)^{s}\partial_{t}+(-\Delta)^{s})v=\widetilde{G}&\text{ in }\Omega_{T},\\ v=0&\text{ in }(\Omega_{e})_{T},\\ v(T)=\partial_{t}v(T)=0&\text{ in }\Omega\end{cases}

(see [DL92, Chapter XVIII, Section 5.3.1]). By [DL92, Chapter XVIII, Section 5.3.4] we know that there holds

(2.44)

\begin{split}v_{\varepsilon}&\overset{\ast}{\rightharpoonup}v\text{ in }L^{\infty}(0,T;\widetilde{H}^{s}(\Omega)),\\ \partial_{t}v_{\varepsilon}&\overset{\ast}{\rightharpoonup}\partial_{t}v\text{ in }L^{\infty}(0,T;\widetilde{L}^{2}(\Omega)),\\ v_{\varepsilon}(t)&\to v(t)\text{ in }\widetilde{H}^{s}(\Omega)\text{ for all }0\leq t\leq T.\end{split}

First, note that the conditions $u\in C^{1}([0,T];H^{-s}(\Omega))$ and $v_{\varepsilon}\in C^{1}([0,T];\widetilde{L}^{2}(\Omega))$ , where the latter follows from the Sobolev embedding, guarantee that $\langle\gamma u,v_{\varepsilon}\rangle\in C^{1}([0,T])$ with

(2.45)

\partial_{t}\langle\gamma u,v_{\varepsilon}\rangle=\langle\partial_{t}u,\gamma v_{\varepsilon}\rangle+\langle u,\gamma\partial_{t}v_{\varepsilon}\rangle_{L^{2}(\Omega)}.

Thus, by the fundamental theorem of calculus we deduce that there holds

\langle\gamma u(T),v_{\varepsilon}(T)\rangle-\langle\gamma u_{0},v_{\varepsilon}(0)\rangle=\int_{0}^{T}\langle\partial_{t}u,\gamma v_{\varepsilon}\rangle+\langle u,\gamma\partial_{t}v_{\varepsilon}\rangle_{L^{2}(\Omega)}\,dt.

By the convergence assertions (2.44) and $v_{\varepsilon}(T)=0$ , we get

-\langle\gamma u_{0},v(0)\rangle=\int_{0}^{T}\langle\partial_{t}u,\gamma v\rangle+\langle u,\gamma\partial_{t}v\rangle_{L^{2}(\Omega)}\,dt.

This proves (2.42). Hence, inserting this into (2.41), we obtain

\begin{split}\int_{0}^{T}\langle\widetilde{G},u\rangle_{L^{2}(\Omega)}\,dt&=\int_{0}^{T}\langle(F-\gamma\partial_{t}u-qu),v\rangle\,dt+\langle u_{1},v(0)\rangle-\langle u_{0},\partial_{t}v(0)\rangle_{L^{2}(\Omega)}\\ &=\int_{0}^{T}\langle F,v\rangle\,dt+\int_{0}^{T}\langle u,\gamma\partial_{t}v\rangle_{L^{2}(\Omega)}\,dt-\int_{0}^{T}\langle u,qv\rangle\,dt\\ &\quad+\langle u_{1},v(0)\rangle-\langle u_{0},\partial_{t}v(0)\rangle_{L^{2}(\Omega)}+\langle\gamma u_{0},v(0)\rangle.\end{split}

As $\widetilde{G}=G+\gamma\partial_{t}v-qv$ this gives

\begin{split}\int_{0}^{T}\langle G,u\rangle_{L^{2}(\Omega)}\,dt&=\int_{0}^{T}\langle F,v\rangle\,dt+\langle u_{1},v(0)\rangle-\langle u_{0},\partial_{t}v(0)\rangle_{L^{2}(\Omega)}+\langle\gamma u_{0},v(0)\rangle.\end{split}

Hence, we observe that $u$ is indeed a solution of (2.36) in the sense of Definition 2.4. By reversing the above arguments one can also observe that if $u$ is a solution in the sense of Definition 2.4, then by (2.42) it is a solution in the sense of (2.41) and thus the solution in the sense of Definition 2.4 is unique. ∎

3. The inverse problem

After establishing the theory of very weak solutions to damped, nonlocal wave equations, we now turn our attention to the inverse problem part. First, in Section 3.1 we prove the optimal Runge approximation theorem (Theorem 3.1) and in Section 3.2 a suitable integral identity. Using these results, we then show in Section 3.3 our first main result dealing with linear perturbations (Theorem 1.2). Finally, in Section 3.4 we prove Theorem 1.3 showing that the damping coefficient and the nonlinearity can be determined simultaneously.

3.1. Runge approximation

With the material from Section 2 at our disposal, we can now show the following Runge approximation theorem, whose proof is very similar to the one of [LTZ24a, Theorem 1.2].

Theorem 3.1 (Runge approximation).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ . Then for any measurement set $W\subset\Omega_{e}$ and initial conditions $(u_{0},u_{1})\in\widetilde{H}^{s}(\Omega)\times\widetilde{L}^{2}(\Omega)$ , the Runge set

(3.1)

\mathcal{R}^{u_{0},u_{1}}_{W}\vcentcolon=\{u_{\varphi}-\varphi\,;\,\varphi\in C_{c}^{\infty}(W_{T})\}

is dense in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ , where $u_{\varphi}$ is the unique solution to

(3.2)

\begin{cases}(L_{\gamma}+q)u=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=u_{0},\,\partial_{t}u(0)=u_{1}&\text{ on }\Omega,\end{cases}

(see Proposition 2.2).

Proof.

First of all note that it is enough to consider the case $(u_{0},u_{1})=0$ . To see this assume that the density holds for $\mathcal{R}_{W}\vcentcolon=\mathcal{R}_{W}^{0,0}$ and let $f\in L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ . Let $v_{0}$ be the unique solution to

(3.3)

\begin{cases}(L_{\gamma}+q)v=0&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(0)=u_{0},\,\partial_{t}v(0)=u_{1}&\text{ on }\Omega\end{cases}

and define $\widetilde{f}\vcentcolon=f-v_{0}\in L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ . By assumption there exists $(\varphi_{k})_{k\in{\mathbb{N}}}\subset C_{c}^{\infty}(W_{T})$ such that $u_{k}-\varphi_{k}\to\widetilde{f}$ in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ as $k\to\infty$ , where $u_{k}$ is the unique solution to

(3.4)

\begin{cases}(L_{\gamma}+q)u=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega\end{cases}

with $\varphi=\varphi_{k}$ . Then $v_{k}\vcentcolon=u_{k}+v_{0}$ is the unique solution to (3.2) with $\varphi=\varphi_{k}$ . The above convergence now implies $v_{k}-\varphi_{k}\to f$ in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ as $k\to\infty$ and we get that $\mathcal{R}_{W}^{u_{0},u_{1}}$ is dense in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ .

Therefore, it remains to show that $\mathcal{R}_{W}$ is dense in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ . As usual, we show this by a Hahn–Banach argument. Thus, suppose that $F\in L^{2}(0,T;H^{-s}(\Omega))$ vanishes on $\mathcal{R}_{W}$ . Let us recall that if $\varphi\in C_{c}^{\infty}(W_{T})$ and $u$ solves (3.4), then by (2.13) and Lemma 2.3 the function $v=(u-\varphi)^{\star}$ satisfies

(3.5)

\begin{cases}(L_{-\gamma}+q)v=-(-\Delta)^{s}\varphi^{\star}&\text{ in }\Omega_{T},\\ v=0&\text{ in }(\Omega_{e})_{T},\\ v(T)=\partial_{t}v(T)=0&\text{ in }\Omega.\end{cases}

Next, let $w$ be the unique solution to

(3.6)

\begin{cases}(L_{\gamma}+q)w=F^{\star}&\text{ in }\Omega_{T},\\ w=0&\text{ in }(\Omega_{e})_{T},\\ w(0)=\partial_{t}w(0)=0&\text{ in }\Omega\end{cases}

(see Theorem 2.8). By testing the equation for $w$ by $v$ , we get

-\int_{0}^{T}\langle(-\Delta)^{s}\varphi^{\star},w\rangle_{L^{2}(\Omega)}\,dt=\int_{0}^{T}\langle F^{\star},v\rangle\,dt=\int_{0}^{T}\langle F,u_{\varphi}-\varphi\rangle\,dt=0

for any $\varphi\in C_{c}^{\infty}(W_{T})$ . This ensures that there holds

(-\Delta)^{s}w=0\quad\text{ in }W_{T}.

Furthermore, by construction $w$ vanishes in $(\Omega_{e})_{T}$ and hence the unique continuation principle for the fractional Laplacian guarantees $w=0$ in ${\mathbb{R}}^{n}_{T}$ (see [GSU20]). As very weak solutions are distributional solutions, we get

\int_{0}^{T}\langle F^{\star},\Phi\rangle\,dt=\int_{0}^{T}\langle(L_{-\gamma}+q)\Phi,w\rangle_{L^{2}(\Omega)}\,dt=0

for all $\Phi\in C_{c}^{\infty}(\Omega_{T})$ . To see that very weak solutions are distributional solutions, one can simply take $G=\chi_{\Omega}(L_{\gamma}+q)\Phi$ with $\Phi\in C_{c}^{\infty}(\Omega\times[0,T))$ in Definition 2.4, where $\chi_{\Omega}$ denotes the characteristic function of $\Omega$ (see also [LTZ24a, Proposition 3.8]). By density of $C_{c}^{\infty}(\Omega_{T})$ in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ we deduce that $F=0$ . This concludes the proof. ∎

As a consequence we have the following lemma.

Lemma 3.2 (Convergence of time derivative).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ . Let $\Phi,\Psi\in L^{2}(0,T;\widetilde{H}^{s}(\Omega))\cap H^{1}(0,T;H^{-s}(\Omega))$ and suppose $(\varphi_{k})_{k\in{\mathbb{N}}}\subset C_{c}^{\infty}((\Omega_{e})_{T})$ is such that

(3.7)

u_{k}-\varphi_{k}\to\Phi\text{ in }L^{2}(0,T;\widetilde{H}^{s}(\Omega))\text{ as }k\to\infty,

where $u_{k}$ solves

(3.8)

\begin{cases}(L_{\gamma}+q)u=0&\text{ in }\Omega_{T},\\ u=\varphi_{k}&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega\end{cases}

for $k\in{\mathbb{N}}$ . If $\Phi,\Psi$ satisfy one of the conditions

(a)

$\Psi(T)=\Phi(0)=0$
(b)

or $\Psi(T)=\Psi(0)=0$ ,

then we have

(3.9)

\lim_{k\to\infty}\int_{0}^{T}\langle\partial_{t}(u_{k}-\varphi_{k}),\Psi\rangle\,dt=\int_{0}^{T}\langle\partial_{t}\Phi,\Psi\rangle\,dt.

Remark 3.3.

Let us note that the same formula (3.9) holds for second order time derivatives under appropriate conditions.

Proof.

Using the integration by parts formula, we may compute

\small\begin{split}&\lim_{k\to\infty}\int_{0}^{T}\langle\partial_{t}(u_{k}-\varphi_{k}),\Psi\rangle\,dt\\ &=\lim_{k\to\infty}\left(\langle(u_{k}-\varphi_{k})(T),\Psi(T)\rangle_{L^{2}(\Omega)}-\langle(u_{k}-\varphi_{k})(0),\Psi(0)\rangle_{L^{2}(\Omega)}-\int_{0}^{T}\langle\partial_{t}\Psi,u_{k}-\varphi_{k}\rangle\,dt\right)\\ &=-\lim_{k\to\infty}\int_{0}^{T}\langle\partial_{t}\Psi,u_{k}-\varphi_{k}\rangle\,dt\\ &=-\int_{0}^{T}\langle\partial_{t}\Psi,\Phi\rangle\,dt\\ &=\langle\Phi(0),\Psi(0)\rangle_{L^{2}(\Omega)}-\langle\Phi(T),\Psi(T)\rangle_{L^{2}(\Omega)}+\int_{0}^{T}\langle\partial_{t}\Phi,\Psi\rangle\,dt\\ &=\int_{0}^{T}\langle\partial_{t}\Phi,\Psi\rangle\,dt.\end{split}

In the first equality sign we used an integration by parts, in the second equality we used (3.8), $\Psi(T)=0$ and (3.8), in the third equality the convergence (3.7), in the fourth equality again an integration by parts and finally in the last equality the conditions a or b. ∎

3.2. DN map and integral identities

Next, we define the Dirichlet to Neumann (DN) map $\Lambda_{\gamma,q}$ related to

(3.10)

\begin{cases}(L_{\gamma}+q)u=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega,\end{cases}

via

(3.11)

\langle\Lambda_{\gamma,q}\varphi,\psi\rangle=\int_{{\mathbb{R}}^{n}_{T}}(-\Delta)^{s/2}u_{\varphi}(-\Delta)^{s/2}\psi\,dx

for all $\varphi,\psi\in C_{c}^{\infty}((\Omega_{e})_{T})$ , where $u_{\varphi}$ is the unique solution to (3.10) with exterior condition $\varphi$ . Using the above preparation, we now establish the following integral identity.

Proposition 3.4 (Integral identity for linear perturbations).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded Lipschitz domain, $T>0$ , $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq\infty$ satisfies (1.6). Assume that we have given coefficients $(\gamma_{j},q_{j})\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ for $j=1,2$ . Let $\varphi_{j}\in C_{c}^{\infty}((\Omega_{e})_{T})$ and denote by $u_{j}$ the corresponding solution of (3.10) with $(\gamma,q)=(\gamma_{j},q_{j})$ . Then there holds

(3.12)

\begin{split}&\langle(\Lambda_{\gamma_{1},q_{1}}-\Lambda_{\gamma_{2},q_{2}})\varphi_{1},\varphi_{2}^{\star}\rangle\\ &\quad=\int_{\Omega_{T}}\{[(\gamma_{1}-\gamma_{2})\partial_{t}+q_{1}-q_{2}](u_{1}-\varphi_{1})\}(u_{2}-\varphi_{2})^{\star}\,dxdt.\end{split}

Proof.

Let $(\Gamma_{j},Q_{j})\in C^{0,\alpha}({\mathbb{R}}^{n})\times L^{p}(\Omega)$ , $j=1,2$ , and suppose $U_{j}$ is the unique solutions of (3.10) with $(\gamma,q)=(\Gamma_{j},Q_{j})$ and exterior condition $\varphi=\psi_{j}$ . Then we may compute

(3.13)

\begin{split}&\int_{\Omega_{T}}\{[(\Gamma_{1}-\Gamma_{2})\partial_{t}+Q_{1}-Q_{2}](U_{1}-\psi_{1})\}(U_{2}-\psi_{2})^{\star}\,dxdt\\ &=\int_{\Omega_{T}}\{[\Gamma_{1}\partial_{t}+Q_{1}](U_{1}-\psi_{1})\}(U_{2}-\psi_{2})^{\star}\,dxdt\\ &\quad-\int_{\Omega_{T}}(U_{1}-\psi_{1})[-\Gamma_{2}\partial_{t}+Q_{2}](U_{2}-\psi_{2})^{\star}\,dxdt\\ &=\int_{0}^{T}\langle L_{\Gamma_{1},Q_{1}}(U_{1}-\psi_{1}),(U_{2}-\psi_{2})^{\star}\rangle\,dt\\ &\quad-\int_{0}^{T}\langle(\partial_{t}^{2}+(-\Delta)^{s})(U_{1}-\psi_{1}),(U_{2}-\psi_{2})^{\star}\rangle\,dt\\ &\quad-\int_{0}^{T}\langle L_{-\Gamma_{2},Q_{2}}(U_{2}-\psi_{2})^{\star},(U_{1}-\psi_{1})\rangle\,dt\\ &\quad+\int_{0}^{T}\langle(\partial_{t}^{2}+(-\Delta)^{s})(U_{2}-\psi_{2})^{\star},(U_{1}-\psi_{1})\rangle\,dt\\ &=-\int_{0}^{T}\langle(-\Delta)^{s}\psi_{1},(U_{2}-\psi_{2})^{\star}\rangle_{L^{2}(\Omega)}\,dt+\int_{0}^{T}\langle(-\Delta)^{s}\psi_{2}^{\star},(U_{1}-\psi_{1})\rangle_{L^{2}(\Omega)}\,dt\\ &=\int_{{\mathbb{R}}^{n}_{T}}((-\Delta)^{s}\psi_{2}^{\star})U_{1}\,dxdt-\int_{{\mathbb{R}}^{n}_{T}}((-\Delta)^{s}\psi_{1})U_{2}^{\star}\,dxdt\\ &=\langle\Lambda_{\Gamma_{1},Q_{1}}\psi_{1},\psi_{2}^{\star}\rangle-\langle\Lambda_{\Gamma_{2},Q_{2}}\psi_{2},\psi_{1}^{\star}\rangle.\end{split}

In the first equality we used that $U_{1}-\psi_{1}$ has vanishing initial conditions, $(U_{2}-\psi_{2})^{\star}$ has vanishing terminal conditions and an integration by parts. In the third equality we used that the PDEs for $U_{1}-\psi_{1}$ and $(U_{2}-\psi_{2})^{\star}$ hold in the sense of $L^{2}(0,T;H^{-s}(\Omega))=(L^{2}(0,T;\widetilde{H}^{s}(\Omega))^{\prime}$ (see Lemma 2.3). In the fourth equality, we used the PDEs for $U_{1}$ and $U_{2}$ , Lemma 2.3 and that there holds

\begin{split}&\int_{0}^{T}\langle(\partial_{t}^{2}+(-\Delta)^{s})(U_{2}-\psi_{2})^{\star},(U_{1}-\psi_{1})\rangle\,dt\\ &=\int_{0}^{T}\langle(\partial_{t}^{2}+(-\Delta)^{s})(U_{1}-\psi_{1}),(U_{2}-\psi_{2})^{\star}\rangle\,dt,\end{split}

which can be established similarly as [LTZ24c, Claim 4.2] (see also the proof of Theorem 2.8). In the last equality, we have made the change of variables $\tau=T-t$ for the second integral. On the one hand, using (3.13) with

\Gamma_{1}=\Gamma_{2}=\gamma_{j}\text{ and }Q_{1}=Q_{2}=q_{j},

we observe that

(3.14)

\langle\Lambda_{\gamma_{j},q_{j}}\psi_{1},\psi_{2}^{\star}\rangle=\langle\Lambda_{\gamma_{j},q_{j}}\psi_{2},\psi_{1}^{\star}\rangle

for all $\psi_{j}\in C_{c}^{\infty}((\Omega_{e})_{T})$ , $j=1,2$ . On the other hand, choosing

\Gamma_{j}=\gamma_{j},\,Q_{j}=q_{j}\text{ and }\psi_{j}=\varphi_{j}

in (3.13) and taking into account the self-adjointness (3.14), we get (3.12). ∎

3.3. Simultaneous determination of damping coefficient and linear perturbations

Proof of Theorem 1.2.

First note that by the integral identity in Proposition 3.4, we may deduce from the condition (1.12) that there holds

(3.15)

\int_{\Omega_{T}}\{[(\gamma_{1}-\gamma_{2})\partial_{t}+q_{1}-q_{2}](u_{1}-\varphi_{1})\}(u_{2}-\varphi_{2})^{\star}\,dxdt=0

for all $\varphi_{j}\in C_{c}^{\infty}((W_{j})_{T})$ , where $u_{j}$ denotes the unique solution to

(3.16)

\begin{cases}(L_{\gamma_{j}}+q_{j})u=0&\text{ in }\Omega_{T},\\ u=\varphi_{j}&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega.\end{cases}

Let $\omega\Subset\Omega$ and choose a cutoff function $\Phi_{1}\in C_{c}^{\infty}(\Omega)$ satisfying $\Phi_{1}=1$ on $\omega$ . Moreover, let $\Phi_{2}\in C_{c}^{\infty}(\omega_{T})$ . By the Runge approximation (Theorem 3.1), there exist sequences $(\varphi_{j}^{k})_{k\in{\mathbb{N}}}\subset C_{c}^{\infty}((W_{j})_{T})$ with corresponding solutions $u_{j}^{k}$ of (3.16) with $\varphi_{j}=\varphi_{j}^{k}$ such that $u_{j}^{k}-\varphi_{j}^{k}\to\Phi_{j}$ in $L^{2}(0,T;\widetilde{H}^{s}(\Omega))$ . Taking $\varphi_{1}=\varphi_{1}^{k}$ and $\varphi_{2}=\varphi_{2}^{\ell}$ in (3.15) gives

\int_{\Omega_{T}}\{[(\gamma_{1}-\gamma_{2})\partial_{t}+q_{1}-q_{2}](u^{k}_{1}-\varphi^{k}_{1})\}(u^{\ell}_{2}-\varphi^{\ell}_{2})^{\star}\,dxdt=0

for all $k,\ell\in{\mathbb{N}}$ . First, we let $\ell\to\infty$ to deduce

(3.17)

\int_{\Omega_{T}}\{[(\gamma_{1}-\gamma_{2})\partial_{t}+q_{1}-q_{2}](u^{k}_{1}-\varphi^{k}_{1})\}\Phi_{2}^{\star}\,dxdt=0

for all $k\in{\mathbb{N}}$ . As $\gamma_{1}-\gamma_{2}\in C^{0,\alpha}({\mathbb{R}}^{n})$ the estimate (2.38) ensures that we can apply Lemma 3.2 under the condition b and so $\partial_{t}\Phi_{1}=0$ shows that the first term in (3.17) goes to zero. So in the limit $k\to\infty$ what remains is

\int_{\Omega_{T}}(q_{1}-q_{2})\Phi_{2}^{\star}\,dxdt=0,

where we used $\Phi_{1}=1$ on $\omega$ . This ensures that $q_{1}=q_{2}$ on $\omega$ . As the set $\omega$ is arbitrary, we get $q_{1}=q_{2}$ in $\Omega$ . Now, the identity (3.15) reduces to

\int_{\Omega_{T}}\{[(\gamma_{1}-\gamma_{2})\partial_{t}](u_{1}-\varphi_{1})\}(u_{2}-\varphi_{2})^{\star}\,dxdt=0

for all $\varphi_{j}\in C_{c}^{\infty}((W_{j})_{T})$ . We choose $\eta\in C_{c}^{\infty}(\Omega_{T})$ , define

\Phi_{1}(x,t)=\int_{0}^{t}\eta(x,\tau)\,d\tau\in C_{c}^{\infty}(\Omega\times(0,T])

and take $\Phi_{2}\in C_{c}^{\infty}(\Omega_{T})$ . Then using $\partial_{t}\Phi_{1}=\eta$ and arguing as above via a Runge approximation and Lemma 3.2, we get from (3.15) the identity

\int_{\Omega_{T}}(\gamma_{1}-\gamma_{2})\eta\Phi_{2}^{\star}\,dxdt=0.

This again implies $\gamma_{1}=\gamma_{2}$ in $\Omega$ . ∎

3.4. Simultaneous determination of damping coefficient and nonlinearity

Before turning to the proof of our second main result, let us recall that the DN map related to the problem

(3.18)

\begin{cases}L_{\gamma}u+f(u)=0&\text{ in }\Omega_{T},\\ u=\varphi&\text{ on }(\Omega_{e})_{T},\\ u(0)=0,\,\partial_{t}u(0)=0&\text{ on }\Omega\end{cases}

is defined by

(3.19)

\langle\Lambda_{\gamma,f}\varphi,\psi\rangle\vcentcolon=\int_{{\mathbb{R}}^{n}_{T}}(-\Delta)^{s/2}u_{\varphi}(-\Delta)^{s/2}\psi\,dxdt,

where $\varphi,\psi\in C_{c}^{\infty}((\Omega_{e})_{T})$ and $u_{\varphi}$ is the unique solution to (3.18) (see [LTZ24c, Proposition 3.7]).

Proof of Theorem 1.3.

Let $\varepsilon>0$ and denote by $u^{(j)}_{\varepsilon}$ the unique solutions to (3.18) with $f=f_{j}$ , $\gamma=\gamma_{j}$ and $\varphi=\varepsilon\eta$ for some fixed $\eta\in C_{c}^{\infty}((W_{1})_{T})$ . Let us observe that the UCP for the fractional Laplacian and the condition (1.15) imply that $u_{\varepsilon}\vcentcolon=u^{(1)}_{\varepsilon}=u^{(2)}_{\varepsilon}$ . Next, let us note that we can write

(3.20)

u_{\varepsilon}=\varepsilon v_{j}+R^{(j)}_{\varepsilon}

for $j=1,2$ , where $v_{j}$ and $R^{(j)}_{\varepsilon}$ are the unique solutions of

(3.21)

\begin{cases}L_{\gamma_{j}}v=0&\text{ in }\Omega_{T},\\ v=\eta&\text{ on }(\Omega_{e})_{T},\\ v(0)=0,\,\partial_{t}v(0)=0&\text{ on }\Omega\end{cases}

and

(3.22)

\begin{cases}L_{\gamma_{j}}R=-f_{j}(u_{\varepsilon})&\text{ in }\Omega_{T},\\ R=0&\text{ on }(\Omega_{e})_{T},\\ R(0)=0,\,\partial_{t}R(0)=0&\text{ on }\Omega,\end{cases}

respectively. This simply follows from the unique solvability of (3.18) and both functions $u_{\varepsilon}$ and $\varepsilon v_{j}+R^{(j)}_{\varepsilon}$ are solutions. Furthermore, we notice that the energy estimate of [LTZ24c, Theorem 3.1], [LTZ24c, eq. (3.18)] and the $r+1$ homogeneity of $f_{j}$ ensure that $R^{(j)}_{\varepsilon}$ satisfies

(3.23)

\begin{split}\|\partial_{t}R^{(j)}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|R^{(j)}_{\varepsilon}(t)\|_{L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))}&\lesssim\|f_{j}(u_{\varepsilon})\|_{L^{2}(\Omega_{T})}\\ &\lesssim\|u_{\varepsilon}\|^{r+1}_{L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))}.\end{split}

Moreover, we may estimate

(3.24)

\begin{split}&\|\partial_{t}u_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{n}))}+\|u_{\varepsilon}\|_{L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))}\\ &\lesssim\|\partial_{t}(u_{\varepsilon}-\varepsilon\eta)\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|u_{\varepsilon}-\varepsilon\eta\|_{L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))}+\varepsilon\|\eta\|_{W^{1,\infty}(0,T;H^{2s}({\mathbb{R}}^{n}))}\\ &\lesssim\varepsilon\|\eta\|_{W^{1,\infty}(0,T;H^{2s}({\mathbb{R}}^{n}))}.\end{split}

This follows from the following observations. If $u$ solves (3.18) for a damping coefficient $\gamma\in C^{0,\alpha}({\mathbb{R}}^{n})$ , a weak nonlinearity $f$ and $\varphi\in C_{c}^{\infty}((\Omega_{e})_{T})$ , then $v=u-\varphi$ solves

(3.25)

\begin{cases}L_{\gamma}v+f(v)=-(-\Delta)^{s}\varphi&\text{ in }\Omega_{T},\\ v=0&\text{ on }(\Omega_{e})_{T},\\ v(0)=0,\,\partial_{t}v(0)=0&\text{ on }\Omega.\end{cases}

Now, we may invoke [LTZ24c, eq. (3.15)] to find that there holds

\begin{split}&\|\partial_{t}v(t)\|_{L^{2}(\Omega)}^{2}+\|v(t)\|_{H^{s}({\mathbb{R}}^{n})}^{2}\\ &\lesssim\int_{0}^{t}|\langle\gamma\partial_{t}v,\partial_{t}v\rangle_{L^{2}(\Omega)}|\,d\tau+\int_{0}^{t}|\langle(-\Delta)^{s}\varphi,\partial_{t}v\rangle_{L^{2}(\Omega)}|\,d\tau\\ &\lesssim\|(-\Delta)^{s}\varphi\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\int_{0}^{t}\|\partial_{t}v\|_{L^{2}(\Omega)}^{2}\,d\tau.\end{split}

Thus, Gronwall’s inequality gives

\|\partial_{t}v(t)\|_{L^{2}(\Omega)}+\|v(t)\|_{H^{s}({\mathbb{R}}^{n})}\lesssim\|(-\Delta)^{s}\varphi\|_{L^{2}(0,t;L^{2}(\Omega))}.

This ensures the validity of the second estimate in (3.24). Next, observe that by subtracting the PDEs for $u^{(1)}_{\varepsilon}$ and $u^{(2)}_{\varepsilon}$ , we deduce that

(3.26)

(\gamma_{1}-\gamma_{2})\partial_{t}u_{\varepsilon}=f_{2}(u_{\varepsilon})-f_{1}(u_{\varepsilon})\text{ in }\Omega_{T}.

By (3.20), we may write

(3.27)

(\gamma_{1}-\gamma_{2})(\varepsilon\partial_{t}v_{1}+\partial_{t}R^{(1)}_{\varepsilon})=f_{2}(u_{\varepsilon})-f_{1}(u_{\varepsilon})\text{ in }\Omega_{T}.

Combining (3.23) and (3.24), we see that

(3.28)

\|\partial_{t}R^{(j)}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|R^{(j)}_{\varepsilon}(t)\|_{L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))}\lesssim\varepsilon^{r+1}.

Multiplying by $\varepsilon^{-1}$ gives

(3.29)

\begin{split}&(\gamma_{1}-\gamma_{2})(\partial_{t}v_{1}+\varepsilon^{-1}\partial_{t}R^{(1)}_{\varepsilon})=f_{2}(\varepsilon^{-1/(r+1)}u_{\varepsilon})-f_{1}(\varepsilon^{-1/(r+1)}u_{\varepsilon})\text{ in }\Omega_{T}.\end{split}

Next, let us focus one the case $2s<n$ as the other one can be treated similarly. As $r>0$ we deduce from (3.24) that $\varepsilon^{-1/(1+r)}u_{\varepsilon}\to 0$ in $L^{\infty}(0,T;H^{s}({\mathbb{R}}^{n}))$ and so by Sobolev’s embedding in $L^{q}(0,T;L^{2_{s}^{*}}(\Omega))$ for all $1\leq q\leq\infty$ and $2_{s}^{*}=\frac{2n}{n-2s}$ . Hence, by our assumptions on $f_{j}$ and [Zim24, Lemma 3.6], we get

(3.30)

f_{j}(\varepsilon^{-1/(r+1)}u_{\varepsilon})\to 0\text{ in }L^{q/(r+1)}(0,T;L^{2_{s}^{*}/(r+1)}(\Omega))

for all $q\geq r+1$ as $\varepsilon\to 0$ . Additionally, using (3.28) we know that

(3.31)

\varepsilon^{-1}\partial_{t}R^{(j)}_{\varepsilon}\to 0\text{ in }L^{\infty}(0,T;L^{2}(\Omega)).

Therefore, from (3.29), (3.30) and (3.31), we infer

(\gamma_{1}-\gamma_{2})\partial_{t}v_{1}=0\text{ in }\Omega_{T}.

In particular, this ensures that there holds

\int_{\Omega_{T}}(\gamma_{1}-\gamma_{2})\partial_{t}(v_{1}-\eta)(w_{2}-\psi)^{\star}\,dxdt=0

for any $\psi\in C_{c}^{\infty}((W_{2})_{T})$ , where $w_{2}$ is the unique solution of

(3.32)

\begin{cases}L_{\gamma_{2}}w=0&\text{ in }\Omega_{T},\\ w=\psi&\text{ on }(\Omega_{e})_{T},\\ w(0)=0,\,\partial_{t}w(0)=0&\text{ on }\Omega.\end{cases}

Now, arguing as in the previous section, we get $\gamma_{1}=\gamma_{2}$ in $\Omega$ . Hence, (3.26) reduces to

(3.33)

f_{1}(u_{\varepsilon})=f_{2}(u_{\varepsilon})\text{ in }\Omega_{T}.

Multiplying this identity by $\varepsilon^{-(r+1)}$ and arguing as before, we deduce that

f_{1}(v)=f_{2}(v)\text{ in }\Omega_{T},

where $v\vcentcolon=v_{1}=v_{2}$ as $\gamma_{1}=\gamma_{2}$ . One can now show $f_{1}(x,\tau)=f_{2}(x,\tau)$ for all $x\in\Omega$ and $\tau\in{\mathbb{R}}$ exactly as described in [LTZ24a, p. 29]. Hence, we can conclude the proof. ∎

Acknowledgments. P. Zimmermann was supported by the Swiss National Science Foundation (SNSF), under grant number 214500.

Statements and Declarations

Data availability statement

No datasets were generated or analyzed during the current study.

Conflict of Interests

Hereby we declare there are no conflict of interests.

References

[CGRU23] Giovanni Covi, Tuhin Ghosh, Angkana Rüland, and Gunther Uhlmann. A reduction of the fractional Calderón problem to the local Calderón problem by means of the Caffarelli-Silvestre extension. arXiv preprint arXiv:2305.04227, 2023.
[CLL19] Xinlin Cao, Yi-Hsuan Lin, and Hongyu Liu. Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators. Inverse Probl. Imaging, 13(1):197–210, 2019.
[CLR20] Mihajlo Cekic, Yi-Hsuan Lin, and Angkana Rüland. The Calderón problem for the fractional Schrödinger equation with drift. Cal. Var. Partial Differential Equations, 59(91), 2020.
[CRTZ22] Giovanni Covi, Jesse Railo, Teemu Tyni, and Philipp Zimmermann. Stability estimates for the inverse fractional conductivity problem, 2022.
[CRTZ24] Giovanni Covi, Jesse Railo, Teemu Tyni, and Philipp Zimmermann. Stability estimates for the inverse fractional conductivity problem. SIAM Journal on Mathematical Analysis, 56(2):2456–2487, 2024.
[DL92] Robert Dautray and Jacques-Louis Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5. Springer-Verlag, Berlin, 1992. Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig.
[Fei21] Ali Feizmohammadi. Fractional Calderón’ problem on a closed Riemannian manifold. arXiv preprint arXiv:2110.07500, 2021.
[FGKU21] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, and Gunther Uhlmann. Fractional anisotropic Calderón problem on closed Riemannian manifolds. arXiv:2112.03480, 2021.
[FKU24] Ali Feizmohammadi, Katya Krupchyk, and Gunther Uhlmann. Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds. arXiv preprint arXiv:2407.16866, 2024.
[FY24] Song-Ren Fu and Yongyi Yu. Well-posedness and inverse problems for the nonlocal third-order acoustic equation with time-dependent nonlinearity, 2024.
[GLX17] Tuhin Ghosh, Yi-Hsuan Lin, and Jingni Xiao. The Calderón problem for variable coefficients nonlocal elliptic operators. Comm. Partial Differential Equations, 42(12):1923–1961, 2017.
[GSU20] Tuhin Ghosh, Mikko Salo, and Gunther Uhlmann. The Calderón problem for the fractional Schrödinger equation. Anal. PDE, 13(2):455–475, 2020.
[KLW22] Pu-Zhao Kow, Yi-Hsuan Lin, and Jenn-Nan Wang. The Calderón problem for the fractional wave equation: uniqueness and optimal stability. SIAM J. Math. Anal., 54(3):3379–3419, 2022.
[KLZ24] Manas Kar, Yi-Hsuan Lin, and Philipp Zimmermann. Determining coefficients for a fractional $p$ -laplace equation from exterior measurements. J. Differential Equations, accepted for publication, 2024.
[LL22] Ru-Yu Lai and Yi-Hsuan Lin. Inverse problems for fractional semilinear elliptic equations. Nonlinear Anal., 216:Paper No. 112699, 21, 2022.
[LL23] Yi-Hsuan Lin and Hongyu Liu. Inverse problems for fractional equations with a minimal number of measurements. Communications and Computational Analysis, 1:72–93, 2023.
[LLU23] Ching-Lung Lin, Yi-Hsuan Lin, and Gunther Uhlmann. The Calderón problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local. arXiv preprint arXiv:2308.09654, 2023.
[LNZ24] Yi-Hsuan Lin, Gen Nakamura, and Philipp Zimmermann. The calderón problem for the schrödinger equation in transversally anisotropic geometries with partial data, 2024.
[LRZ22] Yi-Hsuan Lin, Jesse Railo, and Philipp Zimmermann. The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients. arXiv preprint arXiv:2211.07781, 2022.
[LTZ24a] Yi-Hsuan Lin, Teemu Tyni, and Philipp Zimmermann. Optimal runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities, 2024.
[LTZ24b] Yi-Hsuan Lin, Teemu Tyni, and Philipp Zimmermann. Well-posedness and inverse problems for semilinear nonlocal wave equations. Nonlinear Analysis, 247:113601, 2024.
[LTZ24c] Yi-Hsuan Lin, Teemu Tyni, and Philipp Zimmermann. Well-posedness and inverse problems for semilinear nonlocal wave equations. Nonlinear Analysis, 247:113601, 2024.
[LZ23] Yi-Hsuan Lin and Philipp Zimmermann. Unique determination of coefficients and kernel in nonlocal porous medium equations with absorption term. arXiv preprint arXiv:2305.16282, 2023.
[LZ24] Yi-Hsuan Lin and Philipp Zimmermann. Approximation and uniqueness results for the nonlocal diffuse optical tomography problem. arXiv preprint arXiv:2406.06226, 2024.
[RS20] Angkana Rüland and Mikko Salo. The fractional Calderón problem: low regularity and stability. Nonlinear Anal., 193:111529, 56, 2020.
[RZ23] Jesse Railo and Philipp Zimmermann. Fractional Calderón problems and Poincaré inequalities on unbounded domains. J. Spectr. Theory, 13(1):63–131, 2023.
[RZ24] Jesse Railo and Philipp Zimmermann. Low regularity theory for the inverse fractional conductivity problem. Nonlinear Analysis, 239:113418, 2024.
[SU87] John Sylvester and Gunther Uhlmann. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2), 125(1):153–169, 1987.
[Zim24] Philipp Zimmermann. Calderón problem for nonlocal viscous wave equations: Unique determination of linear and nonlinear perturbations, 2024.

Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results

Abstract.

1. Introduction

1.1. Mathematical model and main results

Question 1.

Definition 1.1.

Question 2.

Theorem 1.2 (Uniqueness for linear perturbations).

Theorem 1.3 (Uniqueness for semilinear perturbations).

Remark 1.4.

1.2. Organization of the article

2. Weak and very weak solutions to damped, nonlocal wave equations

2.1. Weak solutions

Theorem 2.1 (Weak solutions to homogeneous DNWEQ).

Proof.

Proposition 2.2 (Weak solutions to inhomogeneous DNWEQ).

Proof.

Lemma 2.3.

Proof.

2.2. Very weak solutions

Definition 2.4 (Very weak solutions).

Theorem 2.5 (Very weak solutions for γ=q=0\gamma=q=0,[LTZ24a, Theorem 3.6]).

Proposition 2.6 (Solution map).

Proof.

Theorem 2.7.

Proof of Theorem 2.7.

Theorem 2.8 (Very weak solutions to DNWEQ).

Proof.

3. The inverse problem

3.1. Runge approximation

Theorem 3.1 (Runge approximation).

Proof.

Lemma 3.2 (Convergence of time derivative).

Remark 3.3.

Proof.

3.2. DN map and integral identities

Proposition 3.4 (Integral identity for linear perturbations).

Proof.

3.3. Simultaneous determination of damping coefficient and linear perturbations

Proof of Theorem 1.2.

3.4. Simultaneous determination of damping coefficient and nonlinearity

Proof of Theorem 1.3.

Statements and Declarations

Data availability statement

Conflict of Interests

References

Theorem 2.5 (Very weak solutions for $\gamma=q=0$ ,[LTZ24a, Theorem 3.6]).