Criteria of the existence of global solutions to semilinear wave equations with first-order derivatives on exterior domains

Let $n\geq 2$ and $\mathcal{K}\subset\mathbb{R}^{n}$ be a smooth, compact and nontrapping obstacle. We denote $M=\mathbb{R}^{n}\setminus\mathcal{K}$, hence $M=\mathbb{R}^{n}$ when $\mathcal{K}=\emptyset$, and consider the following semilinear wave equation

Here, $\partial u=(\partial_{t}u,\partial_{1}u,\cdots,\partial_{n}u)$, the real-valued function

and real-valued initial data $(f,g)$ satisfy some compatibility conditions when $\mathcal{K}$ is non-empty. By translation and scaling, when $\mathcal{K}$ is non-empty, we can assume that the origin is in the interior of $\mathcal{K}$ and $\mathcal{K}\subset\overline{B_{1}}$. Here, $B_{1}\subset\mathbb{R}^{n}$ is the unit open ball center at the origin. Denote $\rho(\tau)=\sup_{|\mathbf{q}|\leq\tau}|\partial_{\mathbf{q}}^{\kappa}F(\mathbf{q})|$, $0\leq\tau\leq 1$. When there is no danger of misunderstanding, we will omit $\mathbf{q}$ in $\partial_{\mathbf{q}}$ hereby.

The purpose of this paper is to show how the combination of local energy estimates and the control of the $\|\partial u\|_{\infty}$ leads to the criterion of global existence theorems for small amplitude initial data and nonlinear terms only involving first-order derivatives. The problem (1) is a generalized version of Glassey conjecture in the exterior domain; see Hidano-Wang-Yokoyama [5] and the references therein. For spatial dimension $n\geq 3$, we prove the global existence of any small amplitude solutions to (1), if $\int_{0}^{1}\rho(\tau)\tau^{\kappa-3}<\infty$ for $n=3$ and $\kappa>\frac{n}{2}$ for $n\geq 4$. Also, for sample choices of the initial data and nonlinear terms, the solutions will blow up at finite time, when $n=3$, $\kappa=2$, and $\int_{0}^{1}\rho(\tau)\tau^{-1}d\tau=\infty$. When $1\leq\kappa\leq\frac{n}{2}$, for spatial dimension $n\geq 3$, the current technology can only be applied for the radial cases and non-empty obstacles $\mathcal{K}$, that is, as long as $\mathcal{K}$ is a closed ball centered at origin, we can obtain the global existence of any small amplitude radial solution for any nonlinear term $F(\partial_{t}u,\partial_{r}u)$ satisfying $\kappa=1$ and $\int_{0}^{1}\rho(\tau)\tau^{-\frac{2}{n-1}-1}d\tau<\infty$, or $\kappa\geq 2$. Again, for sample choices of initial data and nonlinear terms, the solutions cannot extend to infinity, when $\kappa=1$ and $\int_{0}^{1}\rho(\tau)\tau^{-\frac{2}{n-1}-1}d\tau=\infty$. For spatial dimension $2$, due to the lack of local energy estimates, similar results are only established for $M=\mathbb{R}^{2}$ and $\kappa\geq 2$. Given any small amplitude initial data, global solutions exist when $\int_{0}^{1}\rho(\tau)\tau^{\kappa-4}d\tau<\infty$. For sample choices of initial data and nonlinear terms, the solutions will blow up in finite time, when $\int_{0}^{1}\rho(\tau)\tau^{\kappa-4}d\tau=\infty$. On the whole, the existence of a global solution to (1) with small initial data basically depends on whether the integral $\int_{0}^{1}\rho(\tau)\tau^{\kappa-p_{c}(n)-1}d\tau$, $p_{c}(n)=1+2/(n-1)$, converges or not. Meanwhile, the sharp estimates of the lifespan, the maximal existence time of the solution to (1), are also provided for the blow-up solutions for sample choices of the initial data and nonlinear terms.

Let us recall the history of the Glassey conjecture. For Cauchy problem

with $p>1$ and $a,b\in\mathbb{R}$, it is conjectured that the critical power for the global existence v.s. blow-up is $p_{c}(n)$ in Glassey’s mathematical review [3]; see also Schaeffer [13] and Rammaha [10]. For $a\neq 0$, $b=0$, the global existence part of the conjecture is verified for general initial data in dimension $2,3$ by Hidano-Tsutaya [4] and Tzvetkov [18], independently. For $n\geq 4$, the conjecture is only demonstrated for radial initial data and $p\in(p_{c}(n),1+2/(n-2))$; see Hidano-Wang-Yokoyama [5]. For $a>0$ and sample choices of the initial data, the blow-up result is verified by Rammaha [10] for all spatial dimension $n\geq 2$ expect for critical cases, $p=p_{c}(n)$, in the even dimension. Later, Zhou [21] shows the blow-up for all dimension $n\geq 1$ with $p\in(1,p_{c}(n)]$ where $p_{c}(1)=\infty$, together with upper bound estimate on the lifespan for $p\in(1,p_{c}(n)]$, which is

On the other hand, deduced from the well-posed theory for $p\in(1,p_{c}(n)]$, the lower bound on the lifespan is also obtained for all spatial dimensions, that is,

see Hidano-Wang-Yokoyama [5] for $n\geq 2$ (see also Fang-Wang [2] for the critical case in dimension two) and Kitamura-Morisawa-Takamura [8] for dimension one. Hence, for nonlinear terms $F(\partial u)=|\partial_{t}u|^{p}$, the estimates of the lifespan (3) and (4) are sharp.

As for $a=0,b\neq 0$, the results for the global existence are the same as above, as well as the lower bound estimates for $p\in(1,p_{c}(n)]$. Hence, when $b>0$, one may expect the powers for blow-up results are also $(1,p_{c}(n)]$ and the lifespan estimates (3) and (4) are sharp. For dimension one, the sharp lifespan estimates are verified in Sasaki-Takamatsu-Takamura [12]. For dimension $n=3$, the blow-up result is first established in Sideris [16] for $p=p_{c}(3)=2$, as well as the upper bound estimate $\varlimsup_{\varepsilon\rightarrow 0^{+}}\ln(T(\varepsilon))\varepsilon^{2}<\infty$, which does not match the well-known lower bound verified in John-Klainerman [7], $\varliminf_{\varepsilon\rightarrow 0^{+}}\ln(T(\varepsilon))\varepsilon>0$. For $n=2$, Schaeffer [13] obtains the blow-up result for $p=p_{c}(2)=3$. For $n\geq 4$, Rammaha shows the blow-up result in [10] for all of the critical and subcritical powers excluding the critical cases for even dimensions and studies the upper bound for $n=2,3$ and $p=2$ in [11]. In Takamura, Wang and the author’s recent work [14], we provide a uniform way to deduce the sharp upper bound estimates of the lifespan for all dimension $n\geq 2$ and $p\in(1,p_{c}(n)]$, that is,

There are also similar results when the problem (2) is considered on the asymptotically flat Lorentzian manifolds or the exterior domain; see Wang [20, 19].

Since the critical power $p_{c}(n)$ is the borderline of the existence of the global solution with power-type nonlinearities. To determine the threshold nature of the nonlinear term, one may study the problem

where the function $\mu:[0,\infty)\rightarrow[0,\infty)$ is a non-decreasing sufficiently smooth function with $\mu(0)=0$. The blow-up is established by Chen-Palmieri [1, Theorem 2.1] for all classical solutions to (5) under the assumptions that the function $\tilde{F}:s\in\mathbb{R}\rightarrow|s|^{p_{c}(n)}\mu(|s|)$ is convex, that $\int_{0}^{1}\mu(s)s^{-1}ds=\infty$, and that initial data $\phi,\psi\in C_{c}^{\infty}$ with $\int_{\mathbb{R}^{n}}\psi(x)>0$. Also, they provide the upper bound estimates for the lifespan, that is, for all sufficiently small $\varepsilon$, there exist constants $K_{1}$, $K_{2}$, and $K_{3}$ such that the lifespan $T(\varepsilon)$ of the solution to (5) satisfies

where the function $\mathcal{H}:s\in(0,1]\in\rightarrow\int_{s}^{1}\mu(\tau)\tau^{-1}d\tau$ and $\mathcal{H}^{inv}$ is the inverse function of $\mathcal{H}$. They also obtain results for the existence part when $n=3$ and initial data $\phi,\psi\in C_{c}^{\infty}$ are radial. If $\int_{0}^{1}\mu(s)s^{-1}ds<\infty$, there exists a unique radial global classical solution $u$ to (5) for all sufficiently small $\varepsilon$, provided that $\mu\in C^{2}((0,\infty))$ satisfying $|\mu^{(i)}(\tau)|\lesssim\tau^{-i}\mu(\tau)$, $i=1,2$. When $\int_{0}^{1}\mu(s)s^{-1}ds=\infty$, the lifespan of the radial classical solution to (5) has the lower bound

where $K_{4}$, $K_{5}$, and $K_{6}$ are some positive constants; see [1, Theorem 2.2 and Propsition 2.2].

With these results in hand, it is natural to ask what is the criterion for general non-linear terms $F(\partial u)$. Since $\partial^{\leq\kappa}F(0)=0$, it follows that in some neighborhood of the origin $|F(\mathbf{q})|\lesssim|\mathbf{q}|^{\kappa}\rho(|\mathbf{q}|)$. Heuristically, if the obstacle $K$ is empty, the solutions to (1) could behave like free waves with energy of size $\varepsilon$ and decay rate $(n-1)/2$, for the time interval $[0,T)$, when

Here, $K_{7}$ is some positive constant from the weighted Sobolev embedding and $\langle t\rangle=\sqrt{1+t^{2}}$. With simple calculation, the main term of the left hand side of (6) is

which gives rise to the following expected sharp estimate for the lifespan, provided $\int_{0}^{1}\rho(\tau)\tau^{\kappa-p_{c}(n)-1}d\tau=\infty$,

where $K_{8}$ is some positive constant and $\mathcal{H}_{\kappa,n}$ is

and $\mathcal{H}_{\kappa,n}^{inv}$ is the inverse function of $\mathcal{H}_{\kappa,n}$.

Now, we can state our results.

Theorem 1.1 is mainly established by local energy estimates and Sobolev embedding. Here, we give the definition of the compatibility of order $\kappa+1$.

Using the argument in the proof of Theorem 1.1, we can obtain the following result for $n\geq 3$ and $\kappa>\max\{n/2,2\}$.

Due to the lack of local energy estimates in dimension 2, the problem (1) for non-empty obstacles is still unsettled. However, for $\mathcal{K}=\emptyset$, we will have the following theorem by standard energy estimates, Klainerman-Sobolev inequalities, and trace estimates.

According to Theorem 1.1, Remark 1.3, and Theorem 1.4, as long as $\kappa>\frac{n}{2}$, the convergence of the integral

foreshadows the existence of the global solution. And the divergence of the integral of (9) implies a lower bound of the lifespan. For $n\geq 3$, $\mathcal{K}=\overline{B_{1}}$, and $\kappa=1$, this criterion is still available for the radial solution and we should consider the following nonlinear problem

Here, $F\in C^{\kappa}(\{\mathbf{q}\in\mathbb{R}^{2}:|\mathbf{q}|\leq 1\}),\ \kappa\geq 1,\ \partial_{\mathbf{q}}^{\leq\kappa}F(0)=0$.

Following the argument in [1], we can show that the lower bound estimates of the lifespan are sharp for sample choices of the nonlinear terms and initial data. Before the statement, we give a rigorous definition of the lifespan.

To conclude this section, we give some typical examples of $F(\partial u)$.

• •

  Example 1: $F(\partial u)=|\partial u|^{p}$, $p>1$ for $n\geq 3$ and $p>2$ for $n\geq 2$. We can choose

  $$\kappa=\left\{\begin{aligned} &\text{the integer part of}\ p,\ p\ \text{is not an integer}\\ &p-1,\ p\ \text{is an integer}.\end{aligned}\right.$$

  Then, the criterion is whether $\int_{0}^{1}\tau^{p-p_{c}(n)-1}$ is infinity or not, which coincides with the classical results. For $p<p_{c}(n)$, $H_{\kappa,n}(s)\sim s^{p-p_{c}(n)}-1$, hence, for $\varepsilon\ll 1$,

  $$\varepsilon^{\frac{2}{n-1}}\left[\mathcal{H}_{\kappa,n}^{inv}(K_{9}\varepsilon^{-\frac{2}{n-1}}+\mathcal{H}_{\kappa,n}(K_{10}\varepsilon))\right]^{-\frac{2}{n-1}}\sim\varepsilon^{\frac{2(p-1)}{2-(n-1)(p-1)}},$$

  where $K_{9},K_{10}$ are positive constants given in the lifespan estimates above; for $p=p_{c}(n)$,

  $$H_{\kappa,n}(s)\sim\ln(s^{-1}),$$

  and then, for $\varepsilon\ll 1$

  $$\ln\left(\varepsilon^{\frac{2}{n-1}}\left[\mathcal{H}_{\kappa,n}^{inv}(K_{9}\varepsilon^{-\frac{2}{n-1}}+\mathcal{H}_{\kappa,n}(K_{10}\varepsilon))\right]^{-\frac{2}{n-1}}\right)\sim\varepsilon^{-\frac{2}{n-1}}.$$

• •

  Example 2: $F(\partial u)=|\partial u|^{p_{c}(n)}\mu(|\partial u|)$ with $\mu$ satisfying (13) or (15). We can select $\kappa$ to be the integer part of $p_{c}(n)$. Then, the criterion is whether $\int_{0}^{1}\mu(\tau)\tau^{-1}$ is infinity or not, which coincides with the blow-up results in [1]. By direct calculations,

  $$H_{\kappa,n}(s)\sim\int_{s}^{1}\mu(\tau)\tau^{-1}d\tau,$$

  hence we extend the existence results in [1].

• •

  Example 3: $F(\partial u)=|\partial u|\frac{1}{\log(|\partial u|^{-1})}$ with $n\geq 3$ and $M=\{x:|x|>1\}$. By Theorem 1.5, given radial initial data, we can obtain a unique radial solution locally and provide the lower bound estimate of the lifespan. However, the sharp upper bound estimate is still open.