Local well-posedness of the Cauchy problem for a $p-$adic Nagumo-type equation

Along this article $p$ will denote a prime number. The field of $p-$adic numbers $\mathbb{Q}_{p}$ is defined as the completion of the field of rational numbers $\mathbb{Q}$ with respect to the $p-$adic norm $|\cdot|_{p}$, which is defined as

where $a$ and $b$ are integers coprime with $p$. The integer $\gamma:=ord(x)$, with $ord(0):=+\infty$, is called the $p-$adic order of $x$.

Any $p-$adic number $x\neq 0$ has a unique expansion of the form

where $x_{j}\in\{0,\dots,p-1\}$ and $x_{0}\neq 0$. By using this expansion, we define the fractional part of $x\in\mathbb{Q}_{p}$, denoted $\{x\}_{p}$, as the rational number

For $r\in\mathbb{Z}$, denote by $B_{r}^{n}(a)=\{x\in\mathbb{Q}_{p}^{n};||x-a||_{p}\leq p^{r}\}$ the ball of radius $p^{r}$ with center at $a=(a_{1},\dots,a_{n})\in\mathbb{Q}_{p}^{n}$, and take $B_{r}^{n}(0):=B_{r}^{n}$. Note that $B_{r}^{n}(a)=B_{r}(a_{1})\times\cdots\times B_{r}(a_{n})$, where $B_{r}(a_{i}):=\{x_{i}\in\mathbb{Q}_{p};|x_{i}-a_{i}|_{p}\leq p^{r}\}$ is the one-dimensional ball of radius $p^{r}$ with center at $a_{i}\in\mathbb{Q}_{p}$. The ball $B_{0}^{n}$ equals the product of $n$ copies of $B_{0}=\mathbb{Z}_{p}$, the ring of $p-$adic integers. We also denote by $S_{r}^{n}(a)=\{x\in\mathbb{Q}_{p}^{n};||x-a||_{p}=p^{r}\}$ the sphere of radius $p^{r}$ with center at $a=(a_{1},\dots,a_{n})\in\mathbb{Q}_{p}^{n}$, and take $S_{r}^{n}(0):=S_{r}^{n}$. We notice that $S_{0}^{1}=\mathbb{Z}_{p}^{\times}$ (the group of units of $\mathbb{Z}_{p}$), but $\left(\mathbb{Z}_{p}^{\times}\right)^{n}\subsetneq S_{0}^{n}$. The balls and spheres are both open and closed subsets in $\mathbb{Q}_{p}^{n}$. In addition, two balls in $\mathbb{Q}_{p}^{n}$ are either disjoint or one is contained in the other.

As a topological space $\left(\mathbb{Q}_{p}^{n},||\cdot||_{p}\right)$ is totally disconnected, i.e. the only connected  subsets of $\mathbb{Q}_{p}^{n}$ are the empty set and the points. A subset of $\mathbb{Q}_{p}^{n}$ is compact if and only if it is closed and bounded in $\mathbb{Q}_{p}^{n}$, see e.g. [29, Section 1.3], or [1, Section 1.8]. The balls and spheres are compact subsets. Thus $\left(\mathbb{Q}_{p}^{n},||\cdot||_{p}\right)$ is a locally compact topological space.

Since $(\mathbb{Q}_{p}^{n},+)$ is a locally compact topological group, there exists a Haar measure $d^{n}x$, which is invariant under translations, i.e. $d^{n}(x+a)=d^{n}x$. If we normalize this measure by the condition $\int_{\mathbb{Z}_{p}^{n}}dx=1$, then $d^{n}x$ is unique.

A complex-valued function $\varphi$ defined on $\mathbb{Q}_{p}^{n}$ is called locally constant if for any $x\in\mathbb{Q}_{p}^{n}$ there exist an integer $l(x)\in\mathbb{Z}$ such that

A function $\varphi:\mathbb{Q}_{p}^{n}\rightarrow\mathbb{C}$ is called a Bruhat-Schwartz function (or a test function) if it is locally constant with compact support. Any test function can be represented as a linear combination, with complex coefficients, of characteristic functions of balls. The $\mathbb{C}-$vector space of Bruhat-Schwartz functions is denoted by $\mathcal{D}(\mathbb{Q}_{p}^{n}):=\mathcal{D}$. We denote by $\mathcal{D}_{\mathbb{R}}(\mathbb{Q}_{p}^{n}):=\mathcal{D}_{\mathbb{R}}$ the $\mathbb{R}-$vector space of Bruhat-Schwartz functions. For $\varphi\in\mathcal{D}(\mathbb{Q}_{p}^{n})$, the largest number $l=l(\varphi)$ satisfying (2.1) is called the exponent of local constancy (or the parameter of constancy) of $\varphi$.

We denote by $\mathcal{D}_{m}^{l}(\mathbb{Q}_{p}^{n})$ the finite-dimensional space of test functions from $\mathcal{D}(\mathbb{Q}_{p}^{n})$ having supports in the ball $B_{m}^{n}$ and with parameters  of constancy $\geq l$. We now define a topology on $\mathcal{D}$ as follows. We say that a sequence $\left\{\varphi_{j}\right\}_{j\in\mathbb{N}}$ of functions in $\mathcal{D}$ converges to zero, if the two following conditions hold:

(1) there are two fixed integers $k_{0}$ and $m_{0}$ such that  each $\varphi_{j}\in$ $\mathcal{D}_{m_{0}}^{k_{0}}$;

(2) $\varphi_{j}\rightarrow 0$ uniformly.

$\mathcal{D}$ endowed with the above topology becomes a topological vector space.

Given $\rho\in[1,\infty)$, we denote by $L^{\rho}:=L^{\rho}\left(\mathbb{Q}_{p}^{n}\right):=L^{\rho}\left(\mathbb{Q}_{p}^{n},d^{n}x\right),$ the $\mathbb{C}-$vector space of all the complex-valued functions $g$ satisfying

The corresponding $\mathbb{R}-$vector spaces are denoted as $L_{\mathbb{R}}^{\rho}\allowbreak:=L_{\mathbb{R}}^{\rho}\left(\mathbb{Q}_{p}^{n}\right)=L_{\mathbb{R}}^{\rho}\left(\mathbb{Q}_{p}^{n},d^{n}x\right)$, $1\leq\rho<\infty$.

If $U$ is an open subset of $\mathbb{Q}_{p}^{n}$, $\mathcal{D}(U)$ denotes the space of test functions with supports contained in $U$, then $\mathcal{D}(U)$ is dense in

where $d^{n}x$ is the normalized Haar measure on $\left(\mathbb{Q}_{p}^{n},+\right)$, for $1\leq\rho<\infty$, see e.g. [1, Section 4.3]. We denote by $L_{\mathbb{R}}^{\rho}\left(U\right)$ the real counterpart of $L^{\rho}\left(U\right)$.

Set $\chi_{p}(y)=\exp(2\pi i\{y\}_{p})$ for $y\in\mathbb{Q}_{p}$. The map $\chi_{p}(\cdot)$ is an additive character on $\mathbb{Q}_{p}$, i.e. a continuous map from $\left(\mathbb{Q}_{p},+\right)$ into $S$ (the unit circle considered as multiplicative group) satisfying $\chi_{p}(x_{0}+x_{1})=\chi_{p}(x_{0})\chi_{p}(x_{1})$, $x_{0},x_{1}\in\mathbb{Q}_{p}$.  The additive characters of $\mathbb{Q}_{p}$ form an Abelian group which is isomorphic to $\left(\mathbb{Q}_{p},+\right)$. The isomorphism is given by $\kappa\rightarrow\chi_{p}(\kappa x)$, see e.g. [1, Section 2.3].

Given $\xi=(\xi_{1},\dots,\xi_{n})$ and $y=(x_{1},\dots,x_{n})\allowbreak\in\mathbb{Q}_{p}^{n}$, we set $\xi\cdot x:=\sum_{j=1}^{n}\xi_{j}x_{j}$. The Fourier transform of $\varphi\in\mathcal{D}(\mathbb{Q}_{p}^{n})$ is defined as

where $d^{n}x$ is the normalized Haar measure on $\mathbb{Q}_{p}^{n}$. The Fourier transform is a linear isomorphism from $\mathcal{D}(\mathbb{Q}_{p}^{n})$ onto itself satisfying

see e.g. [1, Section 4.8]. We will also use the notation $\mathcal{F}_{x\rightarrow\xi}\varphi$ and $\widehat{\varphi}$ for the Fourier transform of $\varphi$.

The Fourier transform extends to $L^{2}$. If $f\in L^{2},$ its Fourier transform is defined as

where the limit is taken in $L^{2}$. We recall that the Fourier transform is unitary on $L^{2},$ i.e. $||f||_{2}=||\mathcal{F}f||_{2}$ for $f\in L^{2}$ and that (2.2) is also valid in $L^{2}$, see e.g. [26, Chapter III, Section 2].

The $\mathbb{C}-$vector space $\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$ $:=\mathcal{D}^{\prime}$  of all continuous linear functionals on $\mathcal{D}(\mathbb{Q}_{p}^{n})$ is called the Bruhat-Schwartz space of distributions. Every linear functional on $\mathcal{D}$ is continuous, i.e. $\mathcal{D}^{\prime}$ agrees with the algebraic dual of $\mathcal{D}$, see e.g. [29, Chapter 1, VI.3, Lemma]. We denote by $\mathcal{D}_{\mathbb{R}}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$ $:=\mathcal{D}_{\mathbb{R}}^{\prime}$ the dual space of $\mathcal{D}_{\mathbb{R}}$.

We endow $\mathcal{D}^{\prime}$ with the weak topology, i.e. a sequence $\left\{T_{j}\right\}_{j\in\mathbb{n}}$ in $\mathcal{D}^{\prime}$ converges to $T$ if $\lim_{j\rightarrow\infty}T_{j}\left(\varphi\right)=T\left(\varphi\right)$ for any $\varphi\in\mathcal{D}$.  The map

is a bilinear form which is continuous in $T$ and $\varphi$ separately. We call this map the pairing between $\mathcal{D}^{\prime}$ and $\mathcal{D}$. From now on we will use $\left(T,\varphi\right)$ instead of $T\left(\varphi\right)$.

Every $f$ in $L_{loc}^{1}$ defines a distribution $f\in\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$ by the formula

Such distributions are called regular distributions. Notice that for $f$ $\in L_{\mathbb{R}}^{2}$, $\left(f,\varphi\right)=\left\langle f,\varphi\right\rangle$, where $\left\langle\cdot,\cdot\right\rangle$ denotes the scalar product in $L_{\mathbb{R}}^{2}$.

The Fourier transform $\mathcal{F}\left[T\right]$ of a distribution $T\in\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$ is defined by

The Fourier transform $T\rightarrow\mathcal{F}\left[T\right]$ is a linear (and continuous) isomorphism from $\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$ onto $\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{n}\right)$. Furthermore, $T=\mathcal{F}\left[\mathcal{F}\left[T\right]\left(-\xi\right)\right]$.

Let $\alpha>0$, the Taibleson operator is defined as

for $\varphi\in\mathcal{D}(\mathbb{Q}_{p}^{n})$. This operator admits the extension

to locally constant functions satisfying

The Taibleson operator $\boldsymbol{D}^{\alpha}$ is the $p-$adic analog of the fractional derivative. If $n=1$, $\boldsymbol{D}^{\alpha}$ agrees with the Vladimirov operator. The operator $\boldsymbol{D}^{\alpha}$ does not satisfy the chain rule neither Leibniz formula. We also use the notation $\boldsymbol{D}_{x}^{\alpha}$, when the Taibleson operator acts on functions depending on the variables $x\in\mathbb{Q}_{p}^{n}$ and $t\geq 0$.

Given $0=\delta_{0}<\delta_{1}<\cdots<\delta_{k-1}<\delta_{k}=\delta$, we define

Let $X$, $Y$ Banach spaces, $T_{0}\in(0,\infty)$ and let $F:\left[0,T_{0}\right]\times Y\longrightarrow X$ a continuous function. The Cauchy problem

is locally well-posed in $Y$, if the following conditions are satisfied.

(i) There is $T\in\left(0,T_{0}\right]$ and a function $u\in C([0,T];Y)$, with $u(0)=\phi$, satisfying the differential equation in the following sense:

where the derivatives at $t=0$ and $t=T$ are calculated from the right and left, respectively.

(ii) The initial value problem (4.1) has at most one solution in $C([0,T];Y)$.

(iii) The function $\phi\rightarrow u$ is continuous. That is, let $\left\{\phi_{n}\right\}$ be a sequence in $Y$ such that $\phi_{n}\rightarrow\phi_{\infty}$ in $Y$ and let $u_{n}\in C\left(\left[0,T_{n}\right];Y\right)$, resp. $u_{\infty}\in C\left(\left[0,T_{\infty}\right];Y\right)$, be the corresponding solutions. Let $T\in\left(0,T_{\infty}\right)$, then the solutions $u_{n}$ are defined in $[0,T]$ for all $n$ big enough and

Consider the following Cauchy problem:

where $T$, $\gamma$, $\alpha$, $\beta>0$, and $m$ is a positive integer. The main result of this work is the following:

We denote by $\boldsymbol{V}(t)=e^{-(\gamma\boldsymbol{D}^{\alpha}+\beta\boldsymbol{I})t}$, $t\geq 0$, the semigroup in $L^{2}$ generated by the operator $\boldsymbol{A}=-\gamma\boldsymbol{D}^{\alpha}-\beta\boldsymbol{I}$, that is,