Radial solutions of a pseudo-differential equations associated with a p-adic non-local ultradiffusion operator

Abstract

This paper is dedicated to study some pseudo-differential equations related to the operator $D^{\alpha }_{d_1,d_2}$ with symbol $\mathcal {A}^{\alpha }_{d_1,d_2}(x)=\max \left\{ |x|_p^{d_1},\ |x|_p^{d_2}\right\} ^\alpha$, acting on a radial function $\varphi (|x|_p)$. The importance of this symbol is that it is a generalization of Vladimirov and Bessel operators. Our research provides conditions for the existence of its right inverse and the explicit formulae for both operators, as well as a practical application. The inverse allows solving a Cauchy problem with the operator, which can be reduced to an integral equation whose properties are similar to those of the classical Volterra equation. We also find conditions for the existence and uniqueness of a mild solution for a degenerate nonlinear Cauchy problem.

1 Introduction

Kochubei started the study of pseudo-differential equations over the class of radial functions in [1] with the Cauchy problem

$$\begin{aligned} D^{\alpha } u + a(|x|_K)u&=f(|x|_K), \ x\in K,\end{aligned}$$

(1)

$$\begin{aligned} u(0)&=0 \end{aligned}$$

(2)

where K is a non-Archimedean field, $D^{\alpha }$ is the Vladimirov operator, u is a radial function $u=u(|x|_K)$ and a and f are continuous functions with several conditions. Since the operator has an inverse $I^{\alpha }$, the author considers solutions of the form $u=I^{\alpha }v$ and transforms the original equation into an integral equation

$$\begin{aligned} v(|x|_K)+a(|x|_K)(I^{\alpha }v)(|x|_K)=f(|x|_K), \ x\in K. \end{aligned}$$

The author shows that if some inequalities hold for functions a and f, then the Cauchy problem (1)–(2) has a strong solution.

Next, in [2], Kochubei studies the nonlinear Cauchy problem

$$\begin{aligned} (D^{\alpha }u)(|t|_p) = f(|t|_p, u(|t|_p)), \ 0\ne t\in {\mathbb {Q}}_{p}, \end{aligned}$$

(3)

with the initial condition $u(0)=u_0.$ Under some hypotheses, this problem has a local solution, i.e., there exists a solution in a ball, which can be extended to all $t\in {\mathbb {Q}}_{p}.$

In the paper, Antoniouk et al. [3] study the class of degenerate equations

$$\begin{aligned} |t|_p^{-\gamma } (D^{\alpha }u)(|t|_p)&= f(|t|_p, u(|t|_p)), \ 0\ne t\in {\mathbb {Q}}_{p}, \end{aligned}$$

(4)

$$\begin{aligned} u(0)&=u_0, \end{aligned}$$

(5)

for $\gamma >0$. The authors extend the methods previously given by Kochubei in order to solve this equation. They showed that the problem (4)–(5) has a unique local mild solution.

In this paper, we consider the operator

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)=\mathcal {F}^{-1}\left[ \mathcal {A}^\alpha _{d_1,d_2}(\xi )\mathcal {F}\varphi (\xi )\right] (x), \end{aligned}$$

where the symbol $\mathcal {A}^{\alpha }_{d_1,d_2}(x)$ is defined as

$$\begin{aligned} \mathcal {A}^{\alpha }_{d_1,d_2}(x)=\max \left\{ |x|_p^{d_1},\ |x|_p^{d_2}\right\} ^\alpha , \end{aligned}$$

where $0 \le d_1\le d_2$ are natural numbers, $\alpha >0$, and $\varphi$ is, at first, a test function. This operator was introduced by the authors in [4], and its importance lies in the fact that it is a generalization of Vladimirov and Bessel operators.

The first purpose of this article is to find the right inverse of $D^{\alpha }_{d_1,d_2}$ and give the explicit formulae for it and its inverse $I^{\alpha }_{d_1,d_2}$, following the ideas given in [1]. This is the key fact that lets us solve the problems we consider next, see Theorems 8 and 9 in Sect. 4.

The second objective is to find the solution of the Cauchy problem

$$\begin{aligned} D^{\alpha }_{d_1,d_2} \varphi + a(|x|_K)\varphi =f(|x|_p), \ x\in B_N, \end{aligned}$$

where a and f are continuous and $|x|_p\le p^N$, with initial condition $\varphi (0)=0$. We use the techniques developed by Kochubei in [1, 2] to transform the problem into an integral equation and apply the theory of Volterra-type equations.

Finally, we prove that, under some conditions for $\alpha >0$, $\alpha d_i\ne 1$, $i=1,2$, $\gamma >0$ and $f:p^{{\mathbb {Z}}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, the degenerate nonlinear Cauchy problem

$$\begin{aligned} |t|_p^{\gamma } (D^{\alpha }_{d_1,d_2}\varphi (|t|_p))=f(|t|_p, \varphi (|t|_p)),\ 0\ne t\in {\mathbb {Q}}_{p}\end{aligned}$$

has a unique mild solution, following the ideas given in [3]. Since we already know the inverse operator, we can associate an integral equation to this problem. Then, we construct a sequence ${\varphi _k}$, which converges uniformly to a function $\varphi$, which is indeed a local solution of the integral equation; see Theorem 16 in Sect. 5.

For the sake of completeness, we start with a preliminary section, where we establish the notation and main facts about p-adics.

2 Preliminaries

In this section, we fix the notation and collect some facts about the field of p-adic numbers, that we will use throughout the article. For a detailed exposition on p-adic analysis, the reader may consult [5,6,7].

2.1 The field of p-adic numbers

In 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

$$\begin{aligned} |x|_{p}= {\left\{ \begin{array}{ll} 0 &{} \text {if }x=0,\\ p^{-r} &{} \text {if }x=p^{r}\frac{a}{b}, \end{array}\right. } \end{aligned}$$

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

Any $p-$adic number $x\ne 0$ has a unique expansion $x=p^{ord(x)}\sum _{j=0}^{\infty }x_{j}p^{j}$, where $x_{j}\in \{0,1,2,\dots ,p-1\}$ and $x_{0}\ne 0$. By using this expansion, we define the fractional part of $x\in {\mathbb {Q}}_{p}$, denoted $\{x\}_{p}$, as the rational number

$$\begin{aligned} \{x\}_{p}= {\left\{ \begin{array}{ll} 0 &{} \text {if }x=0\text { or }ord(x)\ge 0,\\ p^{ord(x)}\sum _{j=0}^{-ord(x)-1}x_{j}p^{j} &{} \text {if }ord(x)<0. \end{array}\right. } \end{aligned}$$

For $r\in \mathbb {Z}$, denote by $B_{r}(a)=\{x\in {\mathbb {Q}}_{p}:|x-a|_{p}\le p^{r}\}$ the ball of radius $p^{r}$ with center at $a\in {\mathbb {Q}}_{p}$, and take $B_{r}(0):=B_{r}$. The sphere of radius $p^r$ and center in $a\in {\mathbb {Q}}_{p}$ is defined as $S_r(a)=\{ x\in {\mathbb {Q}}_{p}: |x-a|_p=p^r\}$. The set $\left( {\mathbb {Q}}_{p}, |\cdot |_p\right)$ is a complete ultrametric space.

2.2 The Bruhat-Schwartz space

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

$$\begin{aligned} \varphi (x+x^{\prime })=\varphi (x)\text { for }x^{\prime }\in B_{l(x)}. \end{aligned}$$

(6)

For this function $\varphi$, the largest of such numbers $l:=l(\varphi )$ satisfying (6) is called the exponent of local constancy of $\varphi$. The space of locally constant functions is denoted by $\mathcal {E}({\mathbb {Q}}_{p})$. The characteristic function of the ball $B_r(a)$ is a simple example of a locally constant function:

$$\begin{aligned} \Delta _r(x-a)=\Omega (p^{-r}|x-a|_p), \text { for } a,x\in {\mathbb {Q}}_{p}, \end{aligned}$$

where

$$\begin{aligned} \Omega (t)={\left\{ \begin{array}{ll} 1, &{} 0\le t\le 1\\ 0, &{} t>1. \end{array}\right. } \end{aligned}$$

A function $\varphi :{\mathbb {Q}}_{p}\rightarrow {\mathbb {C}}$ is called a Bruhat-Schwartz function (or a test function) if it is locally constant with compact support. The ${\mathbb {C}}$-vector space of Bruhat-Schwartz functions is denoted by ${{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$. The characteristic function of a ball is also a test function.

Let ${{\,\mathrm{\textbf{S}}\,}}^{\prime }({\mathbb {Q}}_{p})$ denote the set of all functionals (distributions) on ${{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$. All functionals on ${{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$ are continuous; see [7].

2.3 Haar measure

Since $({\mathbb {Q}}_{p}, +)$ is a locally compact topological group, there exists, up to multiplication for a positive constant, a translation invariant measure dx (i.e., $d(x+a)=dx,\ a\in \mathbb Q_{p}$), see [8]. This measure is called the Haar measure of $({\mathbb {Q}}_{p}, +)$. We normalize this measure by the condition $\displaystyle {\int \limits _{|x|_{p}\le 1}dx=1}$, then dx is unique. Furthermore, it is true that

$$\begin{aligned} d(ax)=|a|_pdx, \qquad a\in \mathbb {Q}_p-\{0\}. \end{aligned}$$

A function $f:\mathbb {Q}_p\rightarrow \mathbb {C}$ is called locally integrable, $f\in L^{1}_{loc}({\mathbb {Q}}_{p})$, if for any positive integer N,

$$\begin{aligned} \int \limits _{B_{N}} |f(x)| dx < \infty . \end{aligned}$$

A function $f\in L^{1}_{loc}({\mathbb {Q}}_{p})$ is called integrable on ${\mathbb {Q}}_{p}$, if there exists

$$\begin{aligned} \lim _{N\rightarrow \infty } \int \limits _{B_{N}} f(x)dx=\lim _{N\rightarrow \infty } \sum _{-\infty <r\le N}\int _{S_{r}}f(x)dx. \end{aligned}$$

This limit is called an improper integral of the function f on ${\mathbb {Q}}_{p}$, and it is denoted by $\displaystyle {\int _{{\mathbb {Q}}_{p}}f(x)dx}$, thus

$$\begin{aligned} \int \limits _{{\mathbb {Q}}_{p}}f(x)dx=\sum _{-\infty<r<\infty } \int \limits _{S_{r}} f(x)dx. \end{aligned}$$

In order to clarify the way we integrate, we put the following three examples of integrals.

• 1. The volume of the ball $B_N$ is $\displaystyle \int \limits _{|x|_p\le p^{N}}dx=p^N$, in fact,

  $$\begin{aligned} \int \limits _{|x|_p\le p^{N}}dx=\int \limits _{|x|_p\le 1}d(p^{-N}x)=p^N\int \limits _{|x|_p\le 1}dx=p^N. \end{aligned}$$

• 2. The volume of the sphere $S_r$ is $\displaystyle \int \limits _{|x|_p= p^{N}}dx=p^N(1-p^{-1})$.

  $$\begin{aligned} \int \limits _{|x|_p= p^{N}}dx=\int \limits _{|x|_p\le p^N}dx-\int \limits _{|x|_p\le p^{N-1}}dx=p^N-p^{N-1}. \end{aligned}$$

• 3. Let $\chi _p(y)=\exp (2\pi i\{y\}_{p})$, called the additive standard character, then

  $$\begin{aligned} \displaystyle \int \limits _{|x|_p\le p^{N}}\chi _p(\xi x)dx={\left\{ \begin{array}{ll} p^N&{}\text {if }|\xi |_p\le p^{-N}\\ 0&{}\text {if }|\xi |_p> p^{-N}. \end{array}\right. } \end{aligned}$$

  In fact, if $|\xi |_p\le p^{-N}$, we have $|x\xi |_p\le 1$ and $\chi _p(x\xi )=1$, then

  $$\begin{aligned} \int \limits _{|x|_p\le p^{N}}\chi _p(\xi x)dx=\int \limits _{|x|_p\le p^{N}}dx=p^N. \end{aligned}$$

  If $|\xi |_p>p^{-N}$, then for $x'$ with $|x'|_p=p^N$ we have $|\xi x'|_p>p$. Therefore $\chi _p(\xi x')\ne 1$. Making the change of variable $x=z-x'$, we obtain

  $$\begin{aligned} \int \limits _{|x|_p\le p^{N}}\chi _p(\xi x)dx=\int \limits _{|z-x'|_p\le p^{N}}\chi _p(\xi (z-x'))dz=\chi _p(-\xi x')\int \limits _{|z|_p\le p^{N}}\chi _p(\xi z)dz, \end{aligned}$$

  this implies the second relation.

2.4 The Fourier transform

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 ${\mathbb {Q}}_{p}$ into S (the unit circle) satisfying $\chi _p (y_{0}+y_{1})=\chi _p(y_{0})\chi _p(y_{1})$, $y_{0},y_{1}\in {\mathbb {Q}}_{p}$.

Given $\xi$ and $x\in {\mathbb {Q}}_{p}$, the Fourier transform of $\varphi \in {{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$ is defined as

$$\begin{aligned} ({\mathcal {F}}\varphi )(\xi )=\int \limits _{{\mathbb {Q}}_{p}}\chi _p(-x \xi )\varphi (x)dx\quad \text {for }\xi \in {\mathbb {Q}}_{p}. \end{aligned}$$

The Fourier transform is a linear isomorphism from ${{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$ onto itself satisfying $({\mathcal {F}}({\mathcal {F}}\varphi ))(\xi )=\varphi (-\xi )$, see [7, Ch. 1-Sect.VII].

Example 1

If $\Omega (p^{-N}|x|_p)$ is the characteristic function of $B_N$, then ${\mathcal {F}}(\Omega (p^{-N}|x|_p))(\xi )=p^N\Omega (p^{N}|\xi |_p)$.

By definition

$$\begin{aligned} {\mathcal {F}}(\Omega (p^{-N}|x|_p))(\xi )=\int \limits _{{\mathbb {Q}}_{p}}\chi _p(-x \xi )\Omega (p^{-N}|x|_p)dx=\int \limits _{|x|_p\le p^N}\chi _p(-x \xi )dx. \end{aligned}$$

The result follows from Example 3 of the previous section. In particular ${\mathcal {F}}(\Omega (|x|_p))(\xi )=\Omega (|\xi |_p)$.

The Fourier transform ${\mathcal {F}}\left[ f\right]$ of a distribution $f\in {{\,\mathrm{\textbf{S}}\,}}^{\prime }\left( {\mathbb {Q}}_{p}\right)$ is defined by

$$\begin{aligned} \left( {\mathcal {F}}\left[ f\right] ,\varphi \right) =\left( f,{\mathcal {F}}\left[ \varphi \right] \right) \text { for all }\varphi \in {{\,\mathrm{\textbf{S}}\,}}\left( {\mathbb {Q}}_{p}\right) \text {.} \end{aligned}$$

The Fourier transform $f\rightarrow {\mathcal {F}}\left[ f\right]$ is a linear isomorphism from ${{\,\mathrm{\textbf{S}}\,}}^{\prime }\left( {\mathbb {Q}}_{p}\right)$ onto ${{\,\mathrm{\textbf{S}}\,}}^{\prime }\left( {\mathbb {Q}}_{p}\right)$. Furthermore, $f={\mathcal {F}}\left[ {\mathcal {F}}\left[ f\right] \left( -\xi \right) \right] .$

For more details on p-adic integration and the Fourier transform, the reader may consult Chapters 3 and 4 of [5] or Chapter 1, Sections IV, VI, and VII of [7].

3 The operator $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$

In the article [4], the authors consider the symbol

$$\begin{aligned} \mathcal {A}^{\alpha }_{d_1,d_2}(x)&=\max \left\{ |x|_p^{d_1},\ |x|_p^{d_2}\right\} ^\alpha , \end{aligned}$$

where $0 \le d_1\le d_2$ are natural numbers. One important property of this symbol is that it generalizes the Vladimirov operator’s symbol for a specific choice of numbers $d_i$, and the Bessel operator. The Fourier transform of $\mathcal {A}^{\alpha }_{d_1,d_2}(x)$ motivates the Definition 1 of the distribution $\mathcal {C}_{d_1,d_2}^{\alpha }(\xi )$, since ${\mathcal {F}}\mathcal {A}^{\alpha }_{d_1,d_2}(x)=\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )$.

The operator studied in [4] is written in terms of $\mathcal {A}^{\alpha }_{d_1,d_2}(x)$, but its symbol is $(\mathcal {C}_{d_1,d_2}^{\alpha }(0)-\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi ))$. The term $\mathcal {C}_{d_1,d_2}^{\alpha }(0)$ makes it difficult to prove the semigroup property for the operators.

Instead of that, in this paper, we consider the pseudo-differential operator with symbol $\mathcal {A}^{\alpha }_{d_1,d_2},$ namely

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)=\mathcal {F}^{-1}\left[ \mathcal {A}^\alpha _{d_1,d_2}(\xi )\mathcal {F}(\varphi )(\xi )\right] (x). \end{aligned}$$

(7)

Following the ideas given in [1] and [3], we define its right inverse operator in order to solve the equations (8) and (10).

In order to achieve our goal, we start working with the distribution $\mathcal {C}_{d_1,d_2}^{\alpha }(\xi )$, we see how it acts on a test function; we find its meromorphic continuation and its convolution with test functions.

Definition 1

For ${{\,\mathrm{\text {Re}}\,}}{(\alpha )}>0$ with $\alpha d_j\ne 1+\dfrac{2\pi i}{\ln p}\mathbb {Z},\ j=1,2$, we define a regular distribution as following (acting on $S(\mathbb {Q}_p)$)

$$\begin{aligned} \mathcal {C}^\alpha _{d_1,d_2}(\xi )= & {} \left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) \Omega (|\xi |_p)\\{} & {} +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) |\xi |^{\alpha d_1-1}\left( 1-\Omega (|\xi |_p)\right) , \end{aligned}$$

where $\displaystyle {\Omega (|\xi |_p)={\left\{ \begin{array}{ll} 1 &{}\hbox { if}\ |\xi |_p\le 1\\ 0 &{}\hbox { if}\ |\xi |_p> 1, \end{array}\right. }}$ is the characteristic function of $B_0$.

Theorem 1

The distribution $\mathcal {C}^\alpha _{d_1,d_2}(\xi )$ has a meromorphic continuation to all $\alpha \in \mathbb {C}$ with $\alpha d_j\ne 1+\dfrac{2\pi i}{\ln p}\mathbb {Z},\ j=1,2$, given by

$$\begin{aligned}&\langle \mathcal {C}^\alpha _{d_1,d_2}(\xi ),\varphi (\xi )\rangle \\&\quad =\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) (\varphi (\xi )-\varphi (0))d\xi \\&\qquad +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) \int \limits _{|\xi |_p>1} |\xi |_p^{\alpha d_1-1}\varphi (\xi )d\xi + \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}\varphi (0), \end{aligned}$$

and a simple pole at $\alpha d_1= 1+\dfrac{2\pi i}{\ln p}k,\ k\in \mathbb {Z}$, with the residue

$$\begin{aligned} \frac{p^{-1}-1}{d_1\ln p}\left( \Omega (|x|_p)+|x|_p^{\frac{2\pi k\sqrt{-1}}{\ln p }} (1-\Omega (|x|_p)\right) . \end{aligned}$$

Proof

We prove the first part, the second part is easily obtained from the Definition 1.

$$\begin{aligned}&\langle \mathcal {C}^\alpha _{d_1,d_2}(\xi ),\varphi (\xi )\rangle =\int \limits _{\mathbb {Q}_p}\mathcal {C}^\alpha _{d_1,d_2}(\xi )\varphi (\xi )d\xi \\&\quad =\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) \varphi (\xi ) d\xi \\&\qquad +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) \int \limits _{|\xi |_p>1} |\xi |^{\alpha d_1-1}\varphi (\xi )d\xi \\&\quad =\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) (\varphi (\xi )-\varphi (0)) d\xi \\&\qquad +\varphi (0)\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) d\xi \\&\qquad +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) \int \limits _{|\xi |_p>1} |\xi |^{\alpha d_1-1}d\xi \\&\quad =\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) (\varphi (\xi )-\varphi (0)) d\xi \\&\qquad +\varphi (0)\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}\dfrac{1-p^{-1}}{1-p^{-\alpha d_2-1}}\right) \\&\qquad +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) \int \limits _{|\xi |_p>1} |\xi |^{\alpha d_1-1}d\xi \\&\quad =\int \limits _{|\xi |_p\le 1}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right) (\varphi (\xi )-\varphi (0))d\xi \\&\qquad +\left( \dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}\right) \int \limits _{|\xi |_p>1} |\xi |_p^{\alpha d_1-1}\varphi (\xi )d\xi + \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}\varphi (0). \end{aligned}$$

$\square$

Theorem 2

As distributions, for $\alpha d_j \ne 1+\dfrac{2\pi i}{\ln p}{\mathbb {Z}}, \ j=1,2$ and $\alpha \ne \dfrac{2\pi i}{\ln p}{\mathbb {Z}}$ we have

$$\begin{aligned} {\mathcal {F}}[\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )]= \mathcal {A}^{\alpha }_{d_1,d_2}(x). \end{aligned}$$

Proof

See Theorem 1 and Corollary 1 in [4]. $\square$

Theorem 3

For ${{\,\mathrm{\text {Re}}\,}}(\alpha )>0$

$$\begin{aligned} \int \limits _{|\xi |_p>1}\mathcal {\mathcal {C}}^{-\alpha }_{d_1,d_2}(\xi ) d\xi =\dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}\int \limits _{|\xi |_p>1}|\xi |_p^{-\alpha d_1-1} d\xi =-\dfrac{1-p^{-1}}{1-p^{-1-\alpha d_1}}. \end{aligned}$$

Proof

Note that for $|\xi |_p>1$, $\mathcal {\mathcal {C}}^{-\alpha }_{d_1,d_2}(\xi )=\dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}|\xi |_p^{-\alpha d_1-1}$.

$$\begin{aligned} \dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}\int \limits _{|\xi |_p>1}|\xi |_p^{-\alpha d_1-1}&=\dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}\sum \limits _{k=1}^{\infty }\int \limits _{|\xi |_p=p^k}p^{-k(\alpha d_1+1)}d\xi \\&=\dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}\sum \limits _{k=1}^{\infty }p^{-k(\alpha d_1+1)}p^k(1-p^{-1})\\&=\dfrac{1-p^{\alpha d_1}}{1-p^{-1-\alpha d_1}}\dfrac{p^{-\alpha d_1}}{1-p^{-\alpha d_1}}(1-p^{-1})\\&=-\dfrac{1-p^{-1}}{1-p^{-1-\alpha d_1}}. \end{aligned}$$

$\square$

Theorem 4

For ${{\,\mathrm{\text {Re}}\,}}(\alpha )>0$ and $\varphi \in {{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$

• 1. $\displaystyle {\left\langle \mathcal {C}_{d_1,d_2}^\alpha (\xi ),\varphi (\xi )\right\rangle =\int \limits _{\mathbb {Q}_p}}\mathcal {C}_{d_1,d_2}^\alpha (\xi )\varphi (\xi )d\xi ,\quad \alpha d_j \ne 1+\frac{2\pi i}{\ln p}{\mathbb {Z}}, \ j=1,2$,

• 2. $\displaystyle {\left\langle \mathcal {C}_{d_1,d_2}^{-\alpha }(\xi ),\varphi (\xi )\right\rangle =\int \limits _{\mathbb {Q}_p}}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )(\varphi (\xi )-\varphi (0))d\xi$,

• 3. $\displaystyle {(\mathcal {C}_{d_1,d_2}^\alpha *\varphi )(x)=\int \limits _{\mathbb {Q}_p}}\mathcal {C}_{d_1,d_2}^\alpha (\xi )\varphi (\xi +x)d\xi ,\quad \alpha d_j \ne 1+\frac{2\pi i}{\ln p}{\mathbb {Z}}, \ j=1,2$,

• 4. $\displaystyle {(\mathcal {C}_{d_1,d_2}^{-\alpha }*\varphi )(x)=\int \limits _{\mathbb {Q}_p}}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )(\varphi (\xi +x)-\varphi (x))d\xi$.

Proof

1. follows from the Definition 1. 2. is a consequence of the Theorems 3 and 1. We recall that if $\varphi \in {{\,\mathrm{\textbf{S}}\,}}(\mathbb {Q}_p)$, then

$$\begin{aligned} (\mathcal {C}^\alpha _{d_1,d_2}*\varphi )(x)=\left< \mathcal {C}^\alpha _{d_1,d_2}\xi ), \varphi (x-\xi ) \right>, \end{aligned}$$

and since $\mathcal {C}^\alpha _{d_1,d_2}( -\xi )= \mathcal {C}^\alpha _{d_1,d_2}(\xi )$, we have

$$\begin{aligned} (\mathcal {C}^\alpha _{d_1,d_2}*\varphi )(x)=\left< \mathcal {C}^\alpha _{d_1,d_2}(\xi ),\varphi (x+\xi )\right>. \end{aligned}$$

Therefore 3. follows from 1. and 4. follows from 2. $\square$

There are some ways to define the operator. We choose the integral form, like in the last item, because we can express it as a convolution operator. It is also possible to define the operator as in Theorem 5 and to prove the equivalence with Definition 2.

Definition 2

For ${{\,\mathrm{\text {Re}}\,}}(\alpha )>0$, and $\varphi \in {{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})$

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)=\int \limits _{\mathbb {Q}_p}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )(\varphi (\xi +x)-\varphi (x))d\xi . \end{aligned}$$

In accordance with Theorem 4-(4) we have that

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)=(\mathcal {C}_{d_1,d_2}^{-\alpha }*\varphi )(x), \end{aligned}$$

since $\varphi$ has compact support, the indicated convolution actually exists. The last expression allows us to write $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$ as a pseudo-differential operator with symbol $\mathcal {A}^{\alpha }_{d_1,d_2}(\xi )$, as we desire.

Theorem 5

For $\alpha d_j \ne 1+\dfrac{2\pi i}{\ln p}{\mathbb {Z}}, \ j=1,2$, the operator $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$ is a pseudo-differential operator with symbol $\mathcal {A}^\alpha _{d_1,d_2}(\xi )$, i.e.,

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)=\mathcal {F}^{-1}\left[ \mathcal {A}^\alpha _{d_1,d_2}(\xi )\mathcal {F}(\varphi )(\xi )\right] (x). \end{aligned}$$

Proof

Since the operator can be expressed as a convolution we have

$$\begin{aligned} \mathcal {F}(\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )=\mathcal {F}(\mathcal {C}_{d_1,d_2}^{-\alpha }*\varphi ) =\mathcal {F}(\mathcal {C}_{d_1,d_2}^{-\alpha })\mathcal {F}(\varphi ) =\mathcal {A}_{d_1,d_2}^{\alpha }\mathcal {F}(\varphi ). \end{aligned}$$

$\square$

We already know that the space of test functions is not invariant under the action of some operators, including the functions $|\xi |_p^{\alpha }$, since it is not locally constant on ${\mathbb {Q}}_{p}$. For that reason it is convenient to consider the Lizorkin space, and define the operators over it.

The space

$$\begin{aligned} \Phi ({\mathbb {Q}}_{p})=\left\{ \phi \in {{\,\mathrm{\textbf{S}}\,}}({\mathbb {Q}}_{p})\,\ \int \limits _{{\mathbb {Q}}_{p}}\phi (x)dx=0 \right\} \end{aligned}$$

is called the p-adic Lizorkin space of test functions of the second kind. The most essential property of it is its invariance under the action of some operators, between them, the Vladimirov operator. Therefore, it can be considered the natural domain for some pseudo-differential operators. The following results can be found in [5].

Among other properties, we can prove that the Lizorkin space $\Phi ({\mathbb {Q}}_{p})$ is dense in $L^{\rho }({\mathbb {Q}}_{p})$ for $1<\rho < \infty$. This space plays a crucial role in the theory of pseudo-differential operators, and the theory of p-adic wavelets.

For two pseudo-differential operators A, B with symbols $\mathcal {A}(\xi ), \mathcal {B}(\xi )\in \mathcal {E}({\mathbb {Q}}_{p}-\{0\})$, respectively, the operator AB is well-defined as a distribution (over the p-adic Lizorkin space of the second kind) and it is represented by the formula

$$\begin{aligned} (AB)\phi ={\mathcal {F}}^{-1}[\mathcal {A}\mathcal {B}\ {\mathcal {F}}[\phi ]]. \end{aligned}$$

Also it is possible to define the inverse operator of A,  if $\mathcal {A}(\xi )\ne 0, \xi \in {\mathbb {Q}}_{p}-\{0\}$, as the distribution

$$\begin{aligned} A^{-1}\phi ={\mathcal {F}}^{-1}[\mathcal {A}^{-1} {\mathcal {F}}[\phi ]]. \end{aligned}$$

It is clear that the symbol $\mathcal {A}^{\alpha }_{d_1,d_2}(\xi )$ belongs to $\mathcal {E}({\mathbb {Q}}_{p}-\{0\})$. Therefore we can apply the results of Chapter 9 in [5] and conclude the following results.

Theorem 6

The Lizorkin space of the second kind is invariant under the operator $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$.

Proof

By applying Lemma 9.3.1 in [5] to our pseudodifferential operator $(D^{\alpha }_{d_1,d_2}\varphi )(x)={\mathcal {F}}^{-1}[\mathcal {A}^{\alpha }_{d_1,d_2}(\xi ){\mathcal {F}}[\varphi ](\xi )](x)$, whose symbol $\mathcal {A}^{\alpha }_{d_1,d_2}(\xi )\in \mathcal {E}({\mathbb {Q}}_{p}\setminus \{0\})$, we conclude that it is well-defined and the Lizorkin space $\Phi ({\mathbb {Q}}_{p})$ is invariant under it. $\square$

Theorem 7

The set of operators $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$ form an abelian group.

Proof

According to Proposition 9.3.2 in [5], the set of pseudodifferential operators ${\mathcal {F}}^{-1}[\mathcal {A}^{\alpha }_{d_1,d_2}(\xi ){\mathcal {F}}[\varphi ](\xi )]$ with symbols $\mathcal {A}^{\alpha }_{d_1,d_2}(\xi )\ne 0,$ $\xi \in {\mathbb {Q}}_{p}\setminus \{0\}$ forms an abelian group. $\square$

The natural definition for the inverse operator of $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$ is given by

Definition 3

For $\alpha >0$ and $\alpha d_i \ne 1$

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(x)&=\mathbf {\mathcal {D}}_{d_1,d_2}^{-\alpha }(x)-\mathbf {\mathcal {D}}_{d_1,d_2}^{-\alpha }(0)\\&=\int \limits _{\mathbb {Q}_p}\left[ \mathcal {C}_{d_1,d_2}^{\alpha }(\xi -x)-\mathcal {C}_{d_1,d_2}^{\alpha }(\xi )\right] \varphi (\xi )d\xi . \end{aligned}$$

4 The Cauchy problem for radial functions

In this section, we start by giving the necessary conditions for the existence of the inverse operator in order to solve the Cauchy problem

$$\begin{aligned}{} & {} D^{\alpha }_{d_1,d_2} \varphi + a(|x|_K)\varphi =f(|x|_p), \ x\in B_N, \end{aligned}$$

(8)

$$\begin{aligned}{} & {} \varphi (0)=0, \end{aligned}$$

(9)

where $\alpha >0, \alpha d_i\ne 1$ for $i=1,2$, a and f are continuous functions and $\varphi =\varphi (|x|_p)$ is a radial function.

The strategy for that case is to define the inverse operator of $D^{\alpha }_{d_1,d_2}$, namely $I^{\alpha }_{d_1,d_2}$, see [1]. If the solution has the form $\varphi =I^{\alpha }v$ the equation (8) transforms formally in an integral equation

$$\begin{aligned} v(|x|_p) + a(|x|_p) (I^{\alpha }v)(|x|_p)=f(|x|_p). \end{aligned}$$

(10)

The following result gives the explicit formula for the operator $\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha$ acting on a radial function $\varphi =\varphi (|x|_p)$ and the necessary conditions to have it.

Theorem 8

If a function $\varphi =\varphi (|x|_p)$ satisfies

$$\begin{aligned} \sum _{k=-\infty }^{m-1} |\varphi (p^k)|p^{k}<\infty \end{aligned}$$

(11)

and

$$\begin{aligned} \sum _{k=m+1}^{\infty }|\varphi (p^{k})|p^{-k\alpha d_1}<\infty , \end{aligned}$$

(12)

then for each $m\in {\mathbb {Z}}$ the expression in the right-hand side of the Definition 2 exists for $|x|_p=p^m$, depends only on $|x|_p$, and

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)&=(1-p^{-1})\left[ \mathcal {C}_{d_1,d_2}^{-\alpha }(p^m) \left( \sum _{l=-\infty }^{m-1} \varphi (p^l)p^{l}- \varphi (p^m) \frac{p^{n(m-1)}}{1-p^{-1}}\right) \right. \\&\quad \left. +\sum _{k=m+1}^{\infty }\mathcal {C}_{d_1,d_2}^{-\alpha }(p^k) \varphi (p^k)p^{k}-\varphi (p^m)\sum _{k=m+1}^{\infty } \mathcal {C}_{d_1,d_2}^{-\alpha }(p^k)p^{k} \right] . \\ \end{aligned}$$

Proof

Take $|x|_p=p^m$

$$\begin{aligned}&(\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x) =\int \limits _{{\mathbb {Q}}_{p}}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )[\varphi (|\xi +x|_p)- \varphi (|x|_p)] d\xi \\&\quad =\int \limits _{|\xi |_p < |x|_p}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )[\varphi (|\xi +x|_p)- \varphi (|x|_p)] d\xi \\&\qquad + \int \limits _{|\xi |_p \ge |x|_p}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )[\varphi (|\xi +x|_p)- \varphi (|x|_p)] d\xi . \end{aligned}$$

In the first integral, $|\xi + x|_p=|x|_p$ and the integral vanishes. Then

$$\begin{aligned} (\mathbf {\mathcal {D}}_{d_1,d_2}^\alpha \varphi )(x)&= \int \limits _{|\xi |_p = |x|_p}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )[\varphi (|\xi +x|_p)- \varphi (|x|_p)] d\xi \\&\quad +\int \limits _{|\xi |_p > |x|_p}\mathcal {C}_{d_1,d_2}^{-\alpha }(\xi )[\varphi (|\xi +x|_p)- \varphi (|x|_p)] d\xi :=T+S. \end{aligned}$$

Observe that if $|x|_p=p^m=|\xi |_p$, then $x=p^{-m}(x_0+x_1p+\cdots )$, $\xi =p^{-m}(\xi _0+\xi _1p+\cdots )$. If $x_0\ne \xi _0$ then $|x-\xi |_p=p^{m}=|x|_p$ and the corresponding integral vanishes. Therefore it is enough to consider the integral over the set $\{|x|_p=|\xi |_p; x_0=\xi _0\}$, or $\{ |x-\xi |_p=p^l \text { for }l<m\}$.

The first integral equals

$$\begin{aligned} T&=\int \limits _{|\xi |_p=p^m} \mathcal {C}_{d_1,d_2}^{-\alpha }(\xi ) (\varphi (|\xi +x|_p) - \varphi (|x|_p)) d\xi \\&=\mathcal {C}_{d_1,d_2}^{-\alpha }(p^m) \sum _{l<m} \int \limits _{|\xi +x|_p=p^l} (\varphi (p^l)-\varphi (p^m))d \xi \\&=\mathcal {C}_{d_1,d_2}^{-\alpha }(p^m) \sum _{l=-\infty }^{m-1} (\varphi (p^l)-\varphi (p^m)) p^{l} (1-p^{-1})\\&=\mathcal {C}_{d_1,d_2}^{-\alpha }(p^m) (1-p^{-1}) \left[ \sum _{l=-\infty }^{m-1} \varphi (p^l)p^{l}-\varphi (p^m)\frac{p^{(m-1)}}{1-p^{-1}} \right] . \end{aligned}$$

For the second integral we take $|\xi |_p=p^k$ with $k>m$ and then $|\xi +x|_p=p^k$.

$$\begin{aligned} S&=\int \limits _{|\xi |_p>|x|_p} \mathcal {C}_{d_1,d_2}^{-\alpha }(\xi ) (\varphi (|\xi +x|_p) - \varphi (|x|_p)) d\xi \\&=\sum _{k>m} \int \limits _{|\xi |_p=p^k} \mathcal {C}_{d_1,d_2}^{-\alpha }(p^k) (\varphi (p^k)- \varphi (p^m)) d\xi \\&=\sum _{k=m+1}^{\infty } \mathcal {C}_{d_1,d_2}^{-\alpha }(p^k) (\varphi (p^k)- \varphi (p^m)) p^{k}(1-p^{-1})\\&=(1-p^{-1})\left( \sum _{k=m+1}^{\infty } \mathcal {C}_{d_1,d_2}^{-\alpha }(p^k) \varphi (p^k)p^{k}- \varphi (p^m)\sum _{k=m+1}^{\infty }\mathcal {C}_{d_1,d_2}^{-\alpha }(p^k) p^{k}\right) . \end{aligned}$$

Since the sums in (11) and (12) are finite, the sum $T+S$ converges. $\square$

The expanded form for the inverse operator is given in the next result.

Theorem 9

If the radial function $\varphi =\varphi (|x|_p)$ satisfies

$$\begin{aligned} \sum _{k=-\infty }^{m-1} |\varphi (p^k)| p^{k} <\infty , \end{aligned}$$

(13)

and

$$\begin{aligned} \sum _{k=-\infty }^{m-1}|\varphi (p^k)| p^{k\alpha d_1} <\infty , \end{aligned}$$

(14)

then for $|x|_p=p^m$ we have the following expression for the operator $I^{\alpha }_{d_1,d_2}\varphi$

$$\begin{aligned}&(I^{\alpha }_{d_1,d_2}\varphi )(|x|_p) \nonumber \\&\quad =\int \limits _{|\xi |_p< |x|_p}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|x|_p^{\alpha d_2-1}\right. \nonumber \\&\qquad \left. -\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}|x|_p^{\alpha d_1-1}\right) \Omega (|x|_p)\varphi (\xi )d\xi \nonumber \\&\qquad -\int \limits _{|\xi |_p< |x|_p}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right. \nonumber \\&\qquad \left. -\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}|\xi |_p^{\alpha d_1-1} \right) \Omega (|\xi |_p)\varphi (\xi )d\xi \nonumber \\&\qquad +\varphi (|x|_p)\left[ p^{-\alpha d_2}|x|_p^{\alpha d_2}- p^{-\alpha d_1}|x|_p^{\alpha d_1}\right] \Omega (|x|_p)\nonumber \\&\qquad +\varphi (|x|_p)p^{-\alpha d_1}|x|_p^{\alpha d_1} +\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}} \int \limits _{|\xi |_p< |x|_p}\left[ |x|_p^{\alpha d_1-1}-|\xi |_p^{\alpha d_1-1}\right] \varphi (\xi )d\xi . \end{aligned}$$

(15)

Proof

We take $|x|_p=p^m$ and $|\xi |_p=p^k$. If $|\xi |_p>|x|_p$ then $|\xi -x|_p=|\xi |_p$ and the integral vanishes. Therefore

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(|x|_p)&=\int \limits _{\mathbb {Q}_p}\left[ \mathcal {C}_{d_1,d_2}^{\alpha }(\xi -x)-\mathcal {C}_{d_1,d_2}^{\alpha }(\xi )\right] \varphi (\xi )d\xi \\&=\int \limits _{|\xi |_p\le |x|_p} [\mathcal {C}_{d_1,d_2}^{\alpha }(\xi -x)- \mathcal {C}_{d_1,d_2}^{\alpha }(\xi )] \varphi (\xi ) d \xi . \end{aligned}$$

We split this integral in two parts, in the first one we take $|\xi |_p<|x|_p$, for the second one $|\xi |_p=|x|_p$. In the first case we obtain

$$\begin{aligned}&\int \limits _{|\xi |_p< |x|_p} [\mathcal {C}_{d_1,d_2}^{\alpha }(\xi -x)- \mathcal {C}_{d_1,d_2}^{\alpha }(\xi )] \varphi (\xi ) d \xi \\&\quad =\int \limits _{|\xi |_p< |x|_p}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|x|_p^{\alpha d_2-1}\right. \\&\qquad \left. -\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}|x|_p^{\alpha d_1-1}\right) \Omega (|x|_p)\varphi (\xi )d\xi \\&\qquad -\int \limits _{|\xi |_p< |x|_p}\left( \dfrac{1-p^{-1}}{1-p^{\alpha d_1-1}}-\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}}+\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}|\xi |_p^{\alpha d_2-1}\right. \\&\qquad \left. -\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}|\xi |_p^{\alpha d_1-1} \right) \Omega (|\xi |_p)\varphi (\xi )d\xi \\&\qquad +\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}} \int \limits _{|\xi |_p< |x|_p}\left[ |x|_p^{\alpha d_1-1}-|\xi |_p^{\alpha d_1-1}\right] \varphi (\xi )d\xi . \end{aligned}$$

The integral converges thanks to (13) and (14). For the second integral we suppose $|\xi |_p=|x|_p$ and $|\xi -x|_p=p^s$ with $s<m$, then

$$\begin{aligned} \int \limits _{|\xi |_p= |x|_p}&[\mathcal {C}_{d_1,d_2}^{\alpha }(\xi -x)- \mathcal {C}_{d_1,d_2}^{\alpha }(\xi )] \varphi (\xi ) d \xi \\&=\sum _{s<m} \int \limits _{|\xi -x|_p=p^s} [\mathcal {C}_{d_1,d_2}^{\alpha }(p^s)-\mathcal {C}_{d_1,d_2}^{\alpha }(p^m)] \varphi (p^m)d\xi \\&=\sum _{s<m}[\mathcal {C}_{d_1,d_2}^{\alpha }(p^s)- \mathcal {C}_{d_1,d_2}^{\alpha }(p^m)] \varphi (p^m) p^{s}(1-p^{-1})\\&=(1-p^{-1}) \varphi (p^m) \left[ \sum _{s=-\infty }^{m-1}\mathcal {C}_{d_1,d_2}^{\alpha }(p^s)p^{s} -\mathcal {C}_{d_1,d_2}^{\alpha }(p^m) \sum _{s<m} p^{s}\right] \\&=(1-p^{-1}) \varphi (p^m) \left[ \sum _{s=-\infty }^{m-1}\mathcal {C}_{d_1,d_2}^{\alpha }(p^s)p^{s} -\mathcal {C}_{d_1,d_2}^{\alpha }(p^m)\frac{p^{(m-1)}}{1-p^{-1}}\right] \\&=\varphi (|x|_p)p^{-\alpha d_1}|x|_p^{\alpha d_1}+\varphi (|x|_p)\left[ p^{-\alpha d_2}|x|_p^{\alpha d_2}- p^{-\alpha d_1}|x|_p^{\alpha d_1}\right] \Omega (|x|_p). \end{aligned}$$

Combining these two expressions we obtain the desired result. $\square$

In order to make the notation easier, we write $A=\dfrac{1-p^{-1}}{1-p^{\alpha d_1}-1}$, $B=\dfrac{1-p^{-1}}{1-p^{\alpha d_2-1}},$ $E=\dfrac{1-p^{-\alpha d_2}}{1-p^{\alpha d_2-1}}$ and $F=\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}}$. Therefore, we can re-write the expression (15) as

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(|x|_p)&=\int \limits _{|\xi |_p< |x|_p}\left( A-B+E|x|_p^{\alpha d_2-1}-F|x|_p^{\alpha d_1-1}\right) \Omega (|x|_p)\varphi (\xi )d\xi \nonumber \\&\quad -\int \limits _{|\xi |_p< |x|_p}\left( A-B+E|\xi |_p^{\alpha d_2-1}-F|\xi |_p^{\alpha d_1-1} \right) \Omega (|\xi |_p)\varphi (\xi )d\xi \nonumber \\&\quad +\varphi (|x|_p)\left[ p^{-\alpha d_2}|x|_p^{\alpha d_2}- p^{-\alpha d_1}|x|_p^{\alpha d_1}\right] \Omega (|x|_p)\nonumber \\&\quad +\varphi (|x|_p)p^{-\alpha d_1}|x|_p^{\alpha d_1} +F \int \limits _{|\xi |_p< |x|_p}\left[ |x|_p^{\alpha d_1-1}-|\xi |_p^{\alpha d_1-1}\right] \varphi (\xi )d\xi . \end{aligned}$$

(16)

It follows from the last expression that, if $\displaystyle {|\varphi (|x|_p)| \le K |x|_p^{-\epsilon }}$ for $|x|_p> 1$, then $|I^{\alpha }_{d_1, d_2}\varphi (x)| \le K_1 |x|_p^{\alpha d_1-\epsilon }$ for $|x|_p> 1$, where K and $K_1$ are constants. It means that $I^{\alpha }_{d_1, d_2}\varphi$ is continuous.

Corollary 10

If the radial function $\varphi =\varphi (|x|_p)$ satisfies

$$\begin{aligned} \sum _{k=-\infty }^{m-1}\max (p^k, p^{k\alpha d_1}) |\varphi (p^k)| <\infty , \end{aligned}$$

(17)

and

$$\begin{aligned} \sum _{k=m+1}^{\infty }|\varphi (p^k)| p^{-k\alpha d_1} <\infty , \end{aligned}$$

(18)

then $D^{\alpha }_{d_1, d_2}I^{\alpha }_{d_1, d_2} \varphi (|x|_p) = \varphi (|x|_p)$ for $x\ne 0$ and $\alpha >0$, $\alpha d_i\ne 1$ for $i=1,2$.

Since we are looking for a solution of (8) of the form $\varphi =I^{\alpha }_{d_1,d_2}v$, the equation (10) becomes

$$\begin{aligned} D^{\alpha }_{d_1,d_2} I^{\alpha }_{d_1,d_2}v(|x|_p) + a(|x|_p) I^{\alpha } v(|x|_p)&= f(|x|_p)\\ v(|x|_p) + a(|x|_p) I^{\alpha } v(|x|_p)&= f(|x|_p) \end{aligned}$$

with $v(0)=f(0)$.

Since the inverse is given in terms of the characteristic function of the unit interval, we first analyze the case $|x|_p\le 1$:

$$\begin{aligned}&f(|x|_p)=v(|x|_p)+a(|x|_p) I^{\alpha } v(|x|_p)\\&\quad ={v(|x|_p)+a(|x|_p) \left\{ E\int \limits _{|\xi |_p< |x|_p} \left[ |x|_p^{\alpha d_2-1} - |\xi |_p^{\alpha d_2-1}\right] v(|\xi |_p)d\xi + v(|x|_p) p^{-\alpha d_2}|x|_p^{\alpha d_2} \right\} } \\&\quad =v(|x|_p) \left\{ 1+a(|x|_p) p^{-\alpha d_2}|x|_p^{\alpha d_2} \right\} +a(|x|_p) E \int \limits _{|\xi |_p < |x|_p} \left[ |x|_p^{\alpha d_2-1} - |\xi |_p^{\alpha d_2-1}\right] v(|\xi |_p)d\xi . \end{aligned}$$

Now, if

$$\begin{aligned} F(|x|_p)&=v(|x|_p) + E\int \limits _{|\xi |_p < |x|_p}a(|x|_p) \left\{ 1+a(|x|_p) p^{-\alpha d_2}|x|_p^{\alpha d_2} \right\} ^{-1} \\&\quad \left[ |x|_p^{\alpha d_2-1} - |\xi |_p^{\alpha d_2-1}\right] v(|\xi |_p)d\xi , \end{aligned}$$

then we can write

$$\begin{aligned} v(|x|_p) + \int \limits _{|\xi |_p < |x|_p} k_{\alpha }(x,\xi )v(|\xi |_p)d\xi =F(|x|_p), \end{aligned}$$

(19)

where

$$\begin{aligned} F(|x|_p) = \left\{ 1+a(|x|_p) p^{-\alpha d_2}|x|_p^{\alpha d_2} \right\} ^{-1} f(|x|_p) \end{aligned}$$

and

$$\begin{aligned} k_{\alpha }(x,\xi ) = E a(|x|_p) \left\{ 1+a(|x|_p) p^{-\alpha d_2}|x|_p^{\alpha d_2} \right\} ^{-1} \left[ |x|_p^{\alpha d_2-1} - |\xi |_p^{\alpha d_2-1}\right] . \end{aligned}$$

Next, as in [1], we define the integral operator $\displaystyle {(\mathcal {K} v)(x)=\int \limits _{|\xi |_p < |x|_p} k_{\alpha }(x,\xi )v(|\xi |_p)d\xi }$, which is a compact operator, with no nonzero eigenvectors. By Fredholm’s alternative we conclude that (19) has a unique continuous solution on $B_N$, see [9].

We do not need to consider the case $|x|_p>1$ since in the proof of compactness of the operator we make $|x|_p\rightarrow 0$.

We have proved the following result.

Theorem 11

For $\alpha >0$ the integral equation (19)

$$\begin{aligned} v(|x|_p) + \int \limits _{|\xi |_p < |x|_p} k_{\alpha }(x,\xi )v(|\xi |_p)d\xi =F(|x|_p), \end{aligned}$$

has a unique continuous solution on $B_N$. Therefore the Cauchy problem (8)–(9)

$$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }_{d_1,d_2} \varphi + a(|x|_K)\varphi =f(|x|_p), \ x\in B_N,\\ u(0)=0, \end{array}\right. } \end{aligned}$$

where a and f are continuous functions and $\varphi =\varphi (|x|_p)$ is a radial function, has a unique solution of the form $\varphi (|x|_p)= I^{\alpha }_{d_1,d_2}v(|x|_p)$, for $x\in B_N$.

5 The nonlinear Cauchy problem

Following the results given by Antoniouk et al. [3], we study the degenerate nonlinear Cauchy problem

$$\begin{aligned} |t|_p^{\gamma } (D^{\alpha }_{d_1,d_2}\varphi )(|t|_p) = f(|t|_p, \varphi (|t|_p)), \ 0\ne t\in {\mathbb {Q}}_{p}, \end{aligned}$$

(20)

for $\alpha >0$ and $\gamma >0$, with the initial condition

$$\begin{aligned} \varphi (0)=\varphi _0. \end{aligned}$$

(21)

Since the operator $D^{\alpha }_{d_1, d_2}$ has an inverse $I^{\alpha }_{d_1,d_2}$, we associate the following integral equation

$$\begin{aligned} \varphi (|t|_p) = \varphi _0 + I^{\alpha }_{d_1,d_2} \left[ |\cdot |_p^{-\gamma } f(|\cdot |_p, \varphi (|\cdot |_p))\right] (|t|_p). \end{aligned}$$

(22)

A solution of the equation (22), if it exists, is called a mild solution to the Cauchy problem (20)–(21).

Lemma 12

Let

$$\begin{aligned} I_{\alpha d_2, \sigma }=\int \limits _{|y|_p<|t|_p} \left| |t|_p^{\alpha d_2-1} - |y|_p^{\alpha d_2-1} \right| |y|_p^{\alpha d_2 \sigma }dy, \end{aligned}$$

then, for any $\sigma > \max \{ -\frac{1}{\alpha d_2}, -1\}$ (not necessarily integer):

$$\begin{aligned} I_{\alpha d_2, \sigma } = d_{\alpha d_2, \sigma } |t|_p^{\alpha d_2 (\sigma +1)}, \end{aligned}$$

with $d_{\alpha d_2, \sigma }$ independent of t. Moreover, for any $\epsilon >0$ there exists a constant $A>0$, independent of $\sigma$, such that

$$\begin{aligned} 0< d_{\alpha d_2, \sigma } \le A p^{-\alpha d_2 \sigma }, \text { for any } \sigma \ge \max \left( -\frac{1}{\alpha d_2}, -1\right) +\epsilon . \end{aligned}$$

Proof

The proof is identical to the one given for Lemma 4 in [3]. $\square$

Proposition 13

Let $\gamma < \min \{ 1, \alpha d_1\}$ and suppose that for the radial function $\varphi (|t|_p)$ we have

$$\begin{aligned} |\varphi (|t|_p)| \le \mu |t|_p^{-\gamma }, \end{aligned}$$

(23)

with $\mu >0$. Then the function $I^{\alpha }_{d_1, d_2}\varphi$ exists, it is a radial function, and there exists a constant C such that

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|\le \mu C |t|_p^{\alpha d_2 - \gamma }. \end{aligned}$$

(24)

Proof

We first verify the condition (17) in order to prove that $I^{\alpha }_{d_1,d_2}\varphi$ exists. We have

$$\begin{aligned} \sum _{k=-\infty }^{m-1} \max (p^k, p^{k\alpha d_1}) |\varphi (p^k)|&\le \mu \sum _{k=-\infty }^{m-1} \max (p^k, p^{k\alpha d_1}) p^{-k\gamma }\\&\le \mu \sum _{k=-\infty }^{m-1} \max (p^{k(1-\gamma )}, p^{k(\alpha d_1-\gamma )})<\infty , \end{aligned}$$

since $\gamma <\min (1, \alpha d_1)$.

In order to bound the expression $I^{\alpha }_{d_1,d_2}\varphi$, we first suppose $|t|_p\le 1$, then

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)&= \int \limits _{|\xi |_p< |t|_p}(A-B+E|t|_p^{\alpha d_2-1} -F|t|_p^{\alpha d_1-1}) \varphi (|\xi |_p) d\xi \\&\quad - \int \limits _{|\xi |_p< |t|_p}(A-B+E|\xi |_p^{\alpha d_2-1} -F|\xi |_p^{\alpha d_1-1}) \varphi (|\xi |_p) d\xi \\&\quad + \varphi (|t|_p) \left[ p^{-\alpha d_2} |t|_p^{\alpha d_2} - p^{-\alpha d_1}|t|_p^{\alpha d_1} \right] \\&\quad +\varphi (|t|_p) p^{-\alpha d_1} |t|_p^{\alpha d_1} +\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}} \int \limits _{|\xi |_p< |t|_p}\left[ |t|_p^{\alpha d_1-1}-|\xi |_p^{\alpha d_1-1}\right] \varphi (\xi )d\xi \\&=\int \limits _{|\xi |_p < |t|_p}(E(|t|_p^{\alpha d_2-1}-|\xi |_p^{\alpha d_2-1})\varphi (|\xi |_p)d\xi + \varphi (|t|_p) p^{-\alpha d_2} |t|_p^{\alpha d_2}. \end{aligned}$$

Therefore

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)| \le E \mu \int \limits _{|\xi |_p < |t|_p}\left| |t|_p^{\alpha d_2-1}-|\xi |_p^{\alpha d_2-1}\right| |\xi |_p^{-\gamma }d\xi + \mu |t|_p^{-\gamma } p^{-\alpha d_2} |t|_p^{\alpha d_2}. \end{aligned}$$

then we apply the Lemma 12 in order to bound the three integrals, and we obtain

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|&\le E \mu d_{\alpha d_2, -\frac{\gamma }{\alpha d_2}} |t|_p^{\alpha d_2-\gamma } + \mu p^{-\alpha d_2}|t|_p^{\alpha d_2-\gamma } \\&\le \mu C_1 |t|_p^{\alpha d_2-\gamma }, \end{aligned}$$

where $C_1= Ed_{\alpha d_2, -\frac{\gamma }{\alpha d_2}}+p^{-\alpha d_2}$.

For the case $|t|_p>1$ the expression (16) becomes

$$\begin{aligned}&(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p) = -\int \limits _{|\xi |_p< |t|_p} \left( A-B +E |\xi |_p^{\alpha d_2-1} -F |\xi |_p^{\alpha d_1-1} \right) \Omega (|\xi |_p)\varphi (|\xi |_p)d\xi \\&\quad +\varphi (|t|_p) p^{-\alpha d_1} |t|_p^{\alpha d_1} +\dfrac{1-p^{-\alpha d_1}}{1-p^{\alpha d_1-1}} \int \limits _{|\xi |_p< |t|_p}\left[ |t|^{\alpha d_1-1}-|\xi |^{\alpha d_1-1}\right] \varphi (|\xi |_p)d\xi . \end{aligned}$$

Then

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|&\le \int \limits _{|\xi |_p \le 1} \left| A+B +E |\xi |_p^{\alpha d_2-1} +F |\xi |_p^{\alpha d_1-1} \right| \mu |\xi |_p^{-\gamma } d\xi \\&\quad +\mu p^{-\alpha d_1} |t|_p^{\alpha d_1-\gamma } +F \mu d_{\alpha d_1, -\frac{\gamma }{\alpha d_1}} |t|_p^{\alpha d_1-\gamma }\\&\le \mu C_2 + \mu p^{-\alpha d_1}|t|_p^{\alpha d_1-\gamma } +F d_{\alpha d_1, -\frac{\gamma }{\alpha d_1}} |t|_p^{\alpha d_1-\gamma }\\&\le \mu C_2 |t|_p^{\alpha d_1-\gamma }+ \mu p^{-\alpha d_1}|t|_p^{\alpha d_1-\gamma } +F d_{\alpha d_1, -\frac{\gamma }{\alpha d_1}} |t|_p^{\alpha d_1-\gamma }\\&\le \mu C_3|t|_p^{\alpha d_1-\gamma } \le \mu C_3|t|_p^{\alpha d_2-\gamma }, \end{aligned}$$

where

$$\begin{aligned} C_2&=\int \limits _{|\xi |_p < 1} \left| A+B +E |\xi |_p^{\alpha d_2-1} +F |\xi |_p^{\alpha d_1-1} \right| |\xi |_p^{-\gamma } d\xi \\&=(1-p^{-1})\left[ \frac{A+B}{1-p^{\gamma -1}} +\frac{E}{1-p^{\gamma -\alpha d_2}} + \frac{F}{1-p^{\gamma -\alpha d_2}} \right] . \end{aligned}$$

And $C_3=C_2+p^{-\alpha d_1}+ F d_{\alpha d_1, -\frac{\gamma }{\alpha d_1}}$. By combining these two expressions, with $C=\max (C_1, C_3)$ we obtain the result. $\square$

Proposition 14

Let $n\in {\mathbb {N}}$, $\gamma < \min (1, \alpha d_1)$ and let $\varphi (|t|_p)$ be a radial function such that for all $0\ne t\in {\mathbb {Q}}_{p}$

$$\begin{aligned} |\varphi (|t|_p)| \le \mu |t|_p^{n\alpha d_2 - (n+1)\gamma }. \end{aligned}$$

(25)

Then $I^{\alpha }_{d_1,d_2}\varphi$ exists, is a radial function, and there exists a constant $C_n$ such that

$$\begin{aligned} |(I^{\alpha }_{d_1, d_2}\varphi )(|t|_p)| \le C_n \mu |t|_p^{(n+1)(\alpha d_2-\gamma )}. \end{aligned}$$

(26)

Proof

We first verify the condition (17).

$$\begin{aligned} \sum _{k=-\infty }^{m-1} \max (p^k, p^{k\alpha d_1})|\varphi (p^k)| \le \mu \sum _{k=-\infty }^{m-1} \max (p^k, p^{k\alpha d_1})p^{k(n\alpha d_1-(n+1)\gamma )}. \end{aligned}$$

If the maximum is $p^k$ the right side of the previous line equals

$$\begin{aligned} \mu \sum _{k=-\infty }^{m-1} p^{k(1+n\alpha d_1-(n+1)\gamma )}&=\mu p^{(m-1)[n(\alpha d_1-\gamma )+(1-\gamma )]} \sum _{k=0}^{\infty } p^{-k[n(\alpha d_1-\gamma )+(1-\gamma )]}\\&=\mu p^{(m-1)[n(\alpha d_1-\gamma )+(1-\gamma )]} \frac{1}{1-p^{-[n(\alpha d_1-\gamma )+(1-\gamma )]}}<\infty , \end{aligned}$$

which converges since $\gamma < \min (1, \alpha d_1)$.

If the maximum is $p^{k\alpha d_1}$ then we have

$$\begin{aligned} \mu \sum _{k=-\infty }^{m-1} p^{-k(n+1)(\alpha d_1-\gamma )}&=p^{(m-1)(n+1)(\alpha d_1-\gamma )} \sum _{k=0}^{\infty } p^{-k(n+1)(\alpha d_1-\gamma )}\\&=p^{(m-1)(n+1)(\alpha d_1-\gamma )} \frac{1}{1-p^{-(n+1)(\alpha d_1-\gamma )}}, \end{aligned}$$

which converges since $\gamma <\alpha d_1$.

With this condition we conclude that $I^{\alpha }_{d_1,d_2}\varphi$ exists. As in the proof in the previous proposition we consider two cases.

First, if $|t|_p\le 1$ then

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(|t|_p) =\int \limits _{|\xi |_p < |t|_p}(E(|t|_p^{\alpha d_2-1}-|\xi |_p^{\alpha d_2-1})\varphi (|\xi |_p)d\xi + \varphi (|t|_p) p^{-\alpha d_2} |t|_p^{\alpha d_2}. \end{aligned}$$

Therefore

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|&\le E \mu \int \limits _{|\xi |_p< |t|_p}\left| |t|_p^{\alpha d_2-1}-|\xi |_p^{\alpha d_2-1}\right| |\xi |_p^{n\alpha d_2-(n+1)\gamma }d\xi \\&\quad + \mu |t|_p^{n\alpha d_2-(n+1)\gamma } p^{-\alpha d_2} |t|_p^{\alpha d_2} \\&\le E \mu \int \limits _{|\xi |_p < |t|_p}\left| |t|_p^{\alpha d_2-1}-|\xi |_p^{\alpha d_2-1}\right| |\xi |_p^{\alpha d_2\left( n-\frac{(n+1)\gamma }{\alpha d_2}\right) }d\xi \\&\quad + \mu p^{-\alpha d_2} |t|_p^{(n+1)(\alpha d_2-\gamma )}. \end{aligned}$$

Then we apply Lemma 12,

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|&\le E \mu d_{\alpha d_2, n-\frac{(n+1)\gamma }{\alpha d_2}} |t|_p^{(n+1)(\alpha d_2-\gamma )} +\mu p^{-\alpha d_2} |t|_p^{(n+1)(\alpha d_2-\gamma )}\\&\le \mu C |t|_p^{(n+1)(\alpha d_2-\gamma )}, \end{aligned}$$

where

$$\begin{aligned} C= E d_{\alpha d_2, n-\frac{(n+1)\gamma }{\alpha d_2}} +p^{-\alpha d_2}. \end{aligned}$$

For the case $|t|_p>1$ the expression (16) becomes

$$\begin{aligned} (I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)&= -\int \limits _{|\xi |_p< |t|_p} \left( A-B +E |\xi |_p^{\alpha d_2-1} -F |\xi |_p^{\alpha d_1-1} \right) \Omega (|\xi |_p)\varphi (|\xi |_p)d\xi \\&\quad +\varphi (|t|_p) p^{-\alpha d_1} |t|_p^{\alpha d_1} +F\int \limits _{|\xi |_p< |t|_p}\left[ |t|^{\alpha d_1-1}-|\xi |^{\alpha d_1-1}\right] \varphi (\xi )d\xi . \end{aligned}$$

Then

$$\begin{aligned} |(I^{\alpha }_{d_1,d_2}\varphi )(|t|_p)|&\le \int \limits _{|\xi |_p \le 1} \left| A+B +E |\xi |_p^{\alpha d_2-1} +F |\xi |_p^{\alpha d_1-1} \right| \mu |\xi |_p^{n\alpha d_2-(n+1)\gamma } d\xi \\&\quad +\mu p^{-\alpha d_1} |t|_p^{(n+1)(\alpha d_2-\gamma )} +F \mu d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}} |t|_p^{(n+1)(\alpha d_1-\gamma )}\\&\le \mu C_1 + \mu [p^{-\alpha d_1}+F d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}}]|t|_p^{(n+1)(\alpha d_2-\gamma )}\\&\le \mu C |t|_p^{(n+1)(\alpha d_2-\gamma )}, \end{aligned}$$

where

$$\begin{aligned} C_1&= \int \limits _{|\xi |_p \le 1} \left| A+B +E |\xi |_p^{\alpha d_2-1} +F |\xi |_p^{\alpha d_1-1} \right| |\xi |_p^{n\alpha d_2-(n+1)\gamma } d\xi \\&\le (A+B) \int \limits _{|\xi |_p\le 1} |\xi |_p^{n\alpha d_2 -(n+1)\gamma } d\xi + (E+F) \int \limits _{|\xi |_p\le 1} |\xi |_p^{\alpha d_2(n+1)-(n+1)\gamma -1} d\xi \\&\le (1-p^{-1}) \left[ \frac{A+B}{1-p^{-n\alpha d_2 + (n+1)\gamma -1}} +\frac{E+F}{1-p^{(n+1)(\gamma -\alpha d_2)}}\right] , \end{aligned}$$

and $C=C_1+p^{-\alpha d_1} + F d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}}$.

In both cases, we obtain the result. In summary

$$\begin{aligned} |(I^{\alpha }_{d_1, d_2}\varphi )(|t|_p)| \le C_n \mu |t|_p^{(n+1)(\alpha d_2-\gamma )}, \end{aligned}$$

where $C_n=$

$$\begin{aligned} {\left\{ \begin{array}{ll} E d_{\alpha d_2, n-\frac{(n+1)\gamma }{\alpha d_2}} + p^{-\alpha d_2}, &{} |t|_p\le 1,\\ (1-p^{-1}) \left[ \frac{A+B}{1-p^{-n\alpha d_2 + (n+1)\gamma -1}} +\frac{E+F}{1-p^{(n+1)(\gamma -\alpha d_2)}}\right] +p^{-\alpha d_1} + F d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}}, &{} |t|_p>1. \end{array}\right. } \end{aligned}$$

$\square$

These values $C_n$ can be bounded in both cases, in fact, for $|t|_p\le 1$ we have

$$\begin{aligned} C_n= E d_{\alpha d_2, n-\frac{(n+1)\gamma }{\alpha d_2}} +p^{-\alpha d_2} \le E A_1 p^{(n+1)\gamma - n\alpha d_2}+ p^{-\alpha d_2}:=C, \end{aligned}$$

here $A_1$ is the constant associated to $d_{\alpha d_2, n-\frac{(n+1)\gamma }{\alpha d_2}}$, and $A_2$ is associated to $d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}}$.

For $|t|_p>1$ we have

$$\begin{aligned} C_n&=(1-p^{-1}) \left[ \frac{A+B}{1-p^{-n\alpha d_2 + (n+1)\gamma -1}} +\frac{E+F}{1-p^{(n+1)(\gamma -\alpha d_2)}}\right] +p^{-\alpha d_1} + F d_{\alpha d_1, n-\frac{(n+1)\gamma }{\alpha d_1}}\\&\le (1-p^{-1}) \left[ \frac{A+B}{1-p^{\gamma -1}} +\frac{E+F}{1-p^{\gamma -\alpha d_2}}\right] +p^{-\alpha d_1} + F A_2 p^{(n+1)\gamma -n\alpha d_1}\\&\le (1-p^{-1}) \left[ \frac{A+B}{1-p^{\gamma -1}} +\frac{E+F}{1-p^{\gamma -\alpha d_2}}\right] +p^{-\alpha d_1} + F A_2 p^{\gamma }:=C. \end{aligned}$$

Proposition 15

Let $\gamma < \min (1, \alpha d_1)$. Then the sequence of constants $\{ C_n: n\ge 1\}$ defined in Proposition 14 is bounded.

Proof

$$\begin{aligned} C_n \le C:= {\left\{ \begin{array}{ll} E A_1 p^{(n+1)\gamma - n\alpha d_2}+ p^{-\alpha d_2} &{} \text { for } |t|_p\le 1\\ (1-p^{-1}) \left[ \frac{A+B}{1-p^{\gamma -1}} +\frac{E+F}{1-p^{\gamma -\alpha d_2}}\right] +p^{-\alpha d_1} + F A_2 p^{\gamma } &{} \text { for } |t|_p>1. \end{array}\right. } \end{aligned}$$

$\square$

Following the ideas given in [3], we denote

$$\begin{aligned} {\tilde{f}}(|t|_p, \varphi (|t|_p)) = |t|_p^{-\gamma } f(|t|_p, \varphi (|t|_p)), \ \ 0\ne t\in {\mathbb {Q}}_{p}. \end{aligned}$$

Remember that $f:p^{{\mathbb {Z}}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, and suppose that

$$\begin{aligned} |f(|t|_p, x)|\le M, \end{aligned}$$

(27)

where M is a constant independent of t, x, then

$$\begin{aligned} |\tilde{f}(|t|_p, x)| = |t|_p^{-\gamma } f(|t|_p, x) \le M |t|_p^{-\gamma } \text { for all }t\in {\mathbb {Q}}_{p}, x\in {\mathbb {R}}. \end{aligned}$$

Then, by applying Proposition 13

$$\begin{aligned} |(I^{\alpha }_{d_1, d_2} \tilde{f})(x)|&\le C M |t|_p^{\alpha d_2-\gamma }. \end{aligned}$$

(28)

We also need to impose the condition

$$\begin{aligned} |f(|t|_p, x)- f(|t|_p, y)| \le H|x-y|, \end{aligned}$$

(29)

where H is a constant independent of t, x, y, then

$$\begin{aligned} |\tilde{f}(|t|_p, x)- \tilde{f}(|t|_p, y)| =|t|_p^{-\gamma } |f(|t|_p, x)- f(|t|_p, y)| \le H|x-y| |t|_p^{-\gamma } \end{aligned}$$

(30)

for all $t\in {\mathbb {Q}}_{p}, x,y\in {\mathbb {R}}$.

Then the problem (20) can be written as

$$\begin{aligned} (D^{\alpha }_{d_1,d_2}\varphi )(|t|_p) = |t|_p^{-\gamma } f(|t|_p, \varphi (|t|_p))=\tilde{f}(|t|_p, \varphi (|t|_p)), \ 0\ne t\in {\mathbb {Q}}_{p}. \end{aligned}$$

(31)

With conditions (27) and (29), we can prove the existence and uniqueness of local mild solutions.

Theorem 16

Let $\gamma <\min (1, \alpha d_1)$ and suppose conditions (27) and (29) hold. Then the problem (20)–(21) has a unique local mild solution, i.e., the integral equation (22) has a solution $\varphi (|t|_p)$ defined for $|t|_p\le p^N$, where $N\in {\mathbb {Z}}$ is sufficiently negative, and any other solution ${\bar{\varphi }}(|t|_p)$, if it exists, coincides with $\varphi$ for $|t|_p\le p^K$, where $K\le N$.

Proof

We look for a solution of (22) as the limit of the sequence $\{\varphi _k\}$, with initial value $\varphi _0$:

$$\begin{aligned} \varphi _k(|t|_p) = \varphi _0 + I^{\alpha }_{d_1,d_2} \tilde{f}(|\cdot |_p, \varphi _{k-1}(|\cdot |_p)) (|t|_p). \end{aligned}$$

We will prove by induction that for all $k\ge 0$:

$$\begin{aligned} |\varphi _{k+1}(|t|_p)- \varphi _k(|t|_p)| \le C^{k+1} M H^k |t|_p^{(k+1)(\alpha d_2-\gamma )}. \end{aligned}$$

(32)

First we start looking for a bound for $|\varphi _1(|t|_p)- \varphi _0|$,

$$\begin{aligned} |\varphi _1(|t|_p)- \varphi _0|&=|\varphi _0 + I^{\alpha }_{d_1,d_2}\tilde{f}(|t|_p, \varphi _0)-\varphi _0 | =|I^{\alpha }_{d_1,d_2}\tilde{f}(|t|_p, \varphi _0)| \nonumber \\&\le C_0 M |t|_p^{\alpha d_2-\gamma } \le C M |t|_p^{\alpha d_2-\gamma }, \end{aligned}$$

(33)

where we have used (28).

Base case.

$$\begin{aligned} |\varphi _2(|t|_p)- \varphi _1(|t|_p)|&= |\varphi _0 + I^{\alpha }_{d_1,d_2} \tilde{f}(|t|_p, \varphi _1(|t|_p)) - \varphi _0- I^{\alpha }_{d_1,d_2}\tilde{f}(|t|_p, \varphi _0)|\\&=|I^{\alpha }_{d_1,d_2}(\tilde{f}(|t|_p, \varphi _1(|t|_p))) - I^{\alpha }_{d_1,d_2} \tilde{f}(|t|_p, \varphi _0)|\\&=|I^{\alpha }_{d_1,d_2}[(\tilde{f}(|t|_p, \varphi _1(|t|_p))) - \tilde{f}(|t|_p, \varphi _0)]|. \end{aligned}$$

By using (33) and (30)

$$\begin{aligned} |(\tilde{f}(|t|_p, \varphi _1(|t|_p))) - \tilde{f}(|t|_p, \varphi _0)| \le H|\varphi _1(|t|_p)-\varphi _0| |t|_p^{-\gamma } \le HCM |t|_p^{\alpha d_2 -2\gamma }, \end{aligned}$$

then we can use Proposition 14 with $\mu =HCM$ to get

$$\begin{aligned} I^{\alpha }_{d_1,d_2}[(\tilde{f}(|t|_p, \varphi _1(|t|_p))) - \tilde{f}(|t|_p, \varphi _0)] \le C_1 CHM |t|_p^{2(\alpha d_2-\gamma )} \le C^2 HM |t|_p^{2(\alpha d_2-\gamma )}. \end{aligned}$$

Induction step. We start with

$$\begin{aligned} \tilde{f}(|t|_p, \varphi _{k+1}(|t|_p))- \tilde{f}(|t|_p, \varphi _k(|t|_p))&\le H|\varphi _{k+1}(|t|_p)-\varphi _k(|t|_p)| |t|_p^{-\gamma }\\&\le H C^{k+1} M H^k |t|_p^{(k+1)(\alpha d_2-\gamma )} |t|_p^{-\gamma }\\&\le C^{k+1}H^{k+1} M |t|_p^{(k+1)\alpha d_2- (k+2)\gamma }, \end{aligned}$$

where we have used (30) and the induction hypothesis. Now we apply the Proposition 14 with $\mu = C^{k+1}H^{k+1}M$

$$\begin{aligned} |\varphi _{k+2}(|t|_p)-\varphi _{k+1}(|t|_p)|&=|I^{\alpha }_{d_1,d_2}[\tilde{f}(|t|_p, \varphi _{k+1}(|t|_p))- \tilde{f}(|t|_p, \varphi _k(|t|_p))]|\\&\le C C^{k+1}H^{k+1} M |t|_p^{(k+2)(\alpha d_2-\gamma )}\\&\le C^{k+2}H^{k+1} M |t|_p^{(k+2)(\alpha d_2-\gamma )}. \end{aligned}$$

It means that the sequence $\{\varphi _k\}$ converges uniformly on the ball $|t|_p\le T$, for sufficiently small T, to a function $\varphi (|t|_p)$, in other words

$$\begin{aligned} I^{\alpha }_{d_1, d_2} \tilde{f}(|\cdot |_p, \varphi _k(|\cdot |_p)) (|t|_p) \rightarrow I^{\alpha }_{d_1, d_2} \tilde{f}(|\cdot |_p, \varphi (|\cdot |_p)) (|t|_p), \ \ |t|_p\le T, \end{aligned}$$

according to the definition, $\varphi$ is a local solution of the equation (22).

In order to prove the uniqueness of the solution, it is enough to demonstrate by induction that

$$\begin{aligned} |\varphi (|t|_p)- {\bar{\varphi }}(|t|_p)| \le 2M C^m H^{m-1} |t|_p^{m(\alpha d_2-\gamma )}. \end{aligned}$$

$m=0.$

$$\begin{aligned} |\varphi (|t|_p)- {\bar{\varphi }}(|t|_p)|&= |I^{\alpha }_{d_1,d_2}(\tilde{f}(|t|_p, \varphi (|t|_p)))- I^{\alpha }_{d_1,d_2}(\tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p)))|\\&\le |I^{\alpha }_{d_1,d_2}(\tilde{f}(|t|_p, \varphi (|t|_p)))| + |I^{\alpha }_{d_1,d_2}(\tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p)))|\\&\le 2MC|t|_p^{\alpha d_2-\gamma }, \end{aligned}$$

after applying Proposition 13, since we are assuming (27), i.e., $|\tilde{f}(|t|_p, \varphi (|t|_p))|\le M$, then $|\tilde{f}(|t|_p, \varphi (|t|_p))|\le M|t|_p^{-\gamma }$, and the same is valid for ${\bar{\varphi }}$ instead of $\varphi$.

$m=1$. By using (29)

$$\begin{aligned} |\tilde{f}(|t|_p, \varphi (|t|_p)) -\tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p))| \le H|\varphi (|t|_p), {\bar{\varphi }}(|t|_p)| |t|_p^{-\gamma } \le 2HMC |t|_p^{\alpha d_2-2\gamma }, \end{aligned}$$

and by applying Proposition 14

$$\begin{aligned} |\varphi (|t|_p)- {\bar{\varphi }}(|t|_p)| = |I^{\alpha }_{d_1,d_2}[\tilde{f}(|t|_p, \varphi (|t|_p))- \tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p))]| \le 2HMC^2 |t|_p^{2(\alpha d_2-\gamma )}. \end{aligned}$$

Inductive step. We suppose $|\varphi (|t|_p)-{\bar{\varphi }}(|t|_p)|\le 2MC^m H^{m-1}|t|_p^{m(\alpha d_2-\gamma )}.$ Then

$$\begin{aligned} |\tilde{f}(|t|_p, \varphi (|t|_p)) -\tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p))|&\le H |\varphi (|t|_p)- {\bar{\varphi }}(|t|_p)| |t|_p^{-\gamma }\\&\le H2MC^m H^{m-1} |t|_p^{m(\alpha d_2-\gamma )} |t|_p^{-\gamma }\\&\le 2M C^m H^m |t|_p^{m\alpha d_2- (m+1)\gamma }. \end{aligned}$$

Then

$$\begin{aligned} |\varphi (|t|_p)- {\bar{\varphi }}(|t|_p)|&=|I^{\alpha }_{d_1,d_2}[\tilde{f}(|t|_p, \varphi (|t|_p))- \tilde{f}(|t|_p, {\bar{\varphi }}(|t|_p))]|\\&\le 2M C^m H^m C |t|_p^{(m+1)(\alpha d_2-\gamma )}\\&\le 2M C^{m+1} H^m |t|_p^{(m+1)(\alpha d_2-\gamma )}. \end{aligned}$$

Therefore, for $|t|_p<1$, if $m\rightarrow \infty$ we obtain $\varphi (|t|_p)={\bar{\varphi }}(|t|_p)$. $\square$

6 Conclusion

In summary, we study the pseudo differential operator $D^{\alpha }_{d_1,d_2}$ introduced by the authors in [4], which can be seen as a generalization of Vladimirov operator (taking $d_1=d_2$), and also of Bessel operator (taking $d_1=0$, $d_2\ne 0$). We consider the case $\alpha d_j\ne 1+\dfrac{2\pi i}{\ln p}$, $j=1,2.$

The main contribution of this work is that we found the explicit formulae for the operator and its inverse, and that was essential to solve two key issues: the Cauchy problem, and the degenerate nonlinear Cauchy problem, taking into account that the solution is a radial function.

We applied the techniques used by Kochubei in [1, 2] to solve the Cauchy problem (8)–(9) for radial functions. For the degenerate nonlinear Cauchy problem (20)–(21), we use the techniques given by Antoniouk et al. [3]. In both cases, the strategy was to transform the problem into an integral equation, and then we used some specific tools for that kind of equations.

We would like to express our sincere gratitude to the referee for their insightful comments. Their feedback has been invaluable in improving the quality of our manuscript.