p-Adic Analogue of the Wave Equation

Abstract

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the problem from a modified p-adic integral equation. Moreover, we give an iterative scheme for numerical computation of the regularlized solution.

1 Introduction

p-Adic analysis has attracted many researchers since last century due to its various applications in physics. The definition of derivatives on p-adic field is the most important problem for the study of p-adic equations. In 1960s, Gibbs first gave the definition of logic derivative over dyadic field [8]. Then Butzer, Onneweer, Zheng, Stankovic, Vladimirov, Su, et al. generalized its definition into Vilenkin group and further p-adic field [5, 23, 26,27,28, 32, 38]. Subsequently, Kozyrev, Albeverio, Zuniga-Galindo, Khrennikov et al. have done a lot of work on p-adic pseudo-differential operators [1, 13, 17, 20,21,22, 30, 36, 37]. Up to now, there are a number of monographs and a huge number of papers that cover various problems involving p-adic pseudo-differential equations since they play important role in p-adic theoretical physics, see [2,3,4, 6, 7, 12, 14, 15, 18, 19, 33,34,35]. However. there are few results about nonlinear p-adic equations. Andrei Khrennikov et al. [11, 16] consider a p-adic analog of the porous medium equation by $L^1$-theory of the Vladimirov operator $D^\alpha$. Since nonlinear equations play the important role in mathematical physics, it stimulates us to develop the theory in this field over p-adic field.

In this paper, we consider a p-adic analogue of the wave equation with Lipschitz source

$$\begin{aligned} \frac{\partial ^2 u(t,x)}{\partial t^2}= T^{\alpha } u(t,x)+f(u),\quad 0<t<T, \,\,x\in {\mathbb {Q}}_p, \end{aligned}$$

where f satisfies Lipschitz condition, ${\mathbb {Q}}_p$ is the field of p-adic numbers (prime number $p>1$)and $T^{\alpha }, \alpha >0$ is pseudo-differential operator defined by Su in [27].

It is worth mentioning that this kind of equation has widely studied in Euclidean space since it linked to elliptic sine-Gordon equations arising in molecular biology [9, 10, 24, 31]. According to Ostrowski’s theorem, there exist two parallel “universe”: one is real “universe” based on the field of real numbers ${\mathbb {R}}$, the other is p-adic “universe” based on the field of p-adic numbers ${\mathbb {Q}}_p$. So compare to macro study, it is valuable to consider this problem in microscopic view.

The rest of paper is organized as follows. In Sect. 2, we recall some basic notations in p-adic field and introduce the definition of pseudo-differential operator and corresponding orthonormal base. In Sect. 3, we study the mild solution of Cauchy problem for the p-adic analogue of the wave equation. We define a regularized factor to obtain the stable solution. Then, we prove the existence and the stability of the regularized solution. Further, we consider the error estimation.

2 Preliminaries

We will use the notations and results from Taibleson’s book [29]. Let ${\mathbb {Q}}_p$ be the p-adic field, which p is a prime number. It is a nondiscrete, locally compact, totally disconnected and complete topological field. For any $x,y\in {\mathbb {Q}}_p$, define non-archimedien norm $|\cdot |\,:\,{\mathbb {Q}}_p \rightarrow {\mathbb {R}}^+$ satisfying following properties:

$$\begin{aligned}&\mathrm{{(i)}} \quad |x|\ge 0; |x|=0\Leftrightarrow x=0;\\&\mathrm{{(ii)}} \quad |xy|=|x||y|;\\&\mathrm{{(iii)}} \quad |x+y|\le \max \{|x|, |y|\}. \end{aligned}$$

Denote subset ${\mathfrak {D}}=\{x\in {\mathbb {Q}}_p:\,\,|x|\le 1\}$ as the ring of integers in ${\mathbb {Q}}_p$. It is the unique maximal compact subring in ${\mathbb {Q}}_p$ with the Haar measures $|{\mathfrak {D}}| = 1$. The prime ideal ${B}=\{x\in {\mathbb {Q}}_p:\,\,|x|\le p^{-1}\}$ are balls with center 0 in ${\mathbb {Q}}_p$ satisfying ${B} = \beta {\mathfrak {D}}$ where $\beta =p^{-1}$. It is the unique maximal ideal in ${\mathfrak {D}}$. Then, the fractional ideal can be defined as ${B}_k = \{x \in {\mathbb {Q}}_p : \,\,|x| \le p^{-k}\}$ with the Haar measures $|{B}_k| = p^{-k}$, $k \in {\mathbb {Z}}$.

If $|x|=p^{-t}$, $t \in {\mathbb {Z}}$, then x admits a (unique) canonical representation

$$\begin{aligned} x = x_t\beta ^t+x_{t+1}\beta ^{t+1}+\cdots , \end{aligned}$$

where $\beta =p^{-1}$ and $x_t, x_{t+1},\ldots \in \{0,1,\ldots ,p-1\}$.

For each $l \in {\mathbb {Z}}$, we choose elements $z_{l,i}\in {\mathbb {Q}}_p$, $i \in {\mathbb {Z}}^+$, so that the subsets ${B}_{l,i} = z_{l,i}+{B}_l\subset {\mathbb {Q}}_p$. And if $i\ne j$, then ${B}_{l,i} \cap {B}_{l,j} = \emptyset$, $\cup _{i=0}^\infty {B}_{l,i} = {\mathbb {Q}}_p.$

Define indicative function of Haar measurable subset $E\subset {\mathbb {Q}}_p$ as

$$\begin{aligned} \Phi _E(x)=\left\{ \begin{array}{ll}1,\,\,\, x\in E,\\ 0,\,\,\,x\in E^{c} ,\\ \end{array} \right. \end{aligned}$$

then, the Haar measure of E is $|E|=\int _{E}dx=\int _{{\mathbb {Q}}_p} \Phi _E(x)dx$ where dx denote the Haar measure on ${\mathbb {Q}}_p$ normalized by the condition $\int _{{\mathfrak {D}}}dx=1$.

Define translation operator $\tau _h\,:\, f\rightarrow \tau _h f,\,\,h\in {\mathbb {Q}}_p$ as $\tau _h f(x)=f(x-h),\,\,x\in {\mathbb {Q}}_p$. Then, the test function space $S=S({\mathbb {Q}}_p)$ is defined as

$$\begin{aligned} S({\mathbb {Q}}_p)&=\left\{ \varphi \,:\,{\mathbb {Q}}_p\rightarrow {\mathbb {C}},\,\varphi (x)\right. \\&\left. =\sum _{j=1}^{n}c_j\tau _{h_j}\Phi _{B_{k_j}(x)},\,c_j\in {\mathbb {C}},{h_j}\in \mathbb {Q_p},\,k_j\in {\mathbb {Z}},\,1\le j\le n\right\} , \end{aligned}$$

where the element $\varphi (x)$ is called test function.

For the test function space S, we give the following topology: for $\varphi \in S(\mathbb {Q_p})$, there exists unique integers (k, l) such that the function $\varphi$ is constant on the coset of $B_k$ and with supports in the ball $B_l$; $\lim \limits _{n\rightarrow +\,\infty }\varphi _n(x)=0$ converges uniformly for $x\in \mathbb {Q_p}$. Then, S is complete topological linear spaces.

Denote by $S^\prime =S^\prime ({\mathbb {Q}}_p)$ the distribution space of test function space S. $S^\prime$ is a complete topological linear space under the dual topology.

Let $\chi (x)$ be a fixed non-trivial character of ${\mathbb {Q}}_p$ which is trivial on ${\mathfrak {D}}$. For the p-adic field, $\chi$ can be constructed by the base value [16] as

$$\begin{aligned} \chi (\beta ^{-j})=\left\{ \begin{array}{ll}\mathrm{exp}\left( \frac{2\pi i}{p^j}\right) ,\,\,\,\mathrm{for}\,\, j\in {\mathbb {N}},\\ 1,\,\,\,\quad \quad \quad \quad \quad \mathrm{otherwise},\\ \end{array} \right. \end{aligned}$$

Then for $x = x_t\beta _t+x_{t+1}\beta _{t+1}+\cdots$, $\chi (x)=\exp (2\pi i\sum _{j=t}^{-1}x_jp^j)$ and for $\lambda = \lambda _\tau \beta _\tau +\lambda _{\tau +1}\beta _{\tau +1}+\cdots$

$$\begin{aligned} \chi _\lambda (x)=\chi (\lambda x)=\exp \left( 2\pi i\sum _{k=0}^{-(t+\tau +1)}\left( \sum _{j=0}^k x_{t+\tau -j}\lambda _{t+\tau }\right) p^{t+\tau +k}\right) \end{aligned}$$

For $\varphi \in S({\mathbb {Q}}_p)$, we define its Fourier transform $\varphi ^{\wedge }$ by

$$\begin{aligned} \varphi ^{\wedge }(\xi )=\int _{{\mathbb {Q}}_p}\varphi (x)\overline{\chi _\xi }(x)dx,\,\,\,\,\,\xi \in {\mathbb {Q}}_p, \end{aligned}$$

and inverse Fourier transform $\varphi ^{\vee }$ by

$$\begin{aligned} \varphi ^{\vee }(x)=\int _{{\mathbb {Q}}_p}\varphi (\xi )\chi _{x}(\xi )d\xi ,\,\,\,\,\,x\in {\mathbb {Q}}_p, \end{aligned}$$

In 1992, Su Weiyi [27] has given definitions of the derivative for the p-adic local fields ${\mathbb {Q}}_p$, including derivatives of the fractional orders and real orders.

Definition 2.1

Let $<\xi >=\max \{1,\,|\xi |\},\,\alpha \ge 0$ if for $\varphi \in S({\mathbb {Q}}_p)$, the integral

$$\begin{aligned} T^{\alpha }\varphi (x)=(<\xi >^\alpha \varphi ^{\wedge }(\xi ) )^{\vee }(x) \end{aligned}$$

exists at $x\in {\mathbb {Q}}_p$, where $\chi (x)$ is a fixed non-trivial character of ${\mathbb {Q}}_p$. Then it is called a pointwise derivative of order $\alpha$ of $\varphi$ at x.

Note: The defined domain of $T^\alpha$ in the definition can be extended to the space $S^\prime ({\mathbb {Q}}_p)$, where $S^\prime ({\mathbb {Q}}_p)$ denote the set of all functionals (distributions) on $S({\mathbb {Q}}_p)$.

Let $D(T^\alpha )$ be the domain of $T^\alpha$ defined as

$$\begin{aligned} D(T^\alpha )=\{\varphi \in L^2:\,\,<\xi >^\alpha \varphi ^\wedge \in L^2\}. \end{aligned}$$

We have

Lemma 2.1

[25] Let $\psi (x)=\chi (p^{-1}x)\Phi _{{\mathfrak {D}}}$, then

$$\begin{aligned}T^\alpha \psi (ax+b)= \left\{ \begin{array}{ll}p^\alpha |a|^\alpha \psi (ax+b),&{} { |a|>p^{-1}}\\ \psi (ax+b),&{} { |a|\le p^{-1}}\\ \end{array} \right. \end{aligned}$$

where $a,b\in {\mathbb {Q}}_p$ and $a\ne 0$.

Lemma 2.2

[25] $T^\alpha$ is a positive definite self-adjoint operator on $D(T^\alpha )$, $\{\psi _{NjI}\}$ is an orthonormal base of $L^2$ consisting of eigenfunctions of the operator $T^\alpha$, which defined as follows where $\Phi _{{\mathfrak {D}}}(x)$ is a characteristic function of a unit ball.

and

$$\begin{aligned} T^\alpha \psi _{NjI}(x)= \left\{ \begin{array}{ll}p^{(1-N)\alpha }\psi _{NjI}(x),&{} { N<1}\\ \psi _{NjI}(x),&{} { N\ge 1}\\ \end{array} \right. \end{aligned}$$

Proof

We noted that

$$\begin{aligned} \psi _{NjI}(x)&=p^{-\frac{N}{2}}\chi (p^{-1}(p^{N}jx))\Phi _{{\mathfrak {D}}}(p^Njx-jz_I)\\&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)\chi (p^{-1}(p^Njx-jz_I))\Phi _{{\mathfrak {D}}}(p^Njx-jz_I)\\&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)\psi (p^Njx-jz_I). \end{aligned}$$

For $N<1$, then $|p^N|>p^{-1}$. So we can obtain

$$\begin{aligned} T^\alpha \psi _{NjI}(x)&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)T^\alpha \psi (p^Njx-jz_I)\\&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)p^\alpha |p^N|^\alpha \psi (p^Njx-jz_I)\\&=p^{(1-N)\alpha }\psi _{NjI}(x). \end{aligned}$$

For $N\ge 1$, then $|p^N|\le p^{-1}$. So we can get

$$\begin{aligned} T^\alpha \psi _{NjI}(x)&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)T^\alpha \psi (p^Njx-jz_I)\\&=p^{-\frac{N}{2}}\chi (p^{-1}jz_I)\psi (p^Njx-jz_I)\\&=\psi _{NjI}(x). \end{aligned}$$

To prove the orthogonality of $\{\psi _{NjI}\}$, we consider the inner product $(\psi _{NjI},\psi _{N^\prime j^\prime I^\prime })$ in $L^2$.

$$\begin{aligned} (\psi _{NjI},\psi _{N^\prime j^\prime I^\prime })&=\int _{p^{-N}I\cap p^{-N^\prime } I^\prime } p^{-\frac{N}{2}}\chi (p^{N-1}jx)p^{-\frac{N^\prime }{2}}\chi (p^{N^\prime -1}j^\prime x)dx\\&=\delta _{N N^\prime }\int _{p^{-N}(I\cap I^\prime )} p^{-N}\chi (p^{N-1}jx)\chi (p^{N-1}j^\prime x)dx\\&=\delta _{N N^\prime }\delta _{I I^\prime }\int _{p^{-N}I} p^{-N}\chi (p^{N-1}(j-j^\prime )x)dx\\&=\delta _{N N^\prime }\delta _{I I^\prime }\delta _{j j^\prime }. \end{aligned}$$

To prove the completeness of $\{\psi _{NjI}\}$, we consider the Fourier coefficient of $\Phi _{{\mathfrak {D}}}$.

$$\begin{aligned} (\Phi _{{\mathfrak {D}}}, \psi _{NjI})=p^{-\frac{N}{2}}\int _{{\mathfrak {D}} \cap p^{-N}I}\chi (-p^{N-1}jx)dx. \end{aligned}$$

For $N\le 0$, $(\Phi _{{\mathfrak {D}}}, \psi _{NjI})=0$. While for $N> 0$, $(\Phi _{{\mathfrak {D}}}, \psi _{NjI})=p^{-\frac{N}{2}}\delta _{I{\mathfrak {D}}}$. Then

$$\begin{aligned} \sum \limits _{NjI}|(\Phi _{{\mathfrak {D}}}, \psi _{NjI})|^2=(p-1)\sum _{N=1}^{\infty }p^{-N}=1=||\Phi _{{\mathfrak {D}}}||^2. \end{aligned}$$

So by Parserval equality, we proved the completeness of $\{\psi _{NjI}\}$. $\square$

Remark 2.1

There is another mostly used operator over $p-$adic field called Vladimirov operator $D^\alpha$, $\alpha >0$. It is defined as a pseudo-differential operator with the symbol $|\xi |^\alpha$:

$$\begin{aligned} D^{\alpha }\varphi (x)=(|\xi |^\alpha \varphi ^{\wedge }(\xi ) )^{\vee }(x),\quad \varphi \in S({\mathbb {Q}}_p). \end{aligned}$$

Here, the orthonormal base $\psi _{NjI}(x)$ of $D^\alpha$ is obtained in a different form (see [2]) and $D^\alpha \psi _{NjI}(x)=p^{(1-N)\alpha }\psi _{NjI}(x)$, for any $N\in {\mathbb {N}}$. In this paper, we use operator $T^\alpha$ since from the Lemma 2.2, we can find that for any $N\in {\mathbb {N}}$, the eigenvalues of $T^\alpha$ are no less than 1. This proposition is important for our later estimate.

3 Main Results

3.1 Homogeneous Problem

First, we consider the following homogeneous equation over p-adic field.

$$\begin{aligned} \left\{ \begin{array}{ll}\frac{\partial ^2 u(t,x)}{\partial t^2}= T^{\alpha }_x u(t,x),\\ u(0,x)=\phi (x),\\ u^\prime _t(0,x)=g(x).\\ \end{array} \right. \end{aligned}$$

(1)

We will use the orthonormal base $\{\psi _{NjI}\}$ constructed in Lemma 2.1 to solve the equation. In the following, we will write $\sum$ instead of $\sum _{N,j,I}$ for simplicity.

Let lacunary series $u(t,x)=\sum u_{NjI}(t)\psi _{NjI}(x)$ be the exact form of problem (1). Then

$$\begin{aligned} \frac{\partial ^2 u(t,x)}{\partial t^2}= & {} \sum \frac{d^2 u_{NjI}(t)}{d t^2}\psi _{NjI}(x), \\ T^{\alpha }_{x}u(t,x)= & {} \sum _{N<1}p^{\alpha (1-N)}u_{NjI}(t)\psi _{NjI}(x)+\sum _{N\ge 1}u_{NjI}(t)\psi _{NjI}(x). \end{aligned}$$

From

$$\begin{aligned} \frac{\partial ^2 u(t,x)}{\partial t^2}= T^{\alpha }_x u(t,x), \end{aligned}$$

we get

$$\begin{aligned}&\sum _{N<1}\{ u^{\prime \prime }_{NjI}(t)-p^{\alpha (1-N)}u_{NjI}(t)\}\psi _{NjI}(x)+\sum _{N\ge 1}\{u^{\prime \prime }_{NjI}(t)\\&\quad -\,\,u_{NjI}(t)\}\psi _{NjI}(x)=0, \end{aligned}$$

Due to the orthogonality of $\{\psi _{NjI}(x)\}$, we have

$$\begin{aligned} \left\{ \begin{array}{ll} u^{\prime \prime }_{NjI}(t)=p^{\alpha (1-N)}u_{NjI}(t),&{} { N<1},\\ u^{\prime \prime }_{NjI}(t)=u_{NjI}(t),&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

They are the ordinary differential equations of order 2 on ${\mathbb {R}}$. So the corresponding characteristic equations are given as

$$\begin{aligned} \left\{ \begin{array}{ll}\lambda ^2=p^{\alpha (1-N)},&{} { N<1},\\ \lambda ^2=1,&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

So, the solution of the equation is obtained as

$$\begin{aligned} \left\{ \begin{array}{ll}u_{NjI}(t)=A_{NjI}e^{-tp^{\frac{\alpha }{2}(1-N)}}+B_{NjI}e^{tp^{\frac{\alpha }{2}(1-N)}},&{} { N<1},\\ u_{NjI}(t)=C_{NjI}e^{-t}+D_{NjI}e^{t},&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

To determine the coefficients $A_{NjI}$, $B_{NjI}$, $C_{NjI}$ and $D_{NjI}$, we assume that $\phi \in D(T^\alpha )$ can be expanded as lacunary series $\phi =\sum \phi _{NjI}\psi _{NjI}(x)$, where

$$\begin{aligned} \phi _{NjI}=<\phi (x),\psi _{NjI}(x)>=\int _{{\mathbb {Q}}_p}\phi (x) \overline{\psi _{NjI}(x)}dx,\quad \sum |\phi _{NjI}|^2<+\,\infty . \end{aligned}$$

(2)

and

$$\begin{aligned} T^{\alpha }\phi (x)= & {} \sum _{N<1}p^{\alpha (1-N)}f\phi _{NjI}\psi _{NjI}(x)+\sum _{N\ge 1}\phi _{NjI}\psi _{NjI}(x),\nonumber \\&\sum _{N<1}p^{2\alpha (1-N)}|\phi _{NjI}|^2+\sum _{N\ge 1}|\phi _{NjI}|^2<+\,\infty . \end{aligned}$$

(3)

With the initial condition $u(0,x)=\phi (x)$, then $u_{NjI}(0)=\phi _{NjI}$, we get

$$\begin{aligned} \left\{ \begin{array}{ll} A_{NjI}+B_{NjI}=\phi _{NjI},&{} { N<1},\\ C_{NjI}+D_{NjI}=\phi _{NjI},&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

Similarly, we can also expand $g\in D(T^{\alpha })$ as lacunary series $g=\sum g_{NjI}\psi _{NjI}(x)$,

where

$$\begin{aligned} g_{NjI}=<g(x), \psi _{NjI}(x)>=\int _{{\mathbb {Q}}_p}g(x) \overline{\psi _{NjI}(x)}dx,\quad \sum |g_{NjI}|^2<+\,\infty . \end{aligned}$$

(4)

and

$$\begin{aligned} T^{\alpha }g(x)= & {} \sum _{N<1}p^{\frac{\alpha (1-N)}{2}}g_{NjI}\psi _{NjI}(x)+\sum _{N\ge 1}g_{NjI}\psi _{NjI}(x), \nonumber \\&\sum _{N<1}p^{\alpha (1-N)}|g_{NjI}|^2+\sum _{N\ge 1}|g_{NjI}|^2<+\,\infty . \end{aligned}$$

(5)

With the initial condition $u^\prime (0,x)=g(x)$, then $u^\prime _{NjI}(0)=g_{NjI}$, we get

$$\begin{aligned} \left\{ \begin{array}{ll} p^{\frac{\alpha (1-N)}{2}}(A_{NjI}-B_{NjI})=g_{NjI},&{} { N<1},\\ C_{NjI}-D_{NjI}=g_{NjI},&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

(6)

So, the coefficients are given as

$$\begin{aligned} \left\{ \begin{array}{ll} A_{NjI}=\frac{\phi _{NjI}+g_{NjI}p^{-\frac{\alpha (1-N)}{2}}}{2}, \quad B_{NjI})=\frac{\phi _{NjI}-g_{NjI}p^{-\frac{\alpha (1-N)}{2}}}{2},&{} { N<1},\\ C_{NjI}=\frac{\phi _{NjI}+g_{NjI}}{2}, \quad D_{NjI}=\frac{\phi _{NjI}-g_{NjI}}{2},&{} { N\ge 1}.\\ \end{array} \right. \end{aligned}$$

(7)

Then the exact solution (or mild solution) of the homogeneous equation is

$$\begin{aligned} u(t,x)&=\sum _{N<1}\left[ \phi _{NjI}\cosh \left( tp^{\frac{\alpha (1-N)}{2}}\right) +g_{NjI}p^{-\frac{\alpha (1-N)}{2}}\sinh \left( tp^{\frac{\alpha (1-N)}{2}}\right) \right] \psi _{NjI}(x)\\&\quad + \sum _{N\ge 1}[\phi _{NjI}\cosh (t)+g_{NjI}\sinh (t)]\psi _{NjI}(x). \end{aligned}$$

3.2 Inhomogeneous Problem

Next, we consider the case of inhomogeneous equation with nonlinear Lipschitz source.

$$\begin{aligned} \left\{ \begin{array}{ll}\frac{\partial ^2 u(t,x)}{\partial t^2}= T^{\alpha }_x u(t,x)+f(u),\\ u(0,x)=\phi (x),\\ u^\prime _t(0,x)=g(x),\\ \end{array} \right. \end{aligned}$$

(8)

where $f: {\mathbb {Q}}_p\rightarrow {\mathbb {Q}}_p$ satisfies Lipschitz condition

$$\begin{aligned} ||f(u_1)-f(u_2)||_{L^2({\mathbb {Q}}_p)}\le C||u_1-u_2||_{L^2({\mathbb {Q}}_p)}, \end{aligned}$$

and C is a positive constant independent of $u_1$, $u_2$. By $f(u)=\sum \phi _{NjI}(u)\psi _{NjI}(x)$, the solution u satisfies the following integral equation:

$$\begin{aligned} u(t,x)=&\sum _{N<1}\left[ \phi _{NjI}\cosh \left( tp^{\frac{\alpha (1-N)}{2}}\right) +g_{NjI}p^{-\frac{\alpha (1-N)}{2}}\sinh \left( tp^{\frac{\alpha (1-N)}{2}}\right) \right. \\&\left. +p^{-\frac{\alpha (1-N)}{2}} \int _0^t \sinh \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u)(s)ds\right] \psi _{NjI}(x)\\&+\sum _{N\ge 1}\left[ \phi _{NjI}\cosh (t)+g_{NjI}\sinh (t)\right. \\&\left. + \int _0^t \sinh (t-s)f_{NjI}(u)(s)ds\right] \psi _{NjI}(x). \end{aligned}$$

When $N<1$, we find that terms $\cosh (tp^{\frac{\alpha (1-N)}{2}})$, $\sinh (tp^{\frac{\alpha (1-N)}{2}})$ and $\sinh ((t-s)p^{\frac{\alpha (1-N)}{2}})$ are unstable. So we replace these unstable terms by $\cosh ^\varepsilon (tp^{\frac{\alpha (1-N)}{2}})$, $\sinh ^\varepsilon (tp^{\frac{\alpha (1-N)}{2}})$ and $\sinh ^\varepsilon ((t-s)p^{\frac{\alpha (1-N)}{2}})$, where

$$\begin{aligned} \cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right)&=\frac{E_\varepsilon \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) +\exp \left( -tp^{\frac{\alpha (1-N)}{2}}\right) }{2}, \\ \sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right)&=\frac{E_\varepsilon \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) -\exp \left( -tp^{\frac{\alpha (1-N)}{2}}\right) }{2},\\ \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right)&=\frac{E_\varepsilon \exp \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) -\exp \left( -(t-s)p^{\frac{\alpha (1-N)}{2}}\right) }{2},\\ E_\varepsilon&=\frac{\exp \left( -Tp^{\frac{\alpha (1-N)}{2}}\right) }{\varepsilon +\exp \left( -Tp^{\frac{\alpha (1-N)}{2}}\right) }. \end{aligned}$$

Then, we could approximate the exact solution u by the regularized solution $u_\varepsilon$

$$\begin{aligned} u_\varepsilon (t,x)=&\sum _{N<1}\left[ \phi _{NjI}\cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) +g_{NjI} p^{-\frac{\alpha (1-N)}{2}}\sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) \right. \nonumber \\&\left. +p^{-\frac{\alpha (1-N)}{2}} \int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u_\varepsilon )(s)ds\right] \psi _{NjI}(x)\nonumber \\&+\sum _{N\ge 1}\left[ \phi _{NjI}\cosh (t)+g_{NjI}\sinh (t)+ \int _0^t \sinh (t-s)f_{NjI}(u_\varepsilon )(s)ds\right] \nonumber \\&\times \psi _{NjI}(x). \end{aligned}$$

(9)

Lemma 3.1

$$\begin{aligned} \cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) \le \varepsilon ^{-\frac{t}{T}}, \sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) \le \varepsilon ^{-\frac{t}{T}}, \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) \le \varepsilon ^{\frac{s-t}{T}}. \end{aligned}$$

Proof

We just prove the first inequality.

$$\begin{aligned} E_\varepsilon \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right)&=\frac{\exp \left( (t-T)p^{\frac{\alpha (1-N)}{2}}\right) }{\left( \varepsilon +\exp \left( -Tp^{\frac{\alpha (1-N)}{2}}\right) \right) ^{\frac{T-t}{T}}\left( \varepsilon +\exp \left( -Tp^{\frac{\alpha (1-N)}{2}}\right) \right) ^{\frac{t}{T}}}\\&\le \frac{1}{\left( \varepsilon +\exp \left( -Tp^{\frac{\alpha (1-N)}{2}}\right) \right) ^{\frac{t}{T}}}\le \varepsilon ^{-\frac{t}{T}}. \end{aligned}$$

From $\varepsilon ^{\frac{t}{T}}\le 1\le \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right)$, we obtain

$$\begin{aligned} \cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) =\frac{E_\varepsilon \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) +\exp \left( -tp^{\frac{\alpha (1-N)}{2}}\right) }{2}\le \varepsilon ^{-\frac{t}{T}}. \end{aligned}$$

$\square$

We shall show the existence and uniqueness of the regularized solution.

Theorem 3.1

For any given $\varepsilon >0$, the integral Eq. (9) has a unique solution $u_\varepsilon \in C((0,T),{\mathbb {Q}}_p)$

Proof

For $v(t,x)\in C((0,T),{\mathbb {Q}}_p)$, consider operator F

$$\begin{aligned} Fv=&\sum _{N<1}\left[ \phi _{NjI}\cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) +g_{NjI}p^{-\frac{\alpha (1-N)}{2}}\sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) \right. \\&\left. +p^{-\frac{\alpha (1-N)}{2}} \int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(v)(s)ds\right] \psi _{NjI}(x)\\&+\sum _{N\ge 1}\left[ \phi _{NjI}\cosh (t)+g_{NjI}\sinh (t)+ \int _0^t \sinh (t-s)f_{NjI}(v)(s)ds\right] \psi _{NjI}(x). \end{aligned}$$

Then, we have $||F^{n}v_1-F^{n}v_2||\le \sqrt{(C^2t^2M_\varepsilon ^2)^{n}\frac{1}{n!}}\max \limits _{0\le t\le T}||v_1-v_2||$ for $v_1,v_2\in C([0,T],{\mathbb {Q}}_p)$. Indeed,

$$\begin{aligned} ||Fv_1-Fv_2||^2&=\sum _{N<1}\left[ p^{-\frac{\alpha (1-N)}{2}} \int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) (f_{NjI}(v_1)(s)\right. \\&\quad \left. -f_{NjI}(v_2)(s))ds\right] ^2\\&\quad +\sum _{N\ge 1}\left[ \int _0^t \sinh (t-s)(f_{NjI}(v_1)(s)-f_{NjI}(v_2)(s))ds\right] ^2\\&\le M_\varepsilon ^2t\sum \int _0^t|f_{NjI}(v_1)(s)-f_{NjI}(v_2)(s)|^2ds\\&\le M_\varepsilon ^2t\int _0^t||f(v_1)-f(v_2)||^2ds\le C^2t^2M_\varepsilon ^2\max _{0\le t\le T}||v_1-v_2||^2 \end{aligned}$$

Here we use the relation $\sinh ^\varepsilon ((t-s)p^{\frac{\alpha (1-N)}{2}})\le \varepsilon ^{\frac{s-t}{T}}\le \frac{1}{\varepsilon }$, $\sinh (t-s)\le \exp (T)$ and for any given $\varepsilon$, let $M_{\varepsilon }=\max \{\frac{1}{\varepsilon },\exp (T)\}$. Suppose $n=k$ holds, then

$$\begin{aligned}&||F^{k+1}v_1-F^{k+1}v_2||^2 \\&\quad =\sum _{N<1}\left[ p^{-\frac{\alpha (1{-}N)}{2}} \int _0^t \sinh ^\varepsilon \left( (t{-}s)p^{\frac{\alpha (1-N)}{2}}\right) (f_{NjI}(F^kv_1)(s){-}f_{NjI}(F^kv_2)(s))ds\right] ^2\\&\qquad +\sum _{N\ge 1}\left[ \int _0^t \sinh (t-s)(f_{NjI}(F^kv_1)(s)-f_{NjI}(F^kv_2)(s))ds\right] ^2\\&\quad \le M_\varepsilon ^2t\sum \int _0^t|f_{NjI}(F^kv_1)(s)-f_{NjI}(F^kv_2)(s)|^2ds\\&\quad \le C^2tM_\varepsilon ^2\int _0^t||F^kv_1-F^kv_2||^2ds\\&\quad \le C^2tM_\varepsilon ^2\int _0^t(C^2s^2M^2)^k\frac{1}{k!}\max _{0\le t\le T}||v_1-v_2||^2ds\\&\quad \le \frac{(C^2tM_\varepsilon ^2)^{k+1}}{k!}\max _{0\le t\le T}||v_1-v_2||^2\int _0^t s^k ds \le \frac{(C^2t^2M_\varepsilon ^2)^{k+1}}{(k+1)!}\max _{0\le t\le T}||v_1-v_2||^2. \end{aligned}$$

Since $\lim \limits _{n\rightarrow \infty }\sqrt{(C^2t^2M_\varepsilon ^2)^{n}\frac{1}{n!}}=0$ for any given $\varepsilon$, so for some $n_0$, $F^{n_0}$ is contractive. Then there exists a unique $u_\varepsilon$ satisfying $F^{n_0}u_\varepsilon =u_\varepsilon$. From $Fu_\varepsilon =F(F^{n_0}u_\varepsilon )=F^{n_0}(Fu_\varepsilon )$, so we obtain $Fu_\varepsilon =u_\varepsilon$. $\square$

Next, we will consider the stability of regularized solution.

Theorem 3.2

For any $0<\varepsilon <\exp (-T)$, then

$$\begin{aligned} ||u^1_\varepsilon -u^2_\varepsilon ||^2\le 3\varepsilon ^{-\frac{2t}{T}}\exp (3C^2T^2)[||\phi ^1-\phi ^2||^2+||g^1-g^2||^2], \end{aligned}$$

where the solutions $u^1_\varepsilon$, $u^2_\varepsilon$ subject to the initial condition $(\phi ^1, g^1)$, $(\phi ^2, g^2)$ respectively.

Proof

$$\begin{aligned}&||u^1_\varepsilon -u^2_\varepsilon ||^2 \\&\quad =\sum _{N<1}\left[ (\phi ^1_{NjI}{-}\phi ^2_{NjI})\cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) {+}(g^1_{NjI}-g^2_{NjI})p^{-\frac{\alpha (1-N)}{2}}\sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) \right. \\&\qquad \left. +p^{-\frac{\alpha (1-N)}{2}} \int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) (f_{NjI}(u^1_\varepsilon )(s)-f_{NjI}(u^2_\varepsilon )(s))ds\right] ^2\\&\qquad +\sum _{N\ge 1}[(\phi ^1_{NjI}-\phi ^2_{NjI})\cosh (t)+(g^1_{NjI}-g^2_{NjI})\sinh (t)\\&\qquad +\int _0^t \sinh (t-s)(f_{NjI}(u^1_\varepsilon )(s)-f_{NjI}(u^2_\varepsilon )(s))ds]^2\\&\quad \le \sum _{N<1}3\left[ |\cosh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) |^2|\phi ^1_{NjI}{-}\phi ^2_{NjI}|^2{+}|\sinh ^\varepsilon \left( tp^{\frac{\alpha (1-N)}{2}}\right) |^2|g^1_{NjI}{-}g^2_{NjI}|^2\right. \\&\qquad \left. +\,\,t\int _0^t |\sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) |^2|f_{NjI}(u^1_\varepsilon )(s)-f_{NjI}(u^2_\varepsilon )|^2ds\right] \\&\qquad +\sum _{N\ge 1}3[\cosh ^2(t)|\phi ^1_{NjI}-\phi ^2_{NjI}|^2+\sinh ^2(t)|g^1_{NjI}-g^2_{NjI}|^2\\&\qquad +t\int _0^t |\sinh (t-s)|^2|f_{NjI}(u^1_\varepsilon )(s)-f_{NjI}(u^2_\varepsilon )|^2ds]. \end{aligned}$$

For $0<\varepsilon <\exp (-T)$, we have $\varepsilon ^{-\frac{2t}{T}}>\cosh ^2(t)$, $\varepsilon ^{-\frac{2t}{T}}>\sinh ^2(t)$. Further, by using Lemma 3.1,

$$\begin{aligned} ||u^1_\varepsilon -u^2_\varepsilon ||^2&\le 3\varepsilon ^{-\frac{2t}{T}}\left[ ||\phi ^1-\phi ^2||^2+||g^1-g^2||^2+t\int _0^t\varepsilon ^{\frac{2s}{T}}||f(u^1_\varepsilon )-f(u^2_\varepsilon )||^2ds\right] \\&\le 3\varepsilon ^{-\frac{2t}{T}}\left[ ||\phi ^1-\phi ^2||^2+||g^1-g^2||^2+C^2t\int _0^t\varepsilon ^{\frac{2s}{T}}||u^1_\varepsilon -u^2_\varepsilon ||^2ds\right] . \end{aligned}$$

Then, by using Gronwall’s inequality, we have

$$\begin{aligned} ||u^1_\varepsilon -u^2_\varepsilon ||^2&\le 3\varepsilon ^{-\frac{2t}{T}}\exp (3C^2T^2)\left[ ||\phi ^1-\phi ^2||^2+||g^1-g^2||^2\right] . \end{aligned}$$

$\square$

The following theorem concerns the error estimate of regularized solution $u_\varepsilon$ and the exact solution u.

Theorem 3.3

For any $0<\varepsilon <\exp (-T)$, denote

$$\begin{aligned} G_N= & {} \exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) \phi _{NjI}+g_{NjI}\exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) p^{-\frac{\alpha (1-N)}{2}}\\&+\,\int _0^t\exp \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u)ds, \end{aligned}$$

and suppose $\sum \limits _{N<1}\exp \left( 2(T-t)p^{\frac{\alpha (1-N)}{2}}\right) G^2_N$ converges and satisfies

$$\begin{aligned} \sum _{N<1}\exp \left( 2(T-t)p^{\frac{\alpha (1-N)}{2}}\right) G^2_N\le 2V, \end{aligned}$$

where V is a constant. Then, there exists a regularized solution such that

$$\begin{aligned} ||u_\varepsilon -u||&\le \sqrt{ V\exp (2C^2T^2)}\varepsilon ^{\frac{T-t}{T}}. \end{aligned}$$

Proof

$$\begin{aligned} ||u_\varepsilon -u||^2&=\sum _{N<1}\left[ \frac{E_\varepsilon -1}{2}\exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) \left( \phi _{NjI}+g_{NjI}p^{-\frac{\alpha (1-N)}{2}}\right) \right. \\&\quad \left. \,+p^{-\frac{\alpha (1-N)}{2}} \left( \int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u_\varepsilon )ds\right. \right. \\&\quad \left. \left. \,-\int _0^t \sinh \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u)ds\right) \right] ^2\\&\quad +\sum _{N\ge 1}\left[ \int _0^t \sinh (t-s)(f_{NjI}(u_\varepsilon )(s)-f_{NjI}(u))ds\right] ^2\\&=\sum _{N<1}\left[ \frac{E_\varepsilon -1}{2}(\exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) \phi _{NjI}+g_{NjI}\exp \left( tp^{\frac{\alpha (1-N)}{2}}\right) p^{-\frac{\alpha (1-N)}{2}}\right. \\&\quad \left. \,+\int _0^t\exp \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u)ds)+p^{-\frac{\alpha (1-N)}{2}}\right. \\&\quad \left. \Bigg (\int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) (f_{NjI}(u_\varepsilon )ds-f_{NjI}(u)ds)\right] ^2\\&\quad +\,\sum _{N\ge 1}\left[ \int _0^t \sinh (t-s)(f_{NjI}(u_\varepsilon )(s)-f_{NjI}(u))ds\right] ^2. \end{aligned}$$

With inequality $(a+b)^2\le 2a^2+2b^2$ and

$$\begin{aligned} (E_\varepsilon -1)^2&=\Big (\frac{\varepsilon }{\varepsilon +\exp \Big (-Tp^{\frac{\alpha (1-N)}{2}}\Big )}\Big )^2\\&=\varepsilon ^2\Big (\frac{\exp \Big ((t-T)p^{\frac{\alpha (1-N)}{2}}\Big )}{\varepsilon +\exp \Big (-Tp^{\frac{\alpha (1-N)}{2}}\Big )}\Big )^2\exp \Big (2(T-t)p^{\frac{\alpha (1-N)}{2}}\Big )\\&\le \varepsilon ^{2-\frac{2t}{T}}\exp \Big (2(T-t)p^{\frac{\alpha (1-N)}{2}}\Big ), \end{aligned}$$

we have

$$\begin{aligned} ||u_\varepsilon -u||^2&\le \sum _{N<1}\left[ \frac{(E_\varepsilon -1)^2}{2}G^2_N+2t \int _0^t \left( \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) \right) ^2(f_{NjI}(u_\varepsilon )ds\right. \\&\quad \left. -f_{NjI}(u))^2ds\right] \\&\quad +\sum _{N\ge 1}\left[ t\int _0^t (\sinh (t-s))^2(f_{NjI}(u_\varepsilon )(s)-f_{NjI}(u)))^2ds\right] \\&\le \frac{1}{2}\varepsilon ^{2-\frac{2t}{T}}\sum _{N<1}\exp \left( 2(T-t)p^{\frac{\alpha (1-N)}{2}}\right) G^2_N\\&\quad +2t\int _0^t \varepsilon ^{\frac{2s-2t}{T}}\sum (f_{NjI}(u_\varepsilon )ds-f_{NjI}(u))^2ds\\&\le \varepsilon ^{\frac{-2t}{T}}\left\{ \varepsilon ^2V+2C^2T\int _0^t \varepsilon ^{\frac{2s}{T}}||u_\varepsilon -u||^2ds\right\} . \end{aligned}$$

Then, by using Gronwall’s inequality, we have

$$\begin{aligned} ||u_\varepsilon -u||^2&\le V\exp (2C^2T^2)\varepsilon ^{2-\frac{2t}{T}}. \end{aligned}$$

So,

$$\begin{aligned} ||u_\varepsilon -u||&\le \sqrt{ V\exp (2C^2T^2)}\varepsilon ^{\frac{T-t}{T}}. \end{aligned}$$

$\square$

Remark 3.1

To show the numerical computation of the regularlized solution $u_\varepsilon$, we need to divide the time interval into equal subintervals $t_i=\frac{i}{n}t$, $i=0,1,\ldots ,n$, $n\in {\mathbb {N}}$. Then, the nonlinear term $\int _0^t \sinh ^\varepsilon \left( (t-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u_\varepsilon )(s)ds$ and $\int _0^t \sinh (t-s)f_{NjI}(u_\varepsilon )(s)ds$ in (9) can be approximated by following iterative scheme respectively

$$\begin{aligned}&\sum _{j=1}^i\int _{t_{j-1}}^{t_j} \sinh ^\varepsilon \left( (t_i-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u_\varepsilon )(t_{j-1})ds\\&\sum _{j=1}^i\int _{t_{j-1}}^{t_j} \sinh (t_i-s)f_{NjI}(u_\varepsilon )(t_{j-1})ds. \end{aligned}$$

Then, the regularlized solution $u_\varepsilon$ can be approximated by

$$\begin{aligned} u_\varepsilon (t_i,x)&=\sum _{N<1}\left[ \phi _{NjI}\cosh ^\varepsilon \left( t_ip^{\frac{\alpha (1-N)}{2}}\right) +g_{NjI} p^{-\frac{\alpha (1-N)}{2}}\sinh ^\varepsilon \left( t_ip^{\frac{\alpha (1-N)}{2}}\right) \right. \\&\quad \left. +\,p^{-\frac{\alpha (1-N)}{2}} \sum _{j=1}^i\int _{t_{j-1}}^{t_j} \sinh ^\varepsilon \left( (t_i-s)p^{\frac{\alpha (1-N)}{2}}\right) f_{NjI}(u_\varepsilon )(t_{j-1})ds\right] \psi _{NjI}(x)\\&\quad +\sum _{N\ge 1}[\phi _{NjI}\cosh (t_i)+g_{NjI}\sinh (t_i)\\&\quad + \sum _{j=1}^i\int _{t_{j-1}}^{t_j} \sinh (t_i-s)f_{NjI}(u_\varepsilon )(t_{j-1})ds]\psi _{NjI}(x), \end{aligned}$$

where $u_\varepsilon (t_0,x)=u_\varepsilon (0,x)$ determined by the initial condition.