Cauchy Problems for Evolutionary Pseudodifferential Equations over 𝑝-Adic Field

Abstract

We study a class of evolutionary pseudodifferential equations of the second order in   , where and is pseudodifferential operator in , which defined by Weiyi Su in 1992. We obtained the exact solutions to the equations which belong to mixed classes of real and -adic functions.

1. Introduction

In recent years -adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1–9] and references therein. The definition of pseudodifferential operator is very important in the theory of PDE on -adic field. In 1960s, Gibbs defined logic derivative over dyadic field. Then, Vladimirov et al. [8] generalized logic derivative over -adic field, and we called the operator referred to as Vladimirov pseudodifferential operator. Chuong et al. have done a lot of work on PDE over -adic field using Vladimirov operator; see, for example, [9–12]. However, as a kind of operation, Vladimirov pseudodifferential operator is not closed in the test function space . This makes the definition of Vladimirov operator difficult to be applied to distribution space . In 1992, Su [13] redefined derivative and integral operator over -adic field. The definition makes the operator closed in and can be extended to its dual space . In 2011, Su [14] has applied the differential operator to study two-dimensional wave equations with fractal boundaries.

In this paper, we consider the exact solutions to the pseudodifferential equations of the second order in over -adic field of the type with initial conditions where , , and , , using pseudodifferential operator which was introduced by Su in [14, 15]. Here, , , are functions given by and unknown function is We will give the existence of the solution to (1) and (2) with the form under some assumptions of , , where is an orthonormal base of eigenfunctions of the operator in , which is constructed by Qiu and Su in [17].

2. Preliminaries

We will use the notations and results from Taibleson’s book [16]. Let be the -adic field, in which is a prime number. It is a nondiscrete, locally compact, totally disconnected and complete topological field endowed with nonarchimedean norm satisfying(i); (ii); (iii)

for , , so that it is also ultrametric.

Define as the ring of integers in ; . It is the unique maximal compact subring in with the Haar measures . Define as the prime ideal. There exists a prime element of with such that . It is the unique maximal ideal in . Define the fractional ideal in as with the Haar measures , .

For , it has a unique expression , with . For each , we choose elements , , so that the subsets satisfy if and .

Define indicative function of Haar measurable subset as then, the Haar measure of is where denote the Haar measure on normalized by the condition .

Define translation operator , as , . Then, the test function space is defined as where the element is called test function.

For the test function space , we give the following topology: for , there exists unique integers such that the function is constant on the coset of , with supports in the ball ; converges uniformly for . Then, is complete topological linear spaces.

Denote by the distribution space of test function space . is a complete topological linear space under the dual topology.

Let be a fixed nontrivial character of which is trivial on . For the -adic field, can be constructed by the base value [17] as Then for and for

For , we define its Fourier transform by and inverse Fourier transform by

In 1992, Su [13] has given definitions of the derivative for the -adic local fields , including derivatives of the fractional orders and real orders.

Definition 1. Let , if for , the integral exists at , where is a fixed nontrivial character of . Then it is called a pointwise derivative of order of at .
Note that the defined domain of in the definition can be extended to the space , where denote the set of all functionals (distributions) on .
Let be the domain of defined as We have the following.

Lemma 2. Consider with .

Proof. Let , then with ; thus and we have Then .

Lemma 3 (see [17]). is a positive definite self-adjoint operator on ; is an orthonormal base of consisting of eigenfunctions of the operator , defined as follows: where is a characteristic function of a unit ball.
And

3. Main Results

We will solve the following pseudodifferential equation over -adic field by using the orthonormal base constructed in Lemma 3.

First, we consider the case of homogeneous equation.

Theorem 4. Let where , , , , , , .
Then one has a formal solution and .

Proof. Consider the following.
Step  1. We will write instead of in the following proof.
Let be the exact form of problem (20); it is a lacunary series. Then From we get Due to the orthogonality of , we have Then we obtain an ODE of order 2 on . And the characteristic equation is With , we have The solution of the equation is where , .
To determine the coefficients , , , and , we assume that can be expanded as lacunary series , where With the initial condition and then , we obtain The same as with , we get , where With the initial condition and then , we obtain Then the exact solution of the equation is
Step  2. We will prove that the solution we obtained in Step  1 satisfies the conditions in Theorem 4.(i)Consider that Then the series of converges uniformly in where .
With the assumptions of , , the series is converging uniformly in .(ii)We obtain which converges uniformly in , with , , and Furthermore converges uniformly in where ; then .(iii)Similarly with the above case, the series converges uniformly in where .
Combining (i)–(iii) we obtain .

Next, we will consider the case of nonhomogeneous equation.

Theorem 5. Let where , , , , , , , .
Then there exists an exact solution of equation (40) with the form and .

Proof. Consider the following.
Step  1. Similarly to the proof of Theorem 4, we expand as lacunary series where and we obtain Due to the orthogonality of , we get It is clear that the exact solution of the equation is with
Step  2. It will be proved that the solution satisfies the conditions of Theorem 5.(i)With we obtain where converges in uniformly. Then the series is bounded on .
Furthermore we get where converges in uniformly.(ii)By using Swartz inequality, we obtain and we get with . (iii)Consider Combining (i)–(iii), we obtain .

4. Conclusion

In this work, a class of evolutionary pseudodifferential equations of the second order in over -adic field was investigated where is a -adic pseudodifferential operator defined by Su Weiyi. The exact solution to the equation was obtained and the uniform convergence of the series of the formal solution was constructed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant no. 13KJB110010), the Pre Study Foundation of Nanjing University of Finance & Economics (Grant no. YYJ2013016), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors thank the referee for his (or her) suggestions from which the present paper has benefited.