Cauchy Problems for Evolutionary Pseudodifferential Equations over p-Adic Field

and Applied Analysis 3 Proof. Let φ ∈ D(T), then ∫ Kp 󵄨󵄨󵄨󵄨 φ ∧ (ξ) 󵄨󵄨󵄨󵄨 2 dξ < ∞, ∫ Kp ⟨ξ⟩ 2α󵄨󵄨󵄨󵄨 φ ∧ (ξ) 󵄨󵄨󵄨󵄨 2 dξ < ∞, (15) with ⟨ξ⟩α ≤ ((1 + ⟨ξ⟩2α)/2); thus ∫ Kp ⟨ξ⟩ α󵄨󵄨󵄨󵄨 φ ∧ (ξ) 󵄨󵄨󵄨󵄨 2 dξ ≤ ∫ Kp 1 + ⟨ξ⟩ 2α 2 󵄨󵄨󵄨󵄨 φ ∧ (ξ) 󵄨󵄨󵄨󵄨 2 dξ < ∞,


Introduction
In recent years -adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1][2][3][4][5][6][7][8][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][10][11][12].However, as a kind of operation, Vladimirov pseudodifferential operator is not closed in the test function space (Q  ).This makes the definition of Vladimirov operator difficult to be applied to distribution space   (Q  ).In 1992, Su [13] redefined derivative and integral operator   over -adic field.The definition makes the operator closed in (Q  ) and can be extended to its dual space   (Q  ).In 2011, Su [14] has applied the differential operator to study two-dimensional wave equations with fractal boundaries.

Preliminaries
We will use the notations and results from Taibleson's book [16].Let Q  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 | ⋅ | : for ,  ∈ Q  , so that it is also ultrametric.Define D as the ring of integers in For  ∈ Q  , it has a unique expression  =     +  +1  +1 + ⋅ ⋅ ⋅ ,  ∈ Z with || =  − .For each  ∈ Z, we choose elements  , ∈ Q  ,  ∈ Z + , so that the subsets then, the Haar measure of where  denote the Haar measure on Q  normalized by the condition where the element () is called test function.
For the test function space , we give the following topology: for  ∈ (Q p ), there exists unique integers (, ) such that the function  is constant on the coset of   , with supports in the ball   ; lim  → +∞   () = 0 converges uniformly for  ∈ Q p .Then,  is complete topological linear spaces.
Denote by   =   (Q  ) the distribution space of test function space .  is a complete topological linear space under the dual topology.
Let () be a fixed nontrivial character of Q  which is trivial on D. For the -adic field,  can be constructed by the base value [17] as Then for For  ∈ (Q  ), 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 Q  , including derivatives of the fractional orders and real orders.
exists at  ∈ Q  , where () is a fixed nontrivial character of Q  .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   (Q  ), where   (Q  ) denote the set of all functionals (distributions) on (Q  ).
Let (  ) be the domain of   defined as We have the following.
Lemma 3 (see [17]).  is a positive definite self-adjoint operator on (  ); {  } is an orthonormal base of  2 consisting of eigenfunctions of the operator   , defined as follows: where Φ  0 () is a characteristic function of a unit ball.And

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.
Then one has a formal solution and (, ) Proof.Consider the following.
Step 1.We will write ∑ instead of ∑ ,, in the following proof.
To determine the coefficients   ,   ,   , and   , we assume that  ∈ (  ) can be expanded as lacunary series  = ∑     (), where With the initial condition (0, ) = () and then   (0) =   , we obtain The same as with  ∈ ( /2 ), we get  = ∑     (), where With the initial condition   (0, ) = () and then    (0) =   , 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 With the assumptions of  ∈ (  ),  ∈ ( /2 ), the series is converging uniformly in  2 (Q  ).
Next, we will consider the case of nonhomogeneous equation.Proof.Consider the following.
Step 1. Similarly to the proof of Theorem 4, we expand (, ) as lacunary series where and we obtain  (45) 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.

Conclusion
In this work, a class of evolutionary pseudodifferential equations of the second order in  over -adic field Q  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.