# Cauchy Problems for Evolutionary Pseudodifferential Equations overp-Adic Field

Authors: Bo Wu; Yin Li; Weiyi Su
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study a class of evolutionary pseudodifferential equations of the second order int,  (∂2u(t,x)/∂t2+2a2Tα/2(∂u(t,x)/∂t)+b2Tαu(t,x)+c2u(t,x)=q(t,x)), where t∈(0,z] and Tα is pseudodifferential operator in x∈Qp, which defined by Weiyi Su in 1992. We obtained the exact solutions to the equations which belong to mixed classes of real and p-adic functions.


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## Body

## 1. Introduction

In recent yearsp-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 p-adic field. In 1960s, Gibbs defined logic derivative over dyadic field. Then, Vladimirov et al. [8] generalized logic derivative over p-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 p-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 S(ℚp). This makes the definition of Vladimirov operator difficult to be applied to distribution space S′(ℚp). In 1992, Su [13] redefined derivative and integral operator Tα over p-adic field. The definition makes the operator closed in S(ℚp) and can be extended to its dual space S′(ℚp). 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 int over p-adic field ℚp of the type
(1)∂2u(t,x)∂t2+2a2Tα/2∂u(t,x)∂t+b2Tαu(t,x)+c2u(t,x)=q(t,x),
with initial conditions
(2)u(0,x)=f(x),ut′(0,x)=g(x),
where a, b, c∈ℝ and a4-b2<0, t∈[0,z], using pseudodifferential operator Tα(α>0) which was introduced by Su in [14, 15]. Here, q, f, g are functions given by
(3)q(t,x):[0,z]×ℚp⟶ℂ,f(x):ℚp⟶ℂ,g(x):ℚp⟶ℂ,
and unknown function is
(4)u(t,x):[0,z]×ℚp⟶ℂ.
We will give the existence of the solution u(t,x) to (1) and (2) with the form
(5)u(t,x)=∑N,j,I‍uNjI(t)ψNjI(x),
under some assumptions of f, g, q where ψNjI(x) is an orthonormal base of eigenfunctions of the operator Tα in L2(ℚp), which is constructed by Qiu and Su in [17].


## 2. Preliminaries

We will use the notations and results from Taibleson’s book [16]. Let ℚp be the p-adic field, in which p is a prime number. It is a nondiscrete, locally compact, totally disconnected and complete topological field endowed with nonarchimedean norm |·|:ℚp→ℝ+ satisfying(i)
|x|≥0;|x|=0⇔x=0;(ii)
|xy|=|x||y|;(iii)
|x+y|≤max⁡{|x|,|y|}forx, y∈ℚp, so that it is also ultrametric.Define𝔇 as the ring of integers in ℚp; 𝔇={x∈ℚp:|x|≤1}. It is the unique maximal compact subring in ℚp with the Haar measures |𝔇|=1. Define B={x∈ℚp:|x|≤p-1} as the prime ideal. There exists a prime element β of ℚp with |β|=p-1 such that B=β𝔇. It is the unique maximal ideal in 𝔇. Define the fractional ideal in ℚp as Bk={x∈ℚp:|x|≤p-k} with the Haar measures |Bk|=p-k, k∈ℤ.Forx∈ℚp, it has a unique expression x=xtβt+xt+1βt+1+⋯, t∈ℤ with |x|=p-t. For each l∈ℤ, we choose elements zl,i∈ℚp, i∈ℤ+, so that the subsets Bl,i=zl,i+Bl⊂ℚp satisfy Bl,i∩Bl,j=∅ if i≠j and ∪i=0∞Bl,i=ℚp.Define indicative function of Haar measurable subsetE⊂ℚp as
(6)ΦE(x)={1,x∈E,0,x∈Ec;
then, the Haar measure of E is |E|=∫E‍dx=∫ℚp‍ΦE(x)dx where dx denote the Haar measure on ℚp normalized by the condition ∫𝔇‍dx=1.Define translation operatorτh:f→τhf, h∈ℚp as τhf(x)=f(x-h), x∈ℚp. Then, the test function space S=S(ℚp) is defined as
(7)S(ℚp)={φ:ℚp⟶ℂ,φ(x)=∑j=1n‍cjτhjΦBkj(x),ii∑j=1n‍cj∈ℂ,hj∈ℚ𝕡,kj∈ℤ,1≤j≤n},
where the element φ(x) is called test function.For the test function spaceS, we give the following topology: for φ∈S(ℚ𝕡), there exists unique integers (k,l) such that the function φ is constant on the coset of Bk, with supports in the ball Bl; lim⁡n→+∞φn(x)=0 converges uniformly for x∈ℚ𝕡. Then, S is complete topological linear spaces.Denote byS′=S′(ℚp) the distribution space of test function space S. S′ is a complete topological linear space under the dual topology.Letχ(x) be a fixed nontrivial character of ℚp which is trivial on 𝔇. For the p-adic field, χ can be constructed by the base value [17] as
(8)χ(β-j)={exp⁡(2πipj),forj∈ℕ,1,otherwise.
Then for x=xtβt+xt+1βt+1+⋯,χ(x)=exp⁡(2πi∑j=t-1‍xjpj) and for λ=λτβτ+λτ+1βτ+1+⋯(9)χλ(x)=χ(λx)=exp⁡(2πi∑k=0-(t+τ+1)‍(∑j=0k‍xt+τ-jλt+τ)pt+τ+k).Forφ∈S(ℚp), we define its Fourier transform φ∧ by
(10)φ∧(ξ)=∫ℚpφ(x)χξ¯(x)dx,ξ∈ℚp
and inverse Fourier transform φ∨ by
(11)φ∨(x)=∫ℚpφ(ξ)χx(ξ)dξ,x∈ℚp.In 1992, Su [13] has given definitions of the derivative for the p-adic local fields ℚp, including derivatives of the fractional orders and real orders.Definition 1.
Let〈ξ〉=max⁡{1,|ξ|}, α≥0 if for φ∈S(ℚp), the integral
(12)Tαφ(x)=(〈ξ〉αφ∧(ξ))∨(x)
exists at x∈ℚp, where χ(x) is a fixed nontrivial character of ℚp. Then it is called a pointwise derivative of order α of φ at x.
Note that the defined domain ofTα in the definition can be extended to the space S′(ℚp), where S′(ℚp) denote the set of all functionals (distributions) on S(ℚp).
LetD(Tα) be the domain of Tα defined as
(13)D(Tα)={φ∈L2:〈ξ〉αφ∧∈L2}.
We have the following.Lemma 2.
Consider(14)D(Tα)⊂D(Tα/2),
with α>0.Proof.
Letφ∈D(Tα), then
(15)∫Kp‍|φ∧(ξ)|2dξ<∞,∫Kp‍〈ξ〉2α|φ∧(ξ)|2dξ<∞,
with 〈ξ〉α≤((1+〈ξ〉2α)/2); thus
(16)∫Kp‍〈ξ〉α|φ∧(ξ)|2dξ≤∫Kp‍1+〈ξ〉2α2|φ∧(ξ)|2dξ<∞,
and we have
(17)Tα/2φ∈L2,∥Tα/2φ∥2=∫Kp‍〈ξ〉α|φ∧(ξ)|2dξ.
Then φ∈D(Tα/2).Lemma 3 (see [17]).
T
α is a positive definite self-adjoint operator on D(Tα); {ψNjI} is an orthonormal base of L2 consisting of eigenfunctions of the operator Tα, defined as follows:
(18)ψNjI(x)=p-N/2χj(pN-1x)ΦB0(pNx-zI),iiiiiiiiiiiiiiiiiiiiiN∈ℤ,I=zI+B0∈KpB0,
where ΦB0(x) is a characteristic function of a unit ball.
And(19)TαψNjI(x)={p(1-N)αψNjI(x),N<1,ψNjI(x),N≥1.


## 3. Main Results

We will solve the following pseudodifferential equation overp-adic field by using the orthonormal base {ψNjI} constructed in Lemma 3.First, we consider the case of homogeneous equation.Theorem 4.
Let(20)∂2u(t,x)∂t2+2a2Tα/2∂u(t,x)∂t+b2Tαu(t,x)+c2u(t,x)=0,u(0,x)=f(x),ut′(0,x)=g(x),
where a, b, c∈ℝ, a4-b2<0, α>0, f∈D(Tα), g∈D(Tα/2).
Then one has a formal solution(21)u(t,x)=∑N,j,I‍uNjI(t)ψNjI(x),t∈[0,z],x∈ℚp,
and u(t,x)∈V=C([0,z],D(Tα))∩C1([0,z],D(Tα/2))∩C2([0,z],L2(ℚp)).Proof.
Consider the following.
Step
1. We will write ∑‍ instead of ∑N,j,I‍ in the following proof.
Letu(t,x)=∑‍uNjI(t)ψNjI(x) be the exact form of problem (20); it is a lacunary series. Then
(22)∂2u(t,x)∂t2=∑‍d2uNjI(t)dt2ψNjI(x),Txαu(t,x)=∑N<1‍pα(1-N)uNjI(t)ψNjI(x)+∑N≥1‍uNjI(t)ψNjI(x).
From
(23)∂2u(t,x)∂t2+2a2Tα/2∂u(t,x)∂t+b2Tαu(t,x)+c2u(t,x)=0,
we get
(24)∑N<1‍{uNjI′′(t)+2a2p(α/2)(1-N)uNjI′(t)uNjI′′+b2pα(1-N)uNjI(t)+c2uNjI(t)}ψNjI(x)+∑N≥1‍{uNjI′′(t)+2a2uNjI′(t)uNjI′′(t)+(b2+c2)uNjI(t)}ψNjI(x)=0.
Due to the orthogonality of {ψNjI(x)}, we have
(25)uNjI′′(t)+2a2p(α/2)(1-N)uNjI′(t)+(b2pα(1-N)+c2)uNjI(t)=0,N<1,uNjI′′(t)+2a2uNjI′(t)+(b2+c2)uNjI(t)=0,N≥1.
Then we obtain an ODE of order 2 on ℝ. And the characteristic equation is
(26)λ2+2a2p(α/2)(1-N)λ+b2pα(1-N)+c2=0,N<1,λ2+2a2λ+b2+c2=0,N≥1.
With a4-b2<0, we have
(27)Δ={4[(a4-b2)pα(1-N)-c2],N<1,4[(a4-b2)-c2],N≥1.
The solution of the equation is
(28)uNjI(t)=ANjIe-a2p(α/2)(1-N)tcos⁡(tA)+BNjIe-a2p(α/2)(1-N)tsin(tA),N<1,uNjI(t)=CNjIe-a2tcos⁡(tB)+DNjIe-a2tsin(tB),N≥1,
where A=(b2-a4)pα(1-N)+c2, B=(b2-a4)+c2.
To determine the coefficientsANjI, BNjI, CNjI, and DNjI, we assume that f∈D(Tα) can be expanded as lacunary series f=∑fNjIψNjI(x), where
(29)fNjI=〈f(x),ψNjI(x)〉=∫ℚp‍f(x)ψNjI(x)¯dx,∑‍|fNjI|2<+∞,Tαf(x)=∑N<1‍pα(1-N)fNjIψNjI(x)+∑N≥1‍fNjIψNjI(x),∑N<1‍p2α(1-N)|fNjI|2+∑N≥1‍|fNjI|2<+∞.
With the initial condition u(0,x)=f(x) and then uNjI(0)=fNjI, we obtain
(30)ANjI=fNjI,N<1,CNjI=fNjI,N≥1.
The same as with g∈D(Tα/2), we get g=∑‍gNjIψNjI(x), where
(31)gNjI=〈g(x),ψNjI(x)〉=∫ℚp‍g(x)ψNjI(x)¯dx,∑‍|gNjI|2<+∞,Tαg(x)=∑N<1‍pα(1-N)/2gNjIψNjI(x)+∑N≥1‍gNjIψNjI(x),∑N<1‍pα(1-N)|gNjI|2+∑N≥1‍|gNjI|2<+∞.
With the initial condition u′(0,x)=g(x) and then uNjI′(0)=gNjI, we obtain
(32)BNjI=a2p(α/2)(1-N)AfNjI+1AgNjI,N<1,DNjI=a2BfNjI+1BgNjI,N≥1.
Then the exact solution of the equation is
(33)u(t,x)=∑N<1‍e-a2p(α/2)(1-N)t×[fNjIcos⁡(tA)+(a2p(α/2)(1-N)AfNjI+1AgNjI)(a2p(α/2)(1-N)AfNjI+1AgNjI)×sin(tA)]ψNjI(x)+∑N≥1‍e-a2t[fNjIcos⁡(tB)+(a2BfNjI+1BgNjI)(a2BfNjI+1BgNjI)×sin(tB)]ψNjI(x).
Step
2. We will prove that the solution we obtained in Step  1 satisfies the conditions in Theorem 4.(i)
Consider that(34)0<e-a2p(α/2)(1-N)t≤1;0<e-a2t≤1;|cos⁡(tx)|≤1;|sin(tx)|≤1.Then the series of u(t,x) converges uniformly in L2(ℚp) where t∈[0,z].
With the assumptions off∈D(Tα), g∈D(Tα/2), the series
(35)Txαu(t,x)=∑N<1‍e-a2p(α/2)(1-N)tpα(1-N)×[fNjIcos⁡(tA)+(a2p(α/2)(1-N)AfNjI+1AgNjI)sin(tA)]×ψNjI(x)+∑N≥1‍e-a2t[fNjIcos⁡(tB)+(a2BfNjI+1BgNjI)sin(tB)]×ψNjI(x)
is converging uniformly in L2(ℚp).(ii)
We obtain(36)∂u(t,x)∂t=∑N<1‍e-a2p(α/2)(1-N)t×[1AgNjIcos⁡⁡(tA)-1A(a4pα(1-N)fNjI+a2p(α/2)(1-N)gNjI)×sin(tA)-AfNjIsin(tA)1A]ψNjI(x)+∑N≥1‍e-a2t×[gNjIcos⁡(tB)-1B(a4fNjI+a2gNjI)sin(tB)1B-BfNjIsin(tB)]ψNjI(x)which converges uniformly in L2(ℚp), with f∈D(Tα), g∈D(Tα/2), and
(37)0<e-a2p(α/2)(1-N)t≤1,0<e-a2t≤1.
Furthermore
(38)Txα∂u(t,x)∂t=∑N<1‍e-a2p(α/2)(1-N)tp(α/2)(1-N)×[gNjIcos⁡⁡(tA)-1A(a4pα(1-N)fNjI+a2p(α/2)(1-N)gNjI)×sin(tA)-AfNjIsin⁡(tA)1A]ψNjI(x)+∑N≥1‍e-a2t[gNjIcos⁡(tB)-1B(a4fNjI+a2gNjI)1B×sin(tB)-BfNjIsin(tB)]ψNjI(x)
converges uniformly in L2(ℚp) where t∈[0,z]; then u∈C1([0,z];D(Tα/2)).(iii)
Similarly with the above case, the series(39)∂2u(t,x)∂2t=-(2a2Tα/2∂u(t,x)∂t+b2Tαu(t,x)+c2u(t,x))converges uniformly in L2(ℚp) where t∈[0,z].
Combining (i)–(iii) we obtainu∈V.Next, we will consider the case of nonhomogeneous equation.Theorem 5.
Let(40)∂2u(t,x)∂t2+2a2Tα/2∂u(t,x)∂t+b2Tαu(t,x)+c2u(t,x)=q(t,x),u(0,x)=f(x),∂u(0,x)∂t=g(x),
where a, b, c∈ℝ, a4-b2<0, α>0, f∈D(Tα), g∈D(Tα/2), q(t,x)∈C([0,z],D(Tα)).
Then there exists an exact solutionu(t,x) of equation (40) with the form
(41)u(t,x)=∑N,j,I‍uNjI(t)ψNjI(x),t∈[0,z],x∈ℚp,
and u(t,x)∈V.Proof.
Consider the following.
Step
1. Similarly to the proof of Theorem 4, we expand q(t,x) as lacunary series
(42)q(t,x)=∑‍qNjI(t)ψNjI(x),
where
(43)qNjI(t)=〈q(t,x),ψNjI(x)〉=∫ℚp‍q(t,x)ψNjI(x)¯dx,
and we obtain
(44)∑N<1‍{uNjI′′(t)+2a2p(α/2)(1-N)uNjI′(t)uNjI′′+b2pα(1-N)uNjI(t)+c2uNjI(t)}ψNjI(x)+∑N≥1‍{uNjI′′(t)+2a2uNjI′(t)uNjI′′+(b2+c2)uNjI(t)}ψNjI(x)=∑‍qNjI(t)ψNjI(x).
Due to the orthogonality of {ψNjI(x)}, we get
(45)uNjI′′(t)+2a2p(α/2)(1-N)uNjI′(t)+b2pα(1-N)uNjI(t)+c2uNjI(t)=qNjI(t),N<1,uNjI′′(t)+2a2uNjI′(t)+(b2+c2)uNjI(t)=qNjI(t),N≥1.
It is clear that the exact solution of the equation is
(46)u(t,x)=∑N,j,I‍uNjI(t)ψNjI(x),
with
(47)uNjI(t)=e-a2p(α/2)(1-N)t×[fNjIcos⁡(tA)+(a2p(α/2)(1-N)AfNjI+1AgNjI)+(a2p(α/2)(1-N)AfNjI+1AgNjI)sin(tA)]+1A∫0t‍e-a2p(α/2)(1-N)(t-τ)qNjI(τ)sin((t-τ)A)dτ,0000000000000000000000000000000000000N<1,uNjI(t)=e-a2t×[fNjIcos⁡(tB)+(a2BfNjI+1BgNjI)sin(tB)]+1B∫0t‍e-a2(t-τ)qNjI(τ)sin((t-τ)B)dτ,N≥1.
Step
2. It will be proved that the solution satisfies the conditions of Theorem 5.(i)
Withq(t,x)∈C([0,T],D(Tα)) we obtain
(48)q(t,x)=∑‍qNjI(t)ψNjI(x),where ∑‍|qNjI(t)|2 converges in t∈[0,z] uniformly. Then the series is bounded on [0,z].
Furthermore we get(49)Txαq(t,x)=∑N<1‍pα(1-N)qNjI(t)ψNjI(x)+∑N≥1‍qNjI(t)ψNjI(x),
where
(50)∑N<1‍p2α(1-N)|qNjI(t)|2+∑N≥1‍|qNjI(t)|2
converges in t∈[0,z] uniformly.(ii)
By usingSwartz inequality, we obtain
(51)|∫0t‍e-a2p(α/2)(1-N)(t-τ)qNjI(τ)sin((t-τ)A)dτ|2≤T3A2∫0T‍|qNjI(τ)|2dτ,|∫0t‍e-a2(t-τ)qNjI(τ)sin((t-τ)B)dτ|2≤T3B2∫0T‍|qNjI(τ)|2dτ,and we get
(52)∑N<1‍|1A∫0t‍e-a2p(α/2)(1-N)(t-τ)qNjI(τ)sin((t-τ)A)dτ|2≤T3∫0t‍∑N<1‍|qNjI(τ)|2dτ≤T3∫0t‍Mdτ<∞,(53)∑N≥1‍|1B∫0t‍e-a2(t-τ)qNjI(τ)sin((t-τ)B)dτ|2leqT3×∫0t‍∑N≥1‍|qNjI(τ)|2dτ≤T3∫0t‍Mdτ<∞,
with ∑|qNjI(τ)|2≤M. (iii)
Consider(54)∑N<1‍|pα(1-N)A∫0t‍e-a2p(α/2)(1-N)(t-τ)qNjI(τ)sin((t-τ)A)dτ|2≤T3∫0t‍∑N<1‍(p2α(1-N)|qNjI(τ)|2)dτ<∞.Combining (i)–(iii), we obtain u(t,x)∈V.


## 4. Conclusion

In this work, a class of evolutionary pseudodifferential equations of the second order int over p-adic field ℚp was investigated where Tα is a p-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.



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