solvingnonlinear p-adicpseudo-differentialequations ......recently theory of p-adic wavelets started...

Journal of Fourier Analysis and Applications (2020) 26:70https://doi.org/10.1007/s00041-020-09779-x

Solving Nonlinear p-Adic Pseudo-differential Equations:Combining theWavelet Basis with the Schauder Fixed PointTheorem

Ehsan Pourhadi1 · Andrei Yu. Khrennikov1 · Klaudia Oleschko2 ·María de Jesús Correa Lopez3

Received: 25 December 2019 / Published online: 14 August 2020© The Author(s) 2020

AbstractRecently theory of p-adic wavelets started to be actively used to study of the Cauchyproblem for nonlinear pseudo-differential equations for functions depending on thereal time and p-adic spatial variable. These mathematical studies were motivated byapplications to problems of geophysics (fluids flows through capillary networks inporous disordered media) and the turbulence theory. In this article, using this wavelettechnique in combination with the Schauder fixed point theorem, we study the solv-ability of nonlinear equations with mixed derivatives, p-adic (fractional) spatial andreal time derivatives. Furthermore, in the linear case we find the exact solution for theCauchy problem. Some examples are provided to illustrate the main results.

Keywords Pseudo-differential equations · p-adic field · p-adic wavelet basis ·Schauder fixed point theorem · Arzelà–Ascoli theorem

Communicated by Hans G. Feichtinger.

B Andrei Yu. [email protected]

Ehsan [email protected]

Klaudia [email protected]

María de Jesús Correa [email protected]

1 International Center for Mathematical Modelling in Physics and Cognitive Sciences MSI,Linnaeus University, 351 95 Växjö, Sweden

2 Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico(UNAM), Blvd. Juriquilla 3001, 76230 Querétaro, Mexico

3 Edificio Piramide, Boulevard Adolfo Ruiz Cortines 1202, Oropeza, 86030 Villahermosa, Tabasco,Mexico

http://crossmark.crossref.org/dialog/?doi=10.1007/s00041-020-09779-x&domain=pdf

70 Page 2 of 23 Journal of Fourier Analysis and Applications (2020) 26 :70

Mathematics Subject Classification 35S10 · 42B35 · 47H10

1 Introduction

During recent 30 years, p-adic analysis has received a lot of attention through itsapplications to mathematical physics, string theory, quantum mechanics, dynamicalsystems, turbulence, cognitive sciences, and recently geophysics, see e.g. [5–7,11,13,15,16,19–21,26,28,31,39,40] and references therein. It iswell-known that the theory ofp-adic distributions (generalized functions) and the corresponding Fourier andwaveletanalysis play an important role in solving mathematical problems and applications inaforementioned fields.

In p-adic analysis, which is associated with maps Qp → C, the operation ofdifferentiation is well not defined. For such reason, p-adic modeling widely utilizesthe calculus of pseudo-differential operators. In this calculus, the crucial role is playedby the fractional differentiation operator Dα (the Vladimirov operator). The pseudo-differential equations over p-adic fields have been studied in numerous publications[3,8,12,17–19,22,24,25,27,32,34–37]. But up to now, in almost all models, only linearand semilinear pseudo-differential equations have been considered (see also [2,4,9,10,28,41,42]). For instance, we can mention recent paper [30] devoted to the studyof two classes of semi-linear pseudo-differential equations via the use of the p-adicwavelet functions and the Adomian decomposition method.

It seems that the first nonlinear p-adic pseudo-differential equation was studiedby Kozyrev [26] (at least an equation that is interesting for physical applications—modeling of turbulence). This was also the first application of the p-adic wavelet basisfor study of nonlinear equations.

In paper [16] there was considered a p-adic analogue of one of the most importantfor applications to geophysics nonlinear equations, the porous medium equation (see[38]), that is the equation

∂u

∂t+ Dα(ϕ(u)), u = u(t, x), t > 0, x ∈ Qp,

where ϕ is a strictly monotone increasing continuous real function satisfying |ϕ(s)| ≤Csm for s ∈ R (C > 0,m ≥ 1) and Dα , α > 0 is Vladimirov’s fractional differentia-tion operator.

By the construction of Markov process in the balls, Antoniouk et al. [5] studiedthe Cauchy problem for p-adic nonlinear evolutionary pseudo-differential equationsover the p-adic balls and gave a formula for the solution of these equations. With thehelp of Crandall–Liggett theorem together with the concept of m-accretive nonlinearoperators, they also revealed a result in order to prove the existence of a uniquemild solution for a nonlinear equation including a generator of the semigroup T (t) inL1(Qnp).

In 2019, Pourhadi et al. [31] studied a class of nonlinear p-adic pseudo-differentialequation as the p-adic analogue of the Navier-Stokes equation (see Oleschko et al.[21] for derivation) using the Schauder fixed point theorem together with Adomian

Journal of Fourier Analysis and Applications (2020) 26 :70 Page 3 of 23 70

decomposition method to find some initial terms of the solution. This equation modelsthe propagation of fluid’s flow throughGeo-conduits, including themixture of fractures(as well as fracture’s corridors) and capillary networks.

To proceed our investigation, in the current paper we aim to employ the sametechnique but on the more generalized forms of pseudo-differential equations withnonlinear term F(t, u, x) and initial conditions. This problem is also a generalizedform of the model proposed by Chuong and Co [10]. It is worth pointing out that theterm F has not been observed in the previous results with focus on the wavelet theoryand existence results. We also present an explicit form for the solution in the terms ofwavelet basis for the certain cases.

Throughout this paper, we investigate the solvability of following IVP problem fora class of nonlinear pseudo-differential equations1 over the p-adic field in I = [0, T ]given as

⎧⎪⎪⎨

⎪⎪⎩

Dα∂2u(t, x)

∂t2+ 2aDγ ∂u(t, x)

∂t+ bDβu(t, x) + cu(t, x) = F(t, u, x),

x ∈ Qp, t ∈ (0, T ],u(0, x) = f (x), u′t (0, x) = g(x), x ∈ Qp, t = 0,

(1.1)

such that Dα, Dβ, Dγ are the fractional operators with orders α, β, γ , respectively,and a, b, c ≥ 0 where either 0 ≤ b ≤ a2 < c or 0 ≤ c ≤ a2 < b holds and not both.Besides, let us suppose that

γ := γ (α, β) =

⎧⎪⎨

⎪⎩

α

2a2 < c,

α + β2

a2 < b.

In this work, as special case, when F is independent from the term u, that is, F(t, x),we present the exact solution for the Cauchy problem (1.1).

In Sect. 2, we give some fundamental and auxiliary facts in order to proceed withthe development of our work and conclude our results. Section 3 deals with the presentof the solution to the homogeneous form of the nonlinear pseudo-differential equation(1.1) over the p-adic field Qp. Section 4 dedicates to investigate the study of thesolution for a linear pseudo-differential equation over the p-adic field Qp which isalso deduced by considering F independent from u. Finally, in Sect. 5 we establishthe existence of the solution for IVP (1.1) as our main problem.

2 Preliminaries

In what follows, for a prime number p, we denote by Qp the field of p-adic numbersand by Zp the ring of p-adic integers. Considering x �= 0 inQp, ord(x) ∈ Z∪ {+∞}stands for the valuation of x , i.e. p-adic order of x , and |x |p = p−ord(x) its absolutevalue which possesses the following properties:

1 The linear equations of this class were invented and studied by Chuong and Co [10]


(i) |x |p ≥ 0 for every x ∈ Qp, and |x |p = 0 if and only if x = 0;(ii) |xy|p = |x |p|y|p for every x, y ∈ Qp;(iii) |x + y|p ≤ max{|x |p, |y|p}, for every x, y ∈ Qp, and when |x |p �= |y|p, we

have |x + y|p = max{|x |p, |y|p},which also shows that the norm | · |p is non-Archimedean and the space (Qp, | · |p)

is an ultrametric space.A canonical form of any p-adic number x ∈ Qp, x �= 0, is represented as follows

x =∞∑

j=γx j p

j (2.1)

where γ = γ (x) ∈ Z, and xk = 0, 1, . . . , p − 1, x0 �= 0, k = 0, 1, . . . This seriesconverges in the p-adic norm | · |p to p−γ . The fractional part of a p-adic numberx ∈ Qp defined by (2.1) is given as

{x}p =⎧⎨

⎩

0, if γ (x) ≥ 0 or x = 0,

pγ (x0 + x1 p + x2 p2 + · · · + x|γ |−1 p|γ |−1), if γ (x) < 0.(2.2)

The standard additive character χp of the field Qp is given by

χp(x) = e2π i{x}p , x ∈ Qp.

For the topology induced by | · |p in Qp we assume that

Bγ (a) = {x ∈ Qp : |x − a|p ≤ pγ },Sγ (a) = {x ∈ Qp : |x − a|p = pγ }

are ball and sphere of radius pγ with center at a, respectively. For the convenience,we suppose Bγ (0) = Bγ and Sγ (0) = Sγ . Recall that any point of the ball is itscenter, besides, any two balls inQp are either disjoint or one is contained in the other.Moreover, sets of all balls and spheres are open and closed sets (i.e. clopen) in Qp.

The topological group (Qp,+) is locally compact commutative and thus there isan additive Haar measure dx , which is positive and invariant under the translation,i.e., d(x + a) = dx, a ∈ Qp. This measure is unique by normalizing dx so that

∫

B0dx = 1, d(ax + b) = |a|pdx, a ∈ Q∗p = Qp − {0}.

Further, regarding with the additive normalized character χp(x) on Qp we have

∫

Bγχp(ξ x)dx = pγ (pγ |ξ |p),

where (t) is the characteristic function of the interval [0, 1] ⊂ R.


We say the complex-valued function f defined on Qp is locally constant if forany x ∈ Qp, there exists an integer l(x) ∈ Z such that f (x + y) = f (x), forevery y ∈ Bl(x). We signify by E(Qp) the linear space of such functions in Qp. ByD(Qp) we mean the subspace of E(Qp) consisting of locally constant functions withcompact support (so-called test function). Besides, denote by D′(Qp) the set of alllinear functionals on D(Qp) (see also [39, VI.3]).

The Fourier transform of test function ϕ ∈ D(Qp) is given by

ϕ̂(ξ) = F[ϕ](ξ) =∫

Qp

ϕ(x)χp(ξ x)dx .

Besides, ϕ̂(ξ) ∈ D(Qp) and ϕ(x) = F−1[ϕ](ξ) =∫

Qpϕ̂(ξ)χp(−ξ x)dξ as the

inverse Fourier transform.Suppose L2(Qp) is the set of measurable C-valued functions f on Qp such that

‖ f ‖L2(Qp) =( ∫

Qp

| f (x)|2dx) 1

2

< ∞,

which is clearly a Hilbert space with the inner product

〈 f , g〉 =∫

Qp

f (x)g(x)dx, f , g ∈ L2(Qp),

and ‖ f ‖2L2(Qp)

= 〈 f , f 〉.Hence, there is a linear isomorphism taking D(Qp) onto D(Qp) which also can be

uniquely extended to a linear isomorphism of L2(Qp). Furthermore, the Plancherelequality holds

〈 f , g〉 = 〈 f̂ , ĝ〉, f , g ∈ L2(Qp).

In 1910, Haar [14] initially introduced the wavelet basis by presenting an orthonor-mal basis in L2(R) including dyadic translations and dilations of a single function:

ψHjn(x) = 2−j2 ψH (2− j x − n), x ∈ R, j, n ∈ Z (2.3)

where

ψH (x) = χ[0, 12 )(x) − χ[ 12 ,1](x)

is called a Haar wavelet and χA denotes the characteristic function of a set A ⊂ R.The generalization of Haar basis (2.3) has been studied in various results. In 2002, abasis of complex-valued wavelets with compact support in L2(Qmp ) has been initially


introduced by Kozyrev [23] (see also [24,25,27]). This basis is comparable to the Haarbasis and takes the following form

ψk; jn(x) = p−mj2 χp(p

−1k · (p j x − n))(|p j x − n|p), x ∈ Qmp (2.4)

where k ∈ Jmp := Jp × Jp × · · · × Jp←−−−m−−−→, Jp = {1, 2, . . . , p − 1}, j ∈ Z, and n can be

taken as an element of the m-direct product of factor group

Qp/Zp ={ −1∑

i=ani p

i∣∣∣∣ ni = 0, 1, . . . , p − 1, a ∈ Z−

}

and here,χp and are the standard additive character ofQp and characteristic functionof [0, 1], respectively, as defined before.

Assume the following subspaces of the test functions from D(Qp)

� = �(Qp) = {ψ ∈ D(Qp), ψ(0) = 0},

= (Qp) = {φ : φ = F[ψ], ψ ∈ �}.

It is obvious to see that �, �= ∅. Regarding with the fact that Fourier transformis a linear isomorphism D(Qp) into D(Qp), we get �, ∈ D(Qp). To describe thespace we remark that φ ∈ if and only if φ ∈ D(Qp) and

∫

Qpφ(x)dx = 0. The

space is called the p -adic Lizorkin space of test functions of the first kind whichis a complete space under the topology of the space D(Qp). Furthermore, the space

′ = ′(Qp) is said to be the p -adic Lizorkin space of distributions of the first kindwhich is the topological dual space of (Qp) (see also [1]).

The fractional operator Dα : ϕ → Dαϕ is defined as a convolution of the followingfunctions:

Dαϕ(x) = f−α(x) ∗ ϕ(x) = 〈 f−α(x), ϕ(x − ξ)〉, ϕ ∈ ′(Qp), α ∈ C,

where the distribution fα ∈ ′(Qp) is called the Riesz kernel given by

fα(x) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

|x |α−1p�p(α)

, if α �= 0, 1,

δ(x), if α = 0,p−1 − 1log p

log |x |p, if α = 1,

x ∈ Qp

and �p(α) = 1−pα−11−p−α is the �-function (for more details see [39]).The domain of Dα is given by

M(Dα) = {ϕ ∈ L2(Qp) | Dαϕ ∈ L2(Qp)}.


3 The Homogeneous Cauchy Problem

Throughout this section we are interested in the study of the solution to the homoge-neous form of the nonlinear pseudo-differential equation (1.1).

Theorem 3.1 Suppose that f ∈ M(Dβ), g ∈ M(Dγ ) and fk; jn and gk; jn are thecorresponding coefficients of the series of f , g in terms of the orthonormal functions{ψk; jn(x)}, respectively. Then the homogeneous form of pseudo-differential equation(1.1), that is,

⎧⎨

⎩Dα

∂2u(t, x)

∂t2+ 2aDγ ∂u(t, x)

∂t+ bDβu(t, x) + cu(t, x) = 0, x ∈ Qp, t ∈ (0, T ],

u(0, x) = f (x), u′t (0, x) = g(x), x ∈ Qp, t = 0,(3.1)

where the constants a, b, c are the same as given for Eq. (1.1), possesses a uniquesolution of the form

u(t, x) =∑

J

uk; jn(t)ψk; jn(x), s.t. J := Jp × Z × Qp/Zp

belonging to U = C(I ,M(Dβ)) ∩ C1(I ,M(Dγ )) ∩ C2(I ,M(Dα)), in whichuk; jn(t) = exp(−atp(γ−α)(1− j))

[

fk; jn cos(A j t) +(ap(γ−α)(1− j)

A jfk; jn + 1

A jgk; jn

)

sin(A j t)

]

, ∀(k, j, n) ∈ J ,and

A j := p−α(1− j)√

bp(α+β)(1− j) − a2 p2γ (1− j) + cpα(1− j).Proof Assume that u(t, x) is the solution what we are looking for. Considering u interms of wavelet functions ψk; jn(x) with coefficients uk; jn(t) one can easily arrive atthe following linear differential equation.

pα(1− j)u′′k; jn(t) + 2apγ (1− j)u′k; jn(t)+bpβ(1− j)uk; jn(t) + cuk; jn(t) = 0, ∀(k, j, n) ∈ J . (3.2)

The corresponding characteristic equation is as follows:

pα(1− j)λ2 + 2apγ (1− j)λ + bpβ(1− j) + c = 0.Moreover, the discriminant of above quadratic equation is given as

�′j = a2 p2γ (1− j) − bp(α+β)(1− j) − cpα(1− j).


The definition of γ immediately implies that �′j < 0 and then it follows that

uk; jn(t) = exp(−atp(γ−α)(1− j))[Mk; jn cos(A j t) + Nk; jn sin(A j t)], ∀(k, j, n) ∈ J ,(3.3)

where

A j := p−α(1− j)√

bp(α+β)(1− j) − a2 p2γ (1− j) + cpα(1− j) > 0. (3.4)

Now, imposing the initial conditions in our obtained solution we derive

Mk; jn = fk; jn, Nk; jn = ap(γ−α)(1− j)

A jfk; jn + 1

A jgk; jn (3.5)

where fk; jn and gk; jn are respectively the components of f and g in the correspondingrepresentations based on wavelet functions ψk; jn(x).

Therefore, the solution u(t, x) of the homogeneous equation (3.1) is defined by

u(t, x) =∑

J

exp(−atp(γ−α)(1− j))[

fk; jn cos(A j t) +(ap(γ−α)(1− j)

A jfk; jn + 1

A jgk; jn

)

sin(A j t)

]

ψk; jn(x)

(3.6)

such that A j is given as (3.4). Since t ∈ I and a > 0 one can observe that

0 < exp(−atp(γ−α)(1− j)) ≤ 1. (3.7)

This shows that the series (3.6) converges in L2(Qp) uniformly in t ∈ I .Considering the hypotheses f ∈ M(Dβ), g ∈ M(Dγ )we immediately derive that

Dβx u(t, x) =∑

J

exp(−atp(γ−α)(1− j))

×[

pβ(1− j) fk; jn cos(A j t) + pβ(1− j)(ap(γ−α)(1− j)

A jfk; jn + 1

A jgk; jn

)

sin(A j t)

]

ψk; jn(x),

∂u(t, x)

∂t=

∑

J

exp(−atp(γ−α)(1− j))Sk; jn(t)ψk; jn(x),

where

Sk; jn(t) = gk; jn cos(A j t) − 1A j

(

a2 p2(γ−α)(1− j) fk; jn + ap(γ−α)(1− j)gk; jn)

sin(A j t).


Furthermore, using (3.7) we get

Dγx∂u(t, x)

∂t=

∑

J

pγ (1− j) exp(−atp(γ−α)(1− j))Sk; jn(t)ψk; jn(x),

which means that all the series as above are convergent in L2(Qp) uniformly in t ∈ I .That is, u ∈ C(I ,M(Dβ)) ∩ C1(I ,M(Dγ )). With similar reasoning, one can seethat both series with respect to ∂

2u∂t2

and Dα ∂2u

∂t2converge in L2(Qp) uniformly in t ∈ I

and hence u given as (3.6) is unique and also belongs to U . ��Example 1 Suppose that the problem (3.1) takes the following form over I × Qp:

⎧⎪⎪⎨

⎪⎪⎩

D32∂2u(t, x)

∂t2+ 2D1 ∂u(t, x)

∂t+ 2D 12 u(t, x) + u(t, x) = 0, x ∈ Qp, t ∈ (0, T ],

u(0, x) = u′t (0, x) = (|x |p), x ∈ Qp, t = 0.

Then

fk; jn = gk; jn = 〈g, ψk; jn〉 =∫

Qp

(|x |p) · ψk; jn(x)dx

= p −3 j2∫

Qp

(|p− j (ξ + n)|p) · (|ξ |p) · χp(p−1kξ)dξ

= p −3 j2∫

Qp

(p j max{|ξ |p, |n|p}) · (|ξ |p) · χp(p−1kξ)dξ

where k = 0, 1, 2, . . . , p−1 and n ∈ Qp/Zp. Assuming |n|p = p−γ for some integerγ ≤ −1 together with the fact that (|ξ |p) �= 0 if and only if ξ ∈ Sr for some r ≤ 0,we derive

fk; jn = gk; jn = p−3 j2 (p j−γ )

∑

r≤0

∫

Srχp(p

−1kξ)dξ

= p −3 j2 (p j−γ )∑

r≤0

( ∫

Brχp(p

−1kξ)dξ −∫

Br−1χp(p

−1kξ)dξ)

.

If k = 0 then

f0; jn = g0; jn = p−3 j2 (p j−γ ) = p −3 j2 (p j−ordp(n)).

Otherwise, for the case 1 ≤ k ≤ p − 1, using the formula∫

Brχp(ξ x)dx =

{pr , |ξ |p ≤ p−r ,0, |ξ |p ≥ p−r+1, r ∈ Z,


we see that

fk; jn = gk; jn = p−3 j2 (p j−ordp(n))

(

− p−1 +∑

r≤−1(pr − pr−1)

)

= 0.

Hence the solution of the problem is as follows

u(t, x) =∑

j≤−1

∑

n∈Qp/Zpj≤ordp (n)

u0; jn(t)ψ0; jn(x),

where

u0; jn(t) = exp(−tp−0.5(1− j))p−1.5 j[

cos(A j t) + p0.5(1− j)√

1 + p−0.5(1− j) sin(A j t)]

,

and A j is defined in

A j := p−0.5(1− j)√

1 + p−0.5(1− j).

4 Cauchy Problem for a Linear Pseudo-differential Equation

This section dedicates to investigate the existence of solution for the following linearpseudo-differential equation over the p-adic field Qp in I = [0, T ] given as

⎧⎨

⎩Dα

∂2u(t, x)

∂t2+ 2aDγ ∂u(t, x)

∂t+ bDβu(t, x) + cu(t, x) = F(t, x), t ∈ (0, T ],

u(0, x) = f (x), u′t (0, x) = g(x), x ∈ Qp, t = 0,(4.1)

where the coefficients are defined same as ones given for Eq. (1.1).In order to present the result of this section we need the following lemma.

Lemma 1 For M(Dα) the following inclusion holds true:

M(Dα) ⊆ M(Dβ), 0 < β < α.

Proof Suppose that ϕ ∈ M(Dα), then∫

Qp

|ϕ̂(ξ)|2dξ < ∞ and∫

Qp

|ξ |2αp |ϕ̂(ξ)|2dξ < ∞.


On the other hand, since 0 < β < α we get

∫

Qp

|ξ |2βp |ϕ̂(ξ)|2dξ ≤ max{∫

Qp

|ϕ̂(ξ)|2dξ,∫

Qp

|ξ |2αp |ϕ̂(ξ)|2dξ}

< ∞,

that is,

Dβϕ ∈ L2(Qp), ‖Dβϕ‖2L2(Qp) =∫

Qp

|ξ |2βp |ϕ̂(ξ)|2dξ < ∞.

Hence, ϕ ∈ M(Dβ). ��Theorem 4.1 Suppose that f ∈ M(Dβ), g ∈ M(Dγ ), fk; jn and gk; jn are the cor-responding coefficients of the series of f , g in terms of the orthonormal functions{ψk; jn(x)}, respectively. Further, assume that |γ −α| < β. Then the non-homogeneousequation (4.1) has a unique solution of the form

u(t, x) =∑

J

uk; jn(t)ψk; jn(x), s.t. J := Jp × Z × Qp/Zp,

belonging to U where

uk; jn(t) = exp(−atp(γ−α)(1− j))(

fk; jn cos(A j t) + 1A j

(

ap(γ−α)(1− j) fk; jn + gk; jn)

sin(A j t)

)

+ 1A j

∫ t

0sin(A j (t − r)) exp(a(r − t)p(γ−α)(1− j)) · Fk; jn(r)dr .

Proof As shown in the proof of Theorem 3.1, we have

pα(1− j)u′′k; jn(t) + 2apγ (1− j)u′k; jn(t)+bpβ(1− j)uk; jn(t) + cuk; jn(t) = Fk; jn(t), ∀(k, j, n) ∈ J

where

Fk; jn(t) = 〈F(t, .), ψk; jn〉L2(Qp) =∫

Qp

F(t, x)ψk; jn(x)dx, t ∈ I .

Now, applying the variation of parametersmethod for the solutions (3.3)with constants(3.4) and (3.5) we present the solution of problem by the following

uk; jn(t) = exp(−atp(γ−α)(1− j))[Mk; jn(t) cos(A j t) + Nk; jn(t) sin(A j t)],∀(k, j, n) ∈ J , t ∈ I ,


where Mk; jn(t), Nk; jn(t) are the unknown functions which are found from the fol-lowing system:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

M ′k; jn(t) cos(A j t) + N ′k; jn(t) sin(A j t) = 0,M ′k; jn(t)

(

− A j sin(A j t) − ap(γ−α)(1− j) cos(A j t))

+

N ′k; jn(t)(

A j cos(A j t) − ap(γ−α)(1− j) sin(A j t))

= exp(atp(γ−α)(1− j)) · Fk; jn(t)

which yields that

Mk; jn(t) = −1A j

∫ t

0sin(A jr) exp(arp

(γ−α)(1− j)) · Fk; jn(r)dr + mk; jn,

Nk; jn(t) = 1A j

∫ t

0cos(A jr) exp(arp

(γ−α)(1− j)) · Fk; jn(r)dr + nk; jn .

where

mk; jn = fk; jn, nk; jn = 1A j

(


.

Hence, the unique solution of the problem is

u(t, x) =∑

J

uk; jn(t)ψk; jn(x), (4.2)

such that

uk; jn(t) = exp(−atp(γ−α)(1− j))(

fk; jn cos(A j t) + 1A j

(


sin(A j t)

)

+ 1A j

∫ t

0sin(A j (t − r)) exp(a(r − t)p(γ−α)(1− j)) · Fk; jn(r)dr .

Let us now show that the obtained solution belongs to U .Taking into account that

0 < exp(−atp(γ−α)(1− j)) ≤ 1, 0 ≤{

| sin(A j t)|, | cos(A j t)|}

≤ 1,

it is clear that the series (4.2) is convergent in L2(Qp) uniformly in I . We also notethat since |γ − α| < β, then f ∈ M(D|γ−α|) (see Lemma 1).

On the other hand, since f ∈ M(Dβ), g ∈ M(Dγ ), then the series correspondingto Dβu(t, x) converges in L2(Qp) uniformly in I . Hence u ∈ C(I ,M(Dβ)).


Moving forward, we have

∂u(t, x)

∂t=

∑

J

Sk; jn(t)ψk; jn(x),

where

Sk; jn(t) = exp(−atp(γ−α)(1− j))(

− ap(γ−α)(1− j)[

fk; jn cos(A j t)

+ 1A j

(


sin(A j t)

]

− A j fk; jn sin(A j t) +(


cos(A j t)

)

+ 1A j

∫ t

0exp(a(r − t)p(γ−α)(1− j))

[

A j cos(A j (t − r))

− ap(γ−α)(1− j) sin(A j (t − r))]

· Fk; jn(r)dr

converging in L2(Qp) uniformly over the interval I since f , g ∈ M(D|γ−α|) and

0 <

{

exp(−atp(γ−α)(1− j)), exp(a(r − t)p(γ−α)(1− j))}

< 1.

Besides,

Dγ∂u(t, x)

∂t=

∑

J

pγ (1− j)Sk; jn(t)ψk; jn(x)

is convergent in L2(Qp) uniformly in I , which means u ∈ C1(I ,M(Dγ )). Similarly,the series

Dα∂2u(t, x)

∂t2=

∑

J

pα(1− j)Sk; jn(t)ψk; jn(x)

converges in L2(Qp) uniformly in I where

Sk; jn(t) = exp(−atp(γ−α)(1− j))(

a2 p2(γ−α)(1− j)[

fk; jn cos(A j t)

+ 1A j

(


sin(A j t)

]

− 2ap(γ−α)(1− j)[

− fk; jn A j sin(A j t) +(


cos(A j t)

]


− A2j fk; jn cos(A j t) − A j(


sin(A j t)

)

+ Fk; jn(t)

+ 1A j

∫ t

0exp(a(r − t)p(γ−α)(1− j))

[

− 2ap(γ−α)(1− j)A j cos(A j (t − r))

+ (a2 p2(γ−α)(1− j) − A2j ) sin(A j (t − r))]

· Fk; jn(r)dr , ∀t ∈ I ,

and u ∈ C2(I ,M(Dα)), that is, u ∈ U . Therefore, the function (4.2) is the uniquesolution to the problem (4.1) which satisfies the mentioned initial conditions. ��

Example 2 Consider the problem (4.1) with a similar constants given in Example 1.Suppose that f (x) = g(x) = ln |x |p and non-homogeneity term F(t, x) = t ln |x |p.To write the function F in terms of basis ψk; jn , using notation ξ = p j x − n we have

Fk; jn(t) = 〈F, ψk; jn〉 =∫

Qp

t ln |x |p · ψk; jn(x)dx

= tp −3 j2∫

Qp

ln(|p− j (ξ + n)|p) · (|ξ |p) · χp(p−1kξ)dξ

= tp −3 j2∫

Qp

ln(p j max{|ξ |p, |n|p}) · (|ξ |p) · χp(p−1kξ)dξ

= tp −3 j2 ( j ln p + ln |n|p)∫

Qp

(|ξ |p) · χp(p−1kξ)dξ.

If k = 0 then

F0; jn(t) = tp−3 j2 ln p · ( j − ordp(n)),

otherwise Fk; jn(t) = 0. Similarly, we can discuss on the values of fk; jn, gn; jn andderive

fk; jn = gk; jn = p−3 j2 ln p · ( j − ordp(n)),

for k = 0, otherwise fk; jn = gn; jn = 0. Consequently, the solution is given by thefollowing form:

u(t, x) =∑

j∈Z

∑

n∈Qp/Zpu0; jn(t)ψ0; jn(x),


where

u0; jn(t) = exp(−tp−0.5(1− j))p−3 j2 ln p · ( j − ordp(n))

(

cos(A j t) + 1A j

(

p−0.5(1− j) + 1)

sin(A j t)

)

+ 1A j

p−3 j2 ln p · ( j − ordp(n))

∫ t

0r sin(A j (t − r)) exp((r − t)p−0.5(1− j))dr ,

and A j is defined as Example 1.

5 Cauchy Problem for a Nonlinear Pseudo-differential Equation

Throughout this section we study the existence of solution for the following class ofnonlinear pseudo-differential equation over the p-adic field Qp in I = [0, T ] givenas

⎧⎨

⎩Dα

∂2u(t, x)

∂t2+ 2aDγ ∂u(t, x)

∂t+ bDβu(t, x) + cu(t, x) = F(t, u, x), x ∈ Qp, t ∈ (0, T ],

u(0, x) = f (x), u′t (0, x) = g(x), x ∈ Qp, t = 0,(5.1)

where the coefficients are defined same as before.Moreover, suppose that the constantsof Eq. (5.1) are chosen in the way that A j T < x∗ where x∗ is the only root of

h(x) := x + arctan(e−10x ) − π2

, (5.2)

(see Fig. 1) Considering u in terms of wavelet functions ψk; jn(x) with coefficientsuk; jn(t) one can simply see the following quasilinear differential equation.

pα(1− j)u′′k; jn(t) + 2apγ (1− j)u′k; jn(t) + bpβ(1− j)uk; jn(t) + cuk; jn(t)= Fk; jn(t, u), ∀(k, j, n) ∈ J , (5.3)

where the function u as the solution subjected to the problem is given as

u(t, x) =∑

ı∈Juı (t)ψı (x), ı = (k, j, n) ∈ J := Jp × Z × Qp/Zp,

and suppose that the nonlinear term F takes the following form

F(t, u, x) =∑

ı∈JFı (t, û)ψı (x), û = (uı )ı∈J , ı = (k, j, n) ∈ J . (5.4)


Fig. 1 Function h and x∗ as its only real root

For the convenience of reader, let us remove the index of symbols in infinite system(5.3) and rewrite it as the following matrix differential equation. Taking j ∈ Z asarbitrarily fixed we derive

pα(1− j)û′′(t) + 2apγ (1− j)û′(t) + (bpβ(1− j) + c)û(t) = F(t, û), F = (Fı )ı∈J .(5.5)

It is worth mentioning that all solutions of (5.5) depend on j ∈ Z.On the other hand, by initial conditions we easily see that

uı (0) = 〈 f , ψı 〉, u′ı (0) = 〈g, ψı 〉, ı ∈ J . (5.6)

Let û1(t) and û2(t) form a fundamental system of solutions of the truncated linearequation corresponding to F = 0, that is,

û1(t) := (u1ı (t))ı∈J =(

exp(−atp(γ−α)(1− j)) cos(A j t))

ı∈J,

û2(t) := (u2ı (t))ı∈J =(

exp(−atp(γ−α)(1− j)) sin(A j t))

ı∈J.

(5.7)


Then, considering Eq. (5.5) by its component for any ı ∈ J , we have the followingtransformation

ξı := r(t) = u2ı (t)u1ı (t)

= sin(A j t)sin(A j t) − e10A j T cos(A j t)

= sin(A j t)Bj cos(A j t + r j ) , t ∈ I , ωı =

uıu1ı (t)

, (5.8)

where

Bj =√1 + e20A j T , r j = arctan(e−10A j T ). (5.9)

Here, for some purpose, u1ı is defined as arbitrary special combination of u1ı and u2ıgiven by (5.7). Remark that in definition of ξı , cos(A j t + r j ) �= 0 since

0 < r j ≤ A j t + r j ≤ A j T + r j < π2

, t ∈ I .

More precisely, this is concluded by the fact that h(x) < 0 for x < x∗ ∼= 3.1095where x∗ is the only root of h. And that is satisfied since A j T < x∗.

Based on the imposed condition A j T < x∗, we notice that ξı = r(t) is increasingwith respect to t ∈ I ,

ξı ∈ Ī := [0, r(T )].

The substitutions (5.8) convert Eq. (5.5) into a simpler form

ω′′ξξ = ı (ξı , ωı ), where

ı (ξı , ωı ) = u31ı (t)

[W (u1ı , u2ı )(t)]2 Fı (t, u1ı (t)ωı ), (see also [28, 0.3.2 − 9])(5.10)

whereW (u1ı , u2ı )(t) is the Wronskian of linearly independent functions u1ı , u2ı . Forthe convenience let us ignore the index ı ∈ J , and substitute ρ := ω′ξ then we havethe following integral equation:

{ρ(ξ) = ρ(0) + ∫ ξ0 (s, ω)ds, ξ ∈ Ī ,ω(ξ) = ω(0) + ∫ ξ0 ρ(s)ds, ξ ∈ Ī .

(5.11)

Since t = 0 if and only if ξ = 0 then

ρ(0) = ω′ξ (0) = [ω′t · t ′ξ ](0) =u′(0)u1(0) − u′1(0)u(0)

u21(0)· −e

10A j T

A j

= e−10A j T 〈 f , ψ〉 + 1A j

(

〈g, ψ〉 + ap(γ−α)(1− j)〈 f , ψ〉)

,


and

ω(0) = u(0)u1(0)

= 〈 f , ψ〉.

which both values should be replaced in Eq. (5.11).

Remark 1 To study the solvability of nonlinear system of differential equations (5.3)it only needs to investigate the existence of ω(t) from Eq. (5.11). To do this let us firstpresent the following well-known fixed point result.

Theorem 5.1 (Schauder Fixed Point Theorem [33, Theorem 4.1.1]) Let U be anonempty and convex subset of a normed space E. Let T be a continuous mapping ofU into a compact set K ⊂ U. Then T has a fixed point.Theorem 5.2 Suppose that the following conditions hold:

(i)

|Fı (t, uı ) − Fı (t, vı )| ≤ ϕı (t)|uı − vı |, (t, ı) ∈ I × J (5.12)

where ϕı ∈ L1(I ) with maximum value ϕı and Fı = maxt∈I |Fı (t, 0)|.(ii) There exists a function H j : Ī → R+ for j ∈ Z belonging to ∈ L1( Ī ) such that

∣∣∣∣e

−10A j T 〈 f , ψ〉 + 1A j

(

〈g, ψ〉 + ap(γ−α)(1− j)〈 f , ψ〉)

+∫ ξ

0

(s, 〈 f , ψ〉 +

∫ s

0ρ(r)dr)ds

∣∣∣∣ ≤ H j (ξ), ξ ∈ Ī ,

whenever |ρ(ξ)| ≤ H j (ξ) for any ξ ∈ Ī . Furthermore, suppose that A j T < x∗where x∗ is the only real root of (5.2). Then Eq. (5.11) has a solution ρ(ξ) in C( Ī ,R)bounded above by H j .

Proof Following condition (ii) let us first consider the set S ⊂ C( Ī ,R) defined by

S = {ρ : ρ ∈ C( Ī ,R) and |ρ(ξ)| ≤ H j (ξ) for all ξ ∈ Ī }.

Obviously, the set S is a nonempty, closed, bounded and convex subset of U . Further-more, suppose that

(�ρ)(ξ) := e−10A j T 〈 f , ψ〉 + 1A j

(

〈g, ψ〉 + ap(γ−α)(1− j)〈 f , ψ〉)

+∫ ξ

0

(s, ω)ds

whereω is given by the second relation in (5.11). To prove that Eq. (5.11) has a solutionit only needs to show that the operator � has a fixed point in S. First, we show that Sis �-invariant, that is, �S ⊂ S. This is easily implied by condition (ii).


For the fixed ξ ∈ Ī , suppose ρn → ρ as n → ∞, we get∣∣∣∣(�ρn)(ξ) − (�ρ)(ξ)

∣∣∣∣ ≤

∫ ξ

0

∣∣∣∣(s, ωn) − (s, ω)

∣∣∣∣ds (5.13)

where

ωn(s) = 〈 f , ψ〉 +∫ s

0ρn(r)dr , ω(s) = 〈 f , ψ〉 +

∫ s

0ρ(r)dr , s ∈ Ī .

On the other hand, using (5.10) and (5.12) we see that

∣∣∣∣(s, ωn) − (s, ω)

∣∣∣∣ =

∣∣∣∣

u31(t)

[W (u1, u2)(t)]2∣∣∣∣ ·

∣∣∣∣F(t, u1(t)ωn) − F(t, u1(t)ω)

∣∣∣∣

=∣∣∣∣

[sin(A j t) − e10A j T cos(A j t)]3A2j exp(−atp(γ−α)(1− j)) · e20A j T

∣∣∣∣ ·

∣∣∣∣F(t, u1(t)ωn) − F(t, u1(t)ω)

∣∣∣∣

≤ ϕ(t)A2j · e20A j T

(

sin(A j t) − e10A j T cos(A j t))4

· |ωn(s) − ω(s)|

≤ ϕA2j · e20A j T

(

1 + e20A j T)2

· |ωn(s) − ω(s)|

≤ ϕ · �( Ī )A2j · e20A j T

(

1 + e20A j T)2

· ‖ρn − ρ‖ (5.14)

for any t ∈ I and s ∈ Ī , and ‖ · ‖ is the supremum norm on C( Ī ,R). This togetherwith (5.13) implies that

‖�ρn − �ρ‖ ≤ ϕ · �2( Ī )

A2j · e20A j T(

1 + e20A j T)2

· ‖ρn − ρ‖.

Therefore, the continuity of � is proven. Next, we need to establish that �(S) isequicontinuous. Assume that � > 0 is given, without loss of generality, ξ1 < ξ2 arearbitrarily taken from Ī and ρ ∈ S. Applying condition (i) we have∣∣∣∣(�ρ)(ξ2) − (�ρ)(ξ1)

∣∣∣∣ ≤

∫ ξ2

ξ1

(

|(s, ω) − (s, 0)| + |(s, 0)|)

ds


(

1 + e20A j T)2 ∫ ξ2

ξ1

[

|〈 f , ψ〉| +∫ s

0|ρ(r)|dr + F

]

ds


(

1 + e20A j T)2[

|〈 f , ψ〉| +∫

Ī|H j (r)|dr + F

]

(ξ2 − ξ1)

which vanishes as ξ1 → ξ2. Consequently, we conclude that �(S) is equicontinuouson the compact interval Ī . Moreover, in view of Arzelà–Ascoli theorem one can seethat �(S) is relatively compact. Therefore, all the conditions of Schauder fixed point


theorem are fulfilled and the operator �, as a self-map on S, has a fixed point in thisset. This fact implies that Eq. (5.11) has at least one solution in S. ��An immediate consequence of Theorem 5.2 is given as follows.

Theorem 5.3 Suppose that all the conditions of Theorem 5.2 are satisfied. Then theproblem (5.1) has a solution in U given in Theorem 3.1.Proof From Theorem 5.2, it is possible to find the solutions ρ := ω′ξ and since thenω for Eq. (5.11). This together with (5.8) yields uı = u1ı · ωı exists which means thatthe function

u(t, x) =∑

ı∈Juı (t)ψı (x), ı = (k, j, n) ∈ J := Jp × Z × Qp/Zp,

as a solution of Eq. (5.3) exists under the imposing hypotheses. This completes theproof. ��Example 3 Consider the problem (5.1) with c = 0, b > a2, T < x∗√

b−a2 where x∗ is

the root of (5.2), and f = g = 0. We remark that

A j T =√b − a2 · T < x∗.

Assume Fı , as given in (5.4), has the form Fı (t, u) = φı (t)σı (u)whereφı is continuouson I and σı belongs to � as the class of all increasing convex functions satisfying

‖σ ′ı ‖ < ∞,(

σı (αt) ≤ ασı (t), ∀t ∈ R ⇐⇒ α ≥ 0)

.

Obviously, � �= ∅ since any increasing linear function is contained in �. We notethat

|Fı (t, uı ) − Fı (t, vı )| ≤ |φı (t)| · ‖σ ′ı ‖ · |uı − vı |, (t, ı) ∈ I × J ,that is, the condition (i) in Theorem 5.2 is fulfilled. To check the condition (ii), for thevariable s = r(t) as given in (5.8), we derive

∣∣∣∣

∫ ξ

0

(s,

∫ s

0ρ(r)dr)ds

∣∣∣∣ =

∣∣∣∣

∫ ξ

0

u31ı (t)φı (t)

[W (u1ı , u2ı )(t)]2 σı (u1ı (t)∫ s

0ρ(r)dr)ds

∣∣∣∣

≤∣∣∣∣

u41ı (t0)φı (t0)

[W (u1ı , u2ı )(t0)]2∣∣∣∣ ·

∣∣∣∣

∫ ξ

0σı

( ∫ s

0ρ(r)dr

)

ds

∣∣∣∣,

for t0 ∈ I with r(t0) ∈ [0, ξ ],followed by a variant of the Mean Value Theorem and the fact that σı ∈ �. Now, forany ξ ∈ Ī if |ρ(ξ)| ≤ H j (ξ) then using Jensen’s inequality we conclude that

∣∣∣∣

∫ ξ

0

(s,

∫ s

0ρ(r)dr)ds

∣∣∣∣ ≤

B2j ‖φı‖A2j cos

2(r j )·∫ ξ

0

∫ s

0|(σı ◦ H j )(r)|drds


where the constants are defined by (5.9). Therefore, for given φı and σı if one can finda function H j with nonnegative values on Ī satisfying the following inequality

B2j ‖φı‖A2j cos

2(r j )·∫ ξ

0

∫ s

0|(σı ◦ H j )(r)|drds ≤ H j (ξ), (5.15)

then condition (ii) is fulfilled. To illustrate this, for instance one can take σı = idand H j (x) = eμx with μ = A j cos(r j )Bj‖φı‖ and derive that (5.15) holds. Now, applyingTheorem 5.2we conclude the problem (5.1) with the imposed conditions has a solutionin U .Acknowledgements Open access funding provided by Linnaeus University.

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Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point TheoremAbstract1 Introduction2 Preliminaries3 The Homogeneous Cauchy Problem4 Cauchy Problem for a Linear Pseudo-differential Equation5 Cauchy Problem for a Nonlinear Pseudo-differential EquationAcknowledgementsReferences

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