On a class of nonlocal evolution equations with the p[u(x,t)]-Laplace operator

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Abstract

We study the homogeneous Dirichlet problem for a class of nonlocal singular parabolic equations utdiv|u|p[u]2u=fin Ω×(0,T),where ΩRd, d2, is a smooth bounded domain, p[u]=p(l(u)) is a given function with values in the interval [p,p+](1,2), and l(u)=Ω|u(x,t)|αdx, α[1,2], is a functional of the unknown solution. We find sufficient conditions for global or local in time solvability of the problem, prove the uniqueness, and show that every solution gets extinct in a finite time.

Introduction

In the present work, we study the homogeneous Dirichlet problem utdiv|u|p[u]2u=fin QT,u=0 on Ω×(0,T),u(x,0)=u0 in Ω,where QT=Ω×(0,T], ΩRd, d2, is a bounded domain with the boundary Ω. The exponent of nonlinearity p(s) is a given function, p:R[p,p+](1,2), p±=const. In Eq. (1.1), the argument of p() is the functional l(v)=Ω|v(x,t)|αdx:L(0,T;Lα(Ω))R,α[1,2].To indicate the nonlocal dependence of the exponent p on the function u(x,t) we use the notation p[u]p(l(u)). We prove that problem (1.1) has a unique strong solution, global or local in time, and show that every strong solution vanishes in finite time.

The special form of the functional l(u) in (1.2) is chosen for convenience of presentation, the results easily extend to a wider class of functionals. For example, all the results remain true if l(u) is a functional over Lα(Ω), l(u)=Ωg(x)u(x,t)dx,with some gLα(Ω)α[1,2].

The mathematical modelling of many real-life processes leads to systems of nonlinear equations whose structure may depend on certain components of the sought solution. For example, the stationary thermo-convective flow of a non-Newtonian fluid is described by the following system for the velocity v(x), the pressure p(x) and the temperature θ(x), [1], (v)b(θ)Δθ=g,divv=0in Ω,(v)vdiv(μ(θ)+τ(θ)|S(v)|q(θ)2S(v))+p=f,endowed with the boundary conditions for v, p and θ. Here ΩRd is a bounded domain, g, f are given functions, b, μ, τ and q are given functions of θ, and S(v) is the deformation rate tensor. Another example is the model of the thermistor, [2], [3], div(|u|σ(θ)2u)=g,Δθ=λ|u|σ(θ),where u is the electric potential and θ in the temperature in a conductor in the presence of Joule heating, or the models of electro-rheological fluids in which the character of nonlinearity in the governing Navier–Stokes equations varies according to the applied electromagnetic field [4].

Formally, each of these models can be regarded as a nonlinear equation, or a system of equations, whose nonlinearity depends on the sought solution. For instance, if the equation for θ in (1.3) has a solution for every given u, this dependence defines the nonlocal operator, θθ(u), and the first equation reads div(|u|σ[u]2u)=g,xΩ,where the exponent has the form σ[u]σ(θ(u)).

Functionals with the growth condition depending on the solution or its gradient are successfully used for denoising of digital images — see, e.g., [5], [6], [7] for the models based on minimization of functionals with p(|u|)-growth and [8] for a discussion of the model of denoising of the image f based on the minimization of the functional λfuL2(Ω)2+Ωα1(u)|uξ(u)|p1(u)+α2(u)|uξ(u)|p2(u)dx,where p1(u),p2(u)[1,2], (ξ(u),ξ(u)) is an orthonormal coordinate system such that ξ(u) is approximately parallel to u, wherever u0.

To the best of our knowledge, the equations involving the p[u]-Laplace operators were studied thus far only in papers [9], [10]. Both papers address elliptic equations of the structure (1.4) and consider the cases of local and nonlocal dependence of σ on u, but their approach to the problem is different.

Paper [9] deals with the equation b(u)diva(x,u,u)=f.In this equation b:RR is a nondecreasing function, b(0)=0, and a(x,z,ξ) is a strictly monotone operator of Leray–Lions type which satisfies the growth and coercivity conditions with the variable exponent p(x,u) such that p(x,u)[p,p+](1,), p±=const. This class of equations includes (1.4) as a partial case. A challenging feature of the equations that involve p[u]-Laplacian is that they cannot be interpreted as a duality relation in a fixed Banach space. For this reason, the authors of [9] reduce the study to the L1-setting and use the Young measures to obtain a solution of the degenerate equation of the type (1.5) as the limit of a sequence {un} of solutions of the regularized equations with pn(x) and p+-Laplacian operators. The authors of [10] take another direction and overcome this difficulty adapting the idea of [11] about passing to the limit in a sequence |vk(x)|qk(x). The results of [9] and [10] are obtained under the assumption p=minp>d, which yields compactness of the sequence of solutions of the regularized problems in a space of Hölder-continuous functions. Besides equations (1.4), (1.5) with the exponent p[u]=(pu)(x) defined as a composite function on Ω, the authors of [9], [10] consider the case of nonlocal dependence of p on the solution u and discuss the question of uniqueness.

The nonlocal evolution equations are widely used in modelling of various processes in physics and biology and are intensively studied, see, e.g., [12], [13], [14], [15], [16] and references therein. Eq. (1.1) with α=1 can be regarded as the diffusion equation for the concentration u(x,t), with the diffusion flux |u|p[u]2u which depends on the total mass m(t)=Ωu(x,t)dx at the instant t, or the inverse of the specific volume m(t)|Ω|, |Ω|=measΩ.

Section snippets

The function spaces

For convenience of the reader, we collect here the basic facts on the Lebesgue and Sobolev with variable exponents. For a detailed presentation of the theory of these spaces we refer to the monograph [17], see also [18, Ch. 1].

Let ΩRd be a bounded domain with the Lipschitz-continuous boundary Ω. Given a measurable function p(x):Ω[p,p+](1,), p±=const, the set Lp()(Ω)=f:ΩR:fis measurable on Ω,Ω|f|p(x)dx<equipped with the Luxemburg norm fp(),Ωinfα>0:Ω|fα|p(x)dx1becomes a Banach

Regularized problem

We will obtain a solution of the singular problem (1.1) as the limit when ϵ0 of the family of solutions of the regularized problems uϵt=div(ϵ2+|uϵ|2)p[uϵ]22uϵ+f(z)in QT,uϵ=0 on Ω×(0,T),uϵ(x,0)=u0(x) in Ω,ϵ>0.

Strong solution of the regularized problem

Lemma 4.1

If the data satisfy conditions (2.8), then problem (3.1) has a strong solution uϵ=limu(m) as m. The solution satisfies the estimate uϵt2,QT2+esssup(0,T)uϵ(,t)2,Ω2+QT|uϵxx|p[uϵ]dzCu02,Ω2+QT|f|(p)dz+1.

For the sake of simplicity of notation, throughout this subsection we omit the subindex ϵ and denote by u(z) the limit of the sequence {u(m)}, which approximates the solution of the regularized problem (3.1). The uniform estimates (3.5), (3.6), (3.9), (3.18) allow one to extract from

Uniqueness of strong solutions

Theorem 5.1

Problem (1.1) has at most one strong solution in the class of functions S=v:vC([0,T];L2(Ω))L(0,T;W01,2(Ω)),vtL2(QT).

Proof

Let uiS be two different strong solutions of problem (1.1). Notice that this set in not empty: according to Theorem 2.1 for every u0W01,2(Ω) and fL(p)(QT) problem (1.1) has at least one strong solution uS. Let us denote p1=p[u1],p2=p[u2].The inclusions uiS yield uiWu1(QT)Wu2(QT),which allows one to take the function u=u1u2 for the test-function in the integral

Local in time existence without assumptions on the range of p[u]

The arguments used in the proof of Theorem 2.1 allow one to prove local in time solvability of problem (1.1) in the case when condition (2.8) (d) is removed and substituted by an assumption on u02,Ω. Fix a small δ and consider problem (1.1) with the exponent pδ[s]=2dd+2+δif p[s]2dd+2+δ,p[s]if 2dd+2+2δ<p[s]<22δ,2δif p[s]2δ.One may choose pδC1(R) with supR|pδ|=C(C,δ) with C from condition (2.8) (c). By Theorems 2.1, 5.1 problem (1.1) with the nonlocal exponent pδ[u] has a unique global

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for valuable comments and suggestions which led to the improvement of this work.

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    The first author was supported by the Research Project No. 19-11-00069 of the Russian Science Foundation, Russia and by the Project UID/MAT/ 04561/ 2019 of the Portuguese Foundation for Science and Technology (FCT), Portugal .

    2

    The second author acknowledges the support of the Research Grant MTM2017-87162-P, Spain.

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