On a class of nonlocal evolution equations with the -Laplace operator
Introduction
In the present work, we study the homogeneous Dirichlet problem where , , , is a bounded domain with the boundary . The exponent of nonlinearity is a given function, , . In Eq. (1.1), the argument of is the functional To indicate the nonlocal dependence of the exponent on the function we use the notation . 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 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 is a functional over ,
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 , the pressure and the temperature , [1], endowed with the boundary conditions for , and . Here is a bounded domain, , are given functions, , , and are given functions of , and is the deformation rate tensor. Another example is the model of the thermistor, [2], [3], where is the electric potential and 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 in (1.3) has a solution for every given , this dependence defines the nonlocal operator, , and the first equation reads where the exponent has the form .
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 -growth and [8] for a discussion of the model of denoising of the image based on the minimization of the functional where , is an orthonormal coordinate system such that is approximately parallel to , wherever .
To the best of our knowledge, the equations involving the -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 on , but their approach to the problem is different.
Paper [9] deals with the equation In this equation is a nondecreasing function, , and is a strictly monotone operator of Leray–Lions type which satisfies the growth and coercivity conditions with the variable exponent such that , . This class of equations includes (1.4) as a partial case. A challenging feature of the equations that involve -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 -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 of solutions of the regularized equations with and -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 . The results of [9] and [10] are obtained under the assumption , 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 defined as a composite function on , the authors of [9], [10] consider the case of nonlocal dependence of on the solution 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 can be regarded as the diffusion equation for the concentration , with the diffusion flux which depends on the total mass at the instant , or the inverse of the specific volume , .
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 be a bounded domain with the Lipschitz-continuous boundary . Given a measurable function , , the set equipped with the Luxemburg norm becomes a Banach
Regularized problem
We will obtain a solution of the singular problem (1.1) as the limit when of the family of solutions of the regularized problems
Strong solution of the regularized problem
Lemma 4.1 If the data satisfy conditions (2.8), then problem (3.1) has a strong solution as . The solution satisfies the estimate
For the sake of simplicity of notation, throughout this subsection we omit the subindex and denote by the limit of the sequence , 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
Proof Let be two different strong solutions of problem (1.1). Notice that this set in not empty: according to Theorem 2.1 for every and problem (1.1) has at least one strong solution . Let us denote The inclusions yield which allows one to take the function for the test-function in the integral
Local in time existence without assumptions on the range of
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 . Fix a small and consider problem (1.1) with the exponent One may choose with with from condition (2.8) (c). By Theorems 2.1, 5.1 problem (1.1) with the nonlocal exponent 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.
References (23)
- et al.
Structural stability for variable exponent elliptic problems. II. The -Laplacian and coupled problems
Nonlinear Anal.
(2010) - et al.
Global higher regularity of solutions to singular -parabolic equations
J. Math. Anal. Appl.
(2018) - et al.
On stationary thermo-rheological viscous flows
Ann. Univ. Ferrara Sez. VII Sci. Mat.
(2006) On some variational problems
Russ. J. Math. Phys.
(1997)Solvability of the three-dimensional thermistor problem
Tr. Mat. Inst. Steklova
(2008)Electrorheological fluids: modeling and mathematical theory
- et al.
Variational PDE models and methods for image processing
- et al.
Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion
Adv. Comput. Math.
(2009) - et al.
Variable exponent, linear growth functionals in image restoration
SIAM J. Appl. Math.
(2006) Image denoising using directional adaptive variable exponents model
J. Math. Imaging Vision
(2017)
Some results on the p(u)-laplacian problem
Math. Ann.
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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 .
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The second author acknowledges the support of the Research Grant MTM2017-87162-P, Spain.