Rothe time-discretization method for a nonlinear parabolic p ( u ) -Laplacian problem with Fourier-type boundary conditi
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Rothe time-discretization method for a nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary condition and L1 -data Abdelali Sabri1
· Ahmed Jamea2
Received: 7 September 2020 / Accepted: 17 October 2020 © Università degli Studi di Napoli "Federico II" 2020
Abstract In this paper, we prove the existence and uniqueness results of entropy solutions to a class of nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary conditions and L 1 -data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces. Keywords Entropy solution · p(u)-Laplacian · Semi-discretization · Rothe’s method · Variable exponent Sobolev space Mathematics Subject Classification 35K55 · 35J05 · 35J60
1 Introduction Let ⊂ Rd , (d ≥ 3) be an open bounded domain with a connected Lipschitz boundary ∂, η is the unit outward normal in ∂ and let T be a fixed positive real number. Our aim of this paper is to prove the existence and uniqueness results of entropy solutions for the non-linear parabolic problem ⎧ ∂u ⎪ − div a(x, u(x, t), ∇u(x, t)) + α(u) = f in Q T := ]0, T [×, ⎪ ∂t ⎨ ∂u (1.1) a(x, u(x, t), ∇u(x, t)) + λu = g on T :=]0, T [×∂, ⎪ ⎪ η ⎩ u(., 0) = u 0 in .
B
Abdelali Sabri [email protected]
1
Faculté des Sciences, Université Chouaib Doukkali, El Jadida, Morocco
2
Centre Régional des Métiers de l’Education et de Formation Casablanca-Settat, El Jadida, Morocco
123
A. Sabri, A. Jamea
where α is a strictly increasing continuous real function defined on R and λ is a positive real number, the datum f , g and u 0 are non-regular functions. The operator div a(x, u, ∇u) is called p(u)-Laplacian. It is more complicated than p(x)-Laplacian in the term of nonlinearity. A prototype of this operator is div |∇u| p(u)−2 .∇u . The variable exponent p depends both on the space variable x and on the unknown solution u. In recent years, the study of partial differential equations and variational problems with variable exponent involving p(u)-Laplacian has received considerable attention in many models coming from various branches of mathematical physics, such as elastic mechanics, electrorheological fluid dynamics, image processing and computer vision (see for example [6,10,19,22]). The notion of entropy solutions was introduced by Ph. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez in [9], this notion was then adapted by many authors to study some nonlinear elliptic and parabolic problems with a constant or variable exponent and with Dirichlet, Fourier or Neumann boundary conditions (see for example [1,16,17,20,21]). As far as we know, there are only few important contributions concerning the study of nonlinear problems involving p(u)-Laplacian operator. One is due to Andreianovet-al in [3], where they proved that the problem
α(u) − div a(x, u, ∇u) = f in u=0 on ∂
(1.2)
has a unique kind of weak solutions called narrow and broad weak solutions, under some assmuptions giving in Sect. 3. By using a singular perturbation techni
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