Existence result for nonlinear parabolic problems with L¹-data

Abstract

We study the existence of solutions of the nonlinear parabolic problem ∂u ∂t − div[|Du − Θ(u)| p−2 (Du − Θ(u))] + α(u) = f in ]0, T [ × Ω, (|Du − Θ(u)| p−2 (Du − Θ(u))) • η + γ(u) = g on ]0, T [ × ∂Ω, u(0, •) = u 0 in Ω, with initial data in L 1. We use a time discretization of the continuous problem by the Euler forward scheme.

Existence and stability of solutions to nonlinear parabolic problems with perturbed gradient and measure data

“…Moreover, our main ideas and methods to study this problem come from [1,2]. More precisely, we apply a time discretization of given continuous problem by the Euler forward scheme and study existence, uniqueness and stability questions.…”

“…More precisely, we apply a time discretization of given continuous problem by the Euler forward scheme and study existence, uniqueness and stability questions. Let us recall that this method has been used in the literature for the study of some nonlinear parabolic problems, we refer for example to [2][3][4] for some details. This scheme is usually used to prove existence of solutions as well as to compute numerical approximations.…”

“…The third interesting aspect of the investigation of problems (P ) is motivated amongst others by the following observations: In [2] (for the case p(•) = const, and µ ∈ L 1 (Q T )), the authors proved existence and uniqueness of entropy solution, and the approach used in the stationary case, where the authors in [7] prove that every diffuse measure µ i.e. a measure which does not charge the sets of null p-capacity belongs to L 1 (Ω) + W −1,p ′ (Ω), that permit them to prove the existence and uniqueness of entropy solution for the following problem…”

Entropy solution for a nonlinear parabolic problem with homogeneous Neumann boundary condition involving variable exponents

“…To overcome these difficulties we apply a time discretization of given continuous problem by the Euler forward scheme. Let's recall that this method has been used in the literature for the study CUBO 23, 3 (2021) Entropy solutions for nonlinear parabolic problems with... of some nonlinear parabolic problems, we refer for example to [7,13,16,17] for some details. This scheme is usually used to prove existence of solutions as well as to compute numerical approximations.…”

Weak solutions to nonlinear parabolic problems with variable exponent

“…In the particular case where p(.) = constant, the existence and uniqueness of the solutions to problem (1) have been intensively studied by many authors, we refer for example the reader to the bibliography [4,6,7,9] and references therein.…”