Anisotropic parabolic problems with Orlicz data

Abstract

In this work, we prove the existence of weak solution for a class of anisotropic parabolic problems with Orlicz data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results.

Nonlinear anisotropic parabolic equations in with locally integrable data

“… similar results are established in . In the case of the Dirichlet problem on a bounded domain, existence and regularity results for distributional solutions with L 1 ‐data (or measures data) have been obtained in for a class of anisotropic elliptic and parabolic equations. In this paper, we treated the anisotropic parabolic case with locally integrable right‐hand side f , an unbounded domain, , and s i > p i for i = 1, … , N .…”

Regularity Results for Anisotropic Parabolic Equations With 1 L Data

“…In order to obtain the equality in (10) one has to impose, as in [9], a stronger assumption on the datum f and ; 0 u more precisely, we will require that…”

“…( ). (9) is true for all 1  r and C depends on r and . Ω Possible references on the theory of anisotropic Sobolev spaces are [12,1].…”

“…Anisotropic operators involve today in various branches of applied science. In some case, they provide realistic models for the study of natural phenomena in biology and fluid mechanics (see references of [9]).…”

“…The time variable makes for a shift in the results and critical exponents that we described in some detail. The existence of at least a solution u, with different condition on F, is proved in [10,9].…”