Weak solutions for nonlocal parabolic problems of Kirchhoff type via topological degree theory

Abstract

We consider a class of nonlinear, nonlocal parabolic equations featuring a Kirchhoff-type term and variable exponent growth. These equations naturally arise in the modeling of complex physical processes such as electrorheological fluids and anisotropic diffusion, where the interplay between nonstandard growth conditions and nonlocal effects is particularly significant. Motivated by the recent contribution of M. Ait Hammou [Kragujevac J. Math, 47(4), 523–529 (2023)], who employed topological degree theory to address quasilinear parabolic problems involving the -Laplacian, we broaden this analytical framework to encompass Leray–Lions type operators with gradient dependence, coupled with Kirchhoff-type nonlocal coefficients. Building on this foundation, we employ a carefully refined version of the Berkovits–Mustonen degree theory to establish the existence of weak solutions in variable exponent Sobolev spaces. Our analysis relies on verifying key structural properties of the associated operators—such as continuity, boundedness, and the -condition—which facilitate the application of a fixed point argument to operator sums. This work expands the applicability of topological methods to a broader class of nonlinear parabolic equations, including a wide range of models with nonstandard diffusion and nonlocal effects.