p(x)-kirchhoff-type problem, variable exponent sobolev spaces, weak solution, topological degree theory, 35j60, 47j05, 47h11, Mathematics, QA1-939

This article establishes the existence of a weak solution for a class of p(x)p\left(x)-Kirchhoff-type problem under no-flux boundary conditions with a reaction term depending also on the gradient convection. The proof of the main result is constructed by utilizing the theory of topological degree and the theory of generalized Sobolev spaces.