Some results on existence and regularity for non-linear $p(x)$-parabolic equations with quadratic growth with respect to the gradient and general data

Abstract

We study the regularity of solutions for a certain class of non-linear parabolic equations, which is described as follows:$$\dfrac{\partial b(u)}{\partial t}-div\big[\phi(t,x,u)(1+|u|)^{s(x)}|\nabla u|^{p(x)-2}\nabla u\big]+\zeta(x,t)(1+|u|)^{q(x)-1}u|\nabla u|^{p(x)}=\mu,$$where $\mu$ is a general measure, $b$ is a strictly increasing $C^1$-function, and $−div(\Phi(x, t,\nabla u))$is a Leray–Lions type operator with growth $|\nabla u|^{p(x)-1}$ in $\nabla u$. Our approach involves a combination of convergence and compactness techniques in variable exponent Sobolev spaces by using an approximate problem.