On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions

TL;DR

This paper establishes the continuity of solutions for a broad class of quasilinear parabolic equations with generalized Orlicz growth, under non-logarithmic conditions, including new cases of double-phase equations.

Contribution

It introduces a generalized non-logarithmic Zhikov's condition ensuring solution continuity for complex parabolic equations with $(p,q)$-growth.

Findings

Proves continuity of bounded solutions under new growth conditions

Extends results to include double-phase parabolic equations

Provides conditions under which solutions are continuous

Abstract

We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,\tau,{\rm v}/r), \quad (x,t), (y,\tau)\in Q_{r,r}(x_{0},t_{0}), \quad 0<{\rm v}\leqslant K\lambda(r), $$ $$ \quad \lim\limits_{r\rightarrow0}\lambda(r)=0, \quad \lim\limits_{r\rightarrow0} \frac{\lambda(r)}{r}=+\infty, \quad \int_{0} \lambda(r)\,\frac{dr}{r}=+\infty. $$ In particular, our results cover new cases of double-phase parabolic equations.