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

Abstract

We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$u_t-\mathrm{div}\left(g(x,t,|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)=0,$$ under the generalized non-logarithmic Zhikov's condition $$g(x,t,\mathrm{v}/r)⩽c(K)g(y,\tau ,\mathrm{v}/r),(x,t),(y,\tau )\in Q_{r,r}(x_0,t_0),0<\mathrm{v}⩽K\lambda (r),$$ $$\underset{r\to 0}{\lim }\lambda (r)=0,\underset{r\to 0}{\lim }\frac{\lambda (r)}{r}=+\infty ,{\int }_0\lambda (r)\frac{dr}{r}=+\infty .$$ In particular, our results cover new cases of double-phase parabolic equations.

32 pages