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 $$\begin{aligned} u_{t}-\mathrm{div}\left( g(x,t,|\nabla u|) \,\frac{\nabla u}{|\nabla u|}\right) =0, \end{aligned}$$ under the generalized non-logarithmic Zhikov’s condition $$\begin{aligned}g(x,t,\mathrm{v}/r)&\leqslant c(K)\,g(y,\tau ,\mathrm{v}/r), \quad (x,t), (y,\tau )\in Q_{r,r}(x_{0},t_{0}), \quad 0<\mathrm{v}\leqslant K\lambda (r),\\ &\quad\lim \limits _{r\rightarrow 0}\lambda (r)=0, \quad \lim \limits _{r\rightarrow 0} \frac{\lambda (r)}{r}=+\infty , \quad \int _{0} \lambda (r)\,\frac{dr}{r}=+\infty . \end{aligned}$$ In particular, our results cover new cases of double phase and degenerate double-phase parabolic equations.