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

We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{rm div}left(g(x,t,| abla u|),frac{ abla u}{| abla u|}right)=0, $$ under the generalized non-logarithmic Zhikovs condition $$ g(x,t,{rm v}/r)leqslant c(K),g(y,tau,{rm v}/r), quad (x,t), (y,tau)in Q_{r,r}(x_{0},t_{0}), quad 0<{rm v}leqslant Klambda(r), $$ $$ quad limlimits_{rrightarrow0}lambda(r)=0, quad limlimits_{rrightarrow0} frac{lambda(r)}{r}=+infty, quad int_{0} lambda(r),frac{dr}{r}=+infty. $$ In particular, our results cover new cases of double-phase parabolic equations.