$C^{1,\alpha }$ regularity for quasilinear parabolic equations with nonstandard growth

Abstract

In this paper, we obtain $C^{1,\alpha }$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t-\text{div}(|\nabla u{|}^{p-2}\nabla u+a(t)|\nabla u{|}^{q-2}\nabla u)=0,$$ $$u_t-\text{div}(|\nabla u{|}^{p(t)-2}\nabla u)=0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switching factor $a(t)$ and the variable exponent $p(t)$. Moreover, we are able to take any $p\le q<\infty$ to obtain the desired regularity.

22 pages. arXiv admin note: substantial text overlap with arXiv:2009.02227. Bibliography updated