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

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 $$u_t-\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u+a(t)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u)=0,$$ $$u_t-\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(t)-2}\mathrm{\nabla }u)=0.$$ 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 $p\le q<\mathrm{\infty }$ to obtain the desired regularity.