Doubly degenerate parabolic equations with variable nonlinearity I: Existence of bounded strong solutions

Abstract

We study the homogeneous Dirichlet problem for the anisotropic parabolic equation with double degeneracy \begin{align*} \frac{d}{dt} (|v|^{m(x,t)} \operatorname{sign}\,v ) = \sum_{i=1}^{n}D_{i} ( a_{i}(x,t)|D_{i} v|^{p_{i}(x,t)-2}D_{i} v ) +b(x,t)|v|^{\sigma(x,t)-2}v+g(x,t). \end{align*} The exponents of nonlinearity $m(x,t)>0$, $p_{i}(x,t)>1$, and $\sigma(x,t)>1$ are given bounded continuous functions. It is proved that the problem has a bounded solution in a variable-exponent Sobolev space. The main existence result is local in time and is established under minimal restrictions on the low-order terms. It is shown that under further restrictions on $b$ and $\sigma(x,t)$ the constructed solution can be continued to the arbitrary time interval. The energy estimates are derived.