Existence results for anisotropic and isotropic $p(x)$-Laplace equations

The Dirichlet problem is considered both for degenerate and singular inhomogeneous quasilinear parabolic equations. We prove the existence of a solution $u$ such that $u_t$ belongs to $L_{\mathrm{\infty }}$. The $L_{\mathrm{\infty }}$ estimate of $u_t$ is obtained by introducing a new time variable.