Global existence and gradient estimates for the quasilinear parabolic equations of \(m\)-Laplacian type with a nonlinear convection term (Q1975471)

This paper is concerned with the initial-boundary value problem for the quasilinear parabolic equation of the \(m\)-Laplacian type with a nonlinear convection term under homogeneous Dirichlet boundary conditions in a general bounded domain \(\Omega\) in \(\mathbb{R}^n\) for initial data belonging to \(L^q(\Omega)\) \((q\geq 1)\). The authors assume neither smallness and differentiability conditions on initial data nor geometrical conditions on the boundary \(\partial\Omega\), and they derive precise gradient estimates as well as global existence and uniqueness of solutions. These estimates show certain smoothing effects near \(t=0\) and decay property as \(t\to\infty\).