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

In this paper, we derive precise estimates for ∇u(t) including smoothing effects near t=0 and decay as t→∞ as well as global existence of the solutions u(t) to the initial-boundary value problem in a bounded domain in Rⁿ for the quasilinear parabolic equation of the m Laplacian type with a nonlinear convection term b(u)∇u. For the initial data u₀ we only assume u₀∈L^q(Ω), 1≤q<∞.