Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations

We give existence theorems of global solutions in L_loc ^∞((0,∞);W₀ ^1,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u_t−div{σ(|∇u|²)∇u}=0, where the class of σ(v²) includes the logarithmic case σ(|∇u|²)= log (1+|∇u|²) for a typical example. We assume that the initial data belong to W₀ ^1,p₀ ,p₀≥2, or L^r,r≥1, and we derive precise estimates for ‖∇u(t)‖_∞ near t=0.