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

We give existence theorems of global solutions in Lloc∞((0,∞);W01,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form ut−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 W01,p0,p0≥2, or Lr,r≥1, and we derive precise estimates for ‖∇u(t)‖∞ near t=0.