Smoothing for Nonlinear Parabolic Equations with Nonlinear Boundary Conditions

Of concern are parabolic problems of the form ∂u/∂t=∇·ψ(x,∇u) for (x,t)∈Ω×[0,T] with Ω⊂Rn, −ψ(x,∇u)·ν=β(x,u) for (x,t)∈∂Ω×[0,T],u(x,0)=f(x) forx∈Ω. Under suitable conditions it is shown that forf∈L1(Ω) andt>0, one hasu(·,t)∈L∞(Ω) and ‖u(·,t)‖∞≤C(T)‖f‖1/tn/2and ‖ut(·,t)‖2≤C(T)‖f‖1/tn/4+1 fort∈(0,T] andn≥3. Analogous estimates are obtained with other powers oftin dimensionsn=1,2.