Title

Smoothing for nonlinear parabolic equations with nonlinear boundary conditions

Abstract

Of concern are parabolic problems of the form ∂u/at = ∇ · ψ(x, ∇u) for (x, t) ∈ Ω × [0, T] with Ω ⊂ ℝn, -ψ(x, ∇u) · v = β(x, u) for (x, t) ∈ ∂Ω × [0, T], u(x, 0) = f(x) for x ∈ Ω. Under suitable conditions it is shown that for f ∈ L1(Ω) and t > 0, one has u(·, t) ∈ L∞(Ω) and ∥u(·, t)∥∞ ≤ C(T) ∥f∥1/tn/2 and ∥ut(·, t)∥ ≤ C(Ct)∥f∥1/tn/4 + 1 for t ∈ (0, T] and n ≥ 3. Analogous estimates are obtained with other powers of t in dimensions n = 1, 2. © 1997 Academic Press.

Publication Title

Journal of Mathematical Analysis and Applications

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