"Smoothing for nonlinear parabolic equations with nonlinear boundary co" by Gisèle Ruiz Goldstein and Jerome A. Goldstein
 

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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