The n-dimensional Heat Equation with Nonlinear Generalized Wentzel Boundary Conditions

Gisele Goldstein U of Memphis

Of concern is the Cauchy problem for the heat equation on D ⊂ ℝ^{n}

u_{t} = ∇ · (α(x)∇u) x ∈ D, t > 0
u(x, 0) = f(x) x ∈ ∂D

with nonlinear generalized Wentzel boundary conditions

∇ · (α(x)∇u) + b(x)\frac{∂u}{∂n} ∈ c(x)β(x, u) for x ∈ ∂D, t > 0.

The usual Wentzel boundary condition corresponds to b(x) = c(x) = 0. Here we assume that α ∈ C^{1}(\overline{D}), α > 0 on \overline{D}, b, c ∈ C(∂D), b ≥ 0, c ≤ 0 and β is a maximal monotone graph in ℝ^{2} containing the origin. We present an L^{p} theory for this problem, which gives rise to a nonlinear semigroup on a Banach space X^{p}, 1 ≤ p < ∞ and on C(\overline{D}) in the case p = ∞. Hence, we obtain a solution to the initial boundary value problem. One novel feature of this work is that X^{p} is not L^{p}(D) but rather an L^{p} space based on functions on D and on ∂D, with a measure on ∂D based on the boundary conditions. Questions of regularity and smoothing of the solutions are also obtained via Nash-Moser iteration.

This is joint work with Angelo Favini, Jerry Goldstein, and Silvia Romanelli.