The heat equation with nonlinear general Wentzell boundary condition


Let ω be a bounded subset of RN with a C2,ε boundary ∂ω, α ∈ C2(ω̄) with α > 0 in ω̄ and A the operator defined by Au := ∇. (α∇u) with the nonlinear general Wentzell boundary condition Au + b ∂u/∂n ∈ c β(., u) on ∂ω, where n(x) is the unit outer normal at x, b, c are real-valued functions in C1(∂ω) and β(x,.) is a maximal monotone graph. Then, under additional assumptions on b, c, β, we prove the existence of a contraction semigroup generated by the closure of A on suitable Lp spaces, 1 ≤ p > ∞ and on C(ω̄). Questions involving regularity are settled optimally, using De Giorgi-Nash-Moser iteration.

Publication Title

Advances in Differential Equations

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