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.
Advances in Differential Equations
Favini, A., Goldstein, G., Goldstein, J., & Romanelli, S. (2006). The heat equation with nonlinear general Wentzell boundary condition. Advances in Differential Equations, 11 (5), 481-510. Retrieved from https://digitalcommons.memphis.edu/facpubs/5885