The heat equation with nonlinear generalized Robin boundary conditions

AbstractLet Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H1-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by ∂u∂νdσ+β(x,u)dμ=0 on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))t⩾0 on L2(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p⩾2 and every t>0, one has‖SB(t)u−SB(t)v‖∞⩽C1eC2tt−N2p‖u−v‖p, for some constants C1,C2⩾0. In the second part, we prove that if Ω is a bounded Lipschitz domain, one can also define a realization of the Laplacian with nonlinear Robin boundary conditions on C(Ω¯) and this operator generates a strongly continuous and contractive nonlinear semigroup