The heat equation with nonlinear generalized Robin boundary conditions

Let Omega subset of R-N be a bounded domain and let mu be an admissible measure on partial derivative Omega. We show in the first part that if Omega has the H-1-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by partial derivative u/partial derivative vd sigma + beta(x, u) d mu = 0 on partial derivative Omega, generates a strongly continuous nonlinear submarkovian semigroup S-B = (S-B(t))(t >= 0) on L-2(Omega). We also obtain that this semigroup is ultracontractive in the sense that for every u, v is an element of L-p(Omega), p >= 2 and every t > 0, one has parallel to S-B(t)u - S-B(t)v parallel to(infinity) <= C(1)e(C2t)t(-N/2p) parallel to u - v parallel to(p), for some constants C-1, C-2 >= 0. In the second part, we prove that if Omega is a bounded Lipschitz domain, one can also define a realization of the Laplacian with nonlinear Robin boundary conditions on C((Omega) over bar) and this operator generates a strongly continuous and contractive nonlinear semigroup. (C) 2009 Elsevier Inc. All rights reserved.