The main part of the paper deals with local existence and global existence versus blow-up for solutions of Laplace equation in bounded domains with a nonlinear dynamical boundary condition. More precisely we study the problem consisting on: 1) Laplace equation in (0,\infty)\times\Omega; 2) homogeneous Dirichlet condition on (0,\infty)x Gamma_0; 3) the dynamical boundary condition u_nu=-|u_t|^{m-2}u_t+|u|^{p-2}u on (0,\infty)x Gamma_1; 4) the initial condition u(0,x)=u_0(x) on the boundary of Omega. Here Omega is a regular and bounded domain in R^n, n>=1, Gamma_0 and Gamma_1 endow a measurable partition of the boundary of Omega. Moreover m>1, 2<= p=3, r=\infty when n=1,2, The final part of the paper deals with a refinement of a global nonexistence result by Levine, Park and Serrin, which is applied to previous problem.
On the Laplace equation with non-linear dynamical boundary conditions
VITILLARO, Enzo
2006
Abstract
The main part of the paper deals with local existence and global existence versus blow-up for solutions of Laplace equation in bounded domains with a nonlinear dynamical boundary condition. More precisely we study the problem consisting on: 1) Laplace equation in (0,\infty)\times\Omega; 2) homogeneous Dirichlet condition on (0,\infty)x Gamma_0; 3) the dynamical boundary condition u_nu=-|u_t|^{m-2}u_t+|u|^{p-2}u on (0,\infty)x Gamma_1; 4) the initial condition u(0,x)=u_0(x) on the boundary of Omega. Here Omega is a regular and bounded domain in R^n, n>=1, Gamma_0 and Gamma_1 endow a measurable partition of the boundary of Omega. Moreover m>1, 2<= p=3, r=\infty when n=1,2, The final part of the paper deals with a refinement of a global nonexistence result by Levine, Park and Serrin, which is applied to previous problem.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
