Enzo Vitillaro

The main part of the paper deals with local existence and global existence versus blow-up for solutions of the Laplace equation in bounded domains with a non-linear dynamical boundary condition. More precisely, we study the problem consisting in:
(1) the Laplace equation in $(0, \infty) \times \Omega$;
(2) a homogeneous Dirichlet condition $(0, \infty) \times \Gamma_0$;
(3) the dynamical boundary condition $ \frac {\partial u}{\partial \nu} = - |u_t|^{m-2} u_t + |u|^{p - 2} u$ on $(0, \infty) \times \Gamma_1$;
(4) the initial condition $u(0, x) = u_0 (x)$ on $\partial \Omega$.
Here $\Omega$ is a regular and bounded domain in $\mathbb{R}^n$, with $n \ge 1$, and $\Gamma_0$ and $\Gamma_1$ endow a measurable partition of $\partial \Omega$. Moreover, $m>1$, $2 \le p < r$, where $r = 2 (n - 1) / (n - 2)$ when $n \ge 3$, $r = \infty$ when $n = 1,2$, and $u_0 \in H^{1/2} (\partial \Omega)$, $u_0 = 0$ on $\Gamma_0$.

The final part of the paper deals with a refinement of a global non-existence result by Levine, Park and Serrin, which is applied to the previous problem.

2000 Mathematics Subject Classification: 35K55 (primary), 35K90, 35K77 (secondary).

E-mail:
enzo@dipmat.unipg.it