Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

1997 ◽  
Vol 20 (15) ◽  
pp. 1325-1333 ◽  
Author(s):  
Marek Fila ◽  
Pavol Quittner
Keyword(s):  
Boundary Condition ◽  
Laplace Equation ◽  
Global Solutions ◽  
Nonlinear Dynamical ◽  
Dynamical Boundary Condition

scholarly journals The large diffusion limit for the heat equation with a dynamical boundary condition

2020 ◽  
Vol 23 (01) ◽  
pp. 2050003
Author(s):  
Marek Fila ◽  
Kazuhiro Ishige ◽  
Tatsuki Kawakami
Keyword(s):  
Boundary Condition ◽  
Diffusion Coefficient ◽  
Heat Equation ◽  
Half Space ◽  
Laplace Equation ◽  
Diffusion Limit ◽  
Dynamical Boundary Condition ◽  
Suitable Sense ◽  
Large Diffusion

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

ON THE LAPLACE EQUATION WITH NON-LINEAR DYNAMICAL BOUNDARY CONDITIONS

2006 ◽  
Vol 93 (2) ◽  
pp. 418-446 ◽  
Author(s):  
ENZO VITILLARO
Keyword(s):  
Boundary Condition ◽  
Laplace Equation ◽  
Main Part ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Initial Condition ◽  
Dynamical Boundary Condition ◽  
Non Linear ◽  
Dynamical Boundary Conditions

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.

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