A note on quenching for parabolic equations with dynamic boundary conditions
The author considers the semilinear parabolic equation \(u_t- \Delta u= g(u)\) in for \(x\in\Omega\) and \(t> 0\) under the dynamic boundary condition \(u_t+ u_\nu= g(u)\) for \(x\in\partial\Omega\) and \(t> 0\) and under initial datum \(u(x, 0)= u_0(x)\). Here \(\Omega\) is a bounded domain, \(\nu\) is the exterior normal and \(g\) is positive near zero and blows up at some finite \(b\). If \(u(x, t)\) reaches \(b\), it quenches. The main result states that under suitable assumptions \(u\) quenches in finite time \(T\) and that this time can be estimated in terms of the data.
scientific article; zbMATH DE number 2102593