A qualitative theory for parabolic problems under dynamical boundary conditions
Let \(\Omega\) be an open subset of \(\mathbb R^n\) and let \(F: \Omega\times (0,T) \times \mathbb R\times \mathbb R^n \times \mathbb S^n \to \mathbb R\) be a Carathéodory function (where \(\mathbb S^n\) denotes the set of symmetric \(n\times n\) matrices). This paper considers the differential equation \(u_t= F(x,t,u,Du,D^2u)\) in \(\Omega \times (0,T)\) with the boundary conditions \(u=\varphi\) on \(\partial_1\Omega\times(0,T)\) and \(\sigma u_t +cu_\nu =\rho u\), where \(\partial_1\Omega\) and \(\partial_2\Omega\) are two disjoint subsets of \(\partial\Omega\) whose union is \(\partial\Omega\), and \(\varphi\), \(\sigma\), \(c\), and \(\rho\) are known functions of \(x\) and \(t\) with \(c>0\) and \(\sigma \geq 0\). In addition, \(\partial_2\Omega\) is assumed to be relatively open in \(\partial\Omega\) with \(C^1\) outer normal field \(\nu\). (The model differential equation is, of course, the heat equation \(u _t=\Delta u\).) Under some relatively straightforward assumptions on \(F\) and the coefficients in the boundary condition, the authors prove a number of maximum and comparison principles which are then used to show that this problem with suitable initial conditions has a unique solution. As a special case, the techniques apply to problems which are periodic in time. The authors also discuss how the solution converges to an appropriate steady state solution.
scientific article; zbMATH DE number 1559353