Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension
Consider a non-negative and bounded global weak solution \(u=u(t,x)\) to the (possibly) degenerate parabolic equation \(\partial_t u - \partial_x^2\varphi(u) = f(u)\) for \((t,x)\in (0,+\infty)\times (-L,L)\) with boundary conditions \(\pm\partial_x\varphi(t,\pm L) + \beta(\pm L)\;\varphi(u)(t,\pm L)=h(\pm L)\) and a non-negative initial datum \(u_0\in L^\infty(-L,L)\). Here \(L>0\), \(\varphi\in{\mathcal C}([0,+\infty))\cap {\mathcal C}^3((0,+\infty))\) and \(f\in {\mathcal C}^1([0,+\infty))\) satisfy \(\varphi(0)=f(0)=0\) and \(\varphi'(z)>0\) for \(z>0\), while \(\beta(\pm L)\) and \(h(\pm L)\) are four non-negative real numbers. It is shown that there is a stationary solution \(u_\infty\) such that \(u(t)\) converges uniformly in \([-L,L]\) towards \(u_\infty\) as \(t\to +\infty\). As \(\varphi'(0)\) may vanish, there might be a continuum of stationary solutions, and this is the main difficulty in the proof (if the set of stationary solutions is discrete, standard arguments from the theory of dynamical systems readily imply the result since a Lyapunov functional is associated to the above equation). This difficulty is overcome by a careful study of the set of stationary solutions and the use of the existing theory about the zero level set of solutions to one-dimensional parabolic equations.
scientific article; zbMATH DE number 2101650