On symmetry properties of parabolic equations in bounded domains (Q539888)

The basic idea is the following: if the spatial domain is symmetric, then, regardless of the initial data, solutions to initial-boundary value problems associated to parabolic equations tend to symmetrize as \(t \to +\infty\). To be more precise, let \(\Omega \subset \mathbb R^N\), \(N \geq 1\), be a bounded domain, convex in \(x_1\) and symmetric with respect to the hyperplane \(x_1 = 0\). Furthermore, the boundary \(\partial \Omega\) is assumed to satisfy a condition, which prevents \(\partial \Omega\) from having two ill-displaced flat portions, parallel to the \(x_1\)-axis. In particular, if \(N = 2\) then \(\Omega\) cannot be a rectangle with edges parallel to the coordinate axes, which constitutes a known counterexample to the theorem below. Nevertheless, \(\partial \Omega\) may well have some convenient flat portion, even parallel to the \(x_1\)-axis (examples are depicted). In the space-time cylinder \(\Omega \times (0,+\infty)\), non-negative solutions \(u\) to parabolic problems are considered. Denote by \(\omega(u)\) the \(\omega\)-limit set of \(u\), i.e., the set of all functions \(z\) such that \(u(x,t_k) \to z(x)\) in the supremum norm for some sequence \(t_k \to +\infty\). A solution \(u\) is told asymptotically symmetric if all limit functions \(z\in \omega(u)\) are even in \(x_1\) and non-increasing in the set \(\Omega_0 = \Omega \cap \{x_1> 0\}\). Asymptotic symmetry results are proved. The first theorem deals with any non-negative, bounded, classical solution to \[ \begin{cases} u_t - \Delta u = f(t,u), &(x,t) \in \Omega \times (0, +\infty), \\ u = 0, &(x,t) \in \partial \Omega \times (0, +\infty), \end{cases} \] where the function \(f(t,u)\) is Lipschitz continuous in \(u\), uniformly with respect to \(t\), and \(f(x,0)\) is a bounded function of the space variable \(x\). The conclusion is that every (not identically zero) limit function \(z\in \omega(u)\) is even in \(x_1\) and strictly decreasing in \(\Omega_0\). The theorem is derived from a more general statement which applies to fully nonlinear parabolic equations and takes forth previous results in [\textit{P. Poláčik}, Arch. Ration. Mech. Anal. 183, No. 1, 59--91 (2007; Zbl 1171.35063)]. A weaker statement which holds under less restrictive assumptions is found, too. Proofs involve the moving plane method.