An eigenvalue problem for a periodic parabolic system and applications (Q2366314)

This paper is concerned with the problem of eigenvalues for the parabolic system \[ \widetilde{L} [\vec u]= \lambda\gamma \vec u, \quad \vec u(x,t)= \vec u(x,t+\ell) \quad \text{in } \overline{\Omega}\times \mathbb{R}, \quad \vec u(x,t)\equiv 0 \quad \text{in } \partial \Omega\times \mathbb{R},\tag{1} \] where \(u=\text{col} (u^ 1, u^ 2,\dots, u^ m)\), \(\ell\) is a real fixed number \(>0\), \(\Omega\subset \mathbb{R}^ N\) being a domain whose boundary \(\partial\Omega\) is of class \(C^{2+\alpha}\), \(0<\alpha<1\), \(\widetilde{L} [\vec u]= \text{col} (L[u^ 1], \dots, L[u^ m])\), \(L\) standing for a second order linear parabolic operator on \(\Omega\times \mathbb{R}\). \(\gamma= \{\gamma_{ij}(x,t)\}\) is an \(m\) by \(m\) matrix with continuous entries and \(\ell\)-periodic with respect to \(t\). First, a comparison theorem is obtained for the solutions of two problems similar to (1). Under extra (technical) assumptions on the data, an existence theorem is proven for the smallest eigenvalue of the problem (1). Several auxiliary results are obtained before adequately using the maximum principle. The main result is then applied to a nonlinear system related to (1).