On positive solutions of some periodic parabolic eigenvalue problem with a weight function
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), and let \(m=m(x,t)\) be a \(T\)-periodic in \(t\) real function on \(\Omega\times \mathbb{R}\) whose restriction to \(\Omega\times (0, T)\) is in \(L^r(\Omega\times(0, T))\) for some \(r>n+2\). The paper finds necessary and sufficient conditions on \(m\) for the existence, uniqueness and simplicity of the principal eigenvalue for the following periodic weighted Dirichlet and Neumann parabolic eigenvalue problem \[ {\partial u\over\partial t}-\sum_{i,j}a_{i,j} {\partial^2 u\over\partial x_i\partial x_j}- \sum_j a_j{\partial u\over\partial x_j}= \lambda mu ,\quad B(u)=0 ,\quad u(x,t)=u(x,t+T) , \] where either \(B(u)=u|_{\partial\Omega\times R}\) (Dirichlet condition) or \(B(u)=(\partial u/\partial\nu)|_{\partial\Omega\times R}\) (Neumann condition).
existence, uniqueness and simplicity of the principal eigenvalue
scientific article; zbMATH DE number 1319588
scientific article