Superconvexity of the evolution operator and parabolic eigenvalue problems on \(\mathbb R^N\)

Abstract

The purpose of this paper is to investigate the stability of the zero solution of the equation \[\partial_tu-k(t)\Delta u=\lambda m(x,t)u\] in \(\mathbb R^N\times (0,\infty )\) as the parameter \(\lambda\) varies over \([0,\infty)\) and \(k(t)\) positive and \(T\)-periodic. Assuming that \[\mathcal P(m):=\int_0^T\max_{x\in\mathbb R^N}m(x,\tau )d\tau >0\] we prove the existence of a number \(\lambda_1 (m)>0\), such that the zero solution of the above equation is exponentially stable if \(0<\lambda <\lambda_1(m)\), stable (but not exponentially stable) if \(\lambda =\lambda_1\), and unstable if \(\lambda >\lambda_1\).