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 $${\partial }_{t}u-k(t)\mathrm{\Delta }u=\lambda m(x,t)u$$ in ${\mathbb{R}}^N\times (0,\mathrm{\infty })$ as the parameter $\lambda$ varies over $[0,\mathrm{\infty })$ and $k(t)$ positive and $T$-periodic. Assuming that $$\mathcal{P}(m):={\int }_0^T\underset{x\in {\mathbb{R}}^N}{max}m(x,\tau )d\tau >0$$ 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 $\lambda >{\lambda }_1$.