# Superconvexity of the evolution operator and parabolic eigenvalue problems on $$\mathbb R^N$$

Preprint (PDF) 1993
Differential and Integral Equations 7 (1994), 235-255
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 $$\lambda >\lambda_1$$.