Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part

Abstract

We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+\psi (t,u)$ in $[s,\tau ]\times {\mathbb{R}}^d$, $\tau >s$, in the spaces $C_b([s,\tau ]\times {\mathbb{R}}^d)$ and in $L^p((s,\tau )\times {\mathbb{R}}^d,\nu )$. Here $\nu$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The novelty with respect to the literature is that the coefficients of the operators $\mathcal{A}(t)$ are allowed to be unbounded.