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 $[s,\tau]\times {{\mathbb R}<sup>d}$, $\tau&gt; s $, in the spaces $C_b([s, \tau]\times{{\mathbb R}<sup>d})$ 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 $L<sup>p$ contexts. The novelty with respect to the literature is that the coefficients of the operators $\mathcal{A}(t)$ are allowed to be unbounded.