Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part

We study the Cauchy problem for the semilinear nonautonomous parabolic equation ut=A(t)u+ψ(t,u)ut=A(t)u+ψ(t,u) in [s,τ]×Rd[s,τ]×Rd, τ>sτ>s, in the spaces Cb([s,τ]×Rd)Cb([s,τ]×Rd) and in Lp((s,τ)×Rd,ν)Lp((s,τ)×Rd,ν). Here νν is a Borel measure defined via a tight evolution system of measures for the evolution operator G(t,s)G(t,s) associated to the family of time depending second order uniformly elliptic operators A(t)A(t). Sufficient conditions for existence in the large and stability of the null solution are also given in both CbCb and LpLp contexts. The novelty with respect to the literature is that the coefficients of the operators A(t)A(t) are allowed to be unbounded.