Abstract

We consider nonautonomous semilinear evolution equations of the form dxdt=A(t)x+f(t,x). Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space X and f:R×X→X is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}(t,s)∈Δ such that for every s∈ℝ+ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation x(t)=U(t,s)x+∫tsU(t,τ)f(τ,x(τ))dτ,t≥s. Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈ℝ+, and f(t,0)=0 for all t∈ℝ+) we can generate a (nonlinear) evolution family {X(t,s)}(t,s)∈Δ , in the sense that the map t↦X(t,s)x:[s,∞)→X is the unique solution of Eq. (4), for every x∈X and s∈ℝ+. Considering the Green’s operator (Gf)(t)=∫t0X(t,s)f(s)ds we prove that if the following conditions hold the map Gf lies in Lq(R+,X) for all f∈Lp(R+,X), and G:Lp(R+,X)→Lq(R+,X) is Lipschitz continuous, i.e. there exists K>0 such that ∥Gf−Gg∥q≤K∥f−g∥p,for all f,g∈Lp(R+,X), then the above mild solution will have an exponential decay.