On the asymptotic behavior of the solutions of semilinear nonautonomous equations

Abstract

We consider nonautonomous semilinear evolution equations of the form \label{semilineq} \frac{dx}{dt}= 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 $f: \R\times\X\to\X$ is a (possibly nonlinear) continuous function. We assume that the linear equation \eqref{lineq} is well-posed (i.e. there exists a continuous linear evolution family \Uts such that for every $s\in\R_+$ and $x\in D(A(s))$, the function $x(t) = U(t, s) x$ is the uniquely determined solution of equation \eqref{lineq} satisfying $x(s) = x$). Then we can consider the \defnemph{mild solution} of the semilinear equation \eqref{semilineq} (defined on some interval $[s, s + δ), δ> 0$) as being the solution of the integral equation \label{integreq} x(t) = U(t, s)x + \int_s^t U(t, τ)f(τ, x(τ)) dτ\quad,\quad t\geq 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 $t\in\R_+$, and $f(t,0) = 0$ for all $t\in\R_+$) we can generate a (nonlinear) evolution family \Xts, in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\X$ is the unique solution of equation \eqref{integreq}, for every $x\in\X$ and $s\in\R_+$. Considering the Green's operator $(\G f)(t)=\int_0^t X(t,s)f(s)ds$ we prove that if the following conditions hold \bullet \quad the map $\G f$ lies in $L^q(\R_+,\X)$ for all $f\in L^{p}(\R_+,\X)$, and \bullet \quad $\G:L^{p}(\R_+,\X)\to L^{q}(\R_+,\X)$ is Lipschitz continuous, i.e. there exists $K>0$ such that $$|\G f-\G g|_{q} \leq K\|f-g\|_{p}, for all f,g\in L^p(\R_+,\X),$$ then the above mild solution will have an exponential decay.