On the asymptotic behavior of the solutions of semilinear nonautonomous equations

TL;DR

This paper investigates the long-term behavior of solutions to nonautonomous semilinear evolution equations in Banach spaces, establishing conditions under which solutions decay exponentially based on properties of the Green's operator.

Contribution

It introduces new conditions involving the Green's operator that guarantee exponential decay of solutions in nonautonomous semilinear equations.

Findings

Green's operator maps $L^p$ to $L^q$ spaces

Lipschitz continuity of Green's operator implies decay

Exponential decay of solutions under specified conditions

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 $[s, s + \delta), \delta > 0$) as being the solution of the integral equation \label{integreq} x(t) = U(t, s)x + \int_s^t U(t, \tau)f(\tau, x(\tau)) d\tau \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.