, Volume 87, Issue 1, pp 18-34
Date: 04 Jan 2013

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

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Abstract

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space \(\mathbb{X}\) and \(f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}\) 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 xD(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 $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ 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∈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 \(t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}\) is the unique solution of Eq. (4), for every \(x\in\mathbb{X}\) and s∈ℝ+.

Considering the Green’s operator \((\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds\) we prove that if the following conditions hold

  • the map \(\mathbb{G}{f}\) lies in \(L^{q}(\mathbb{R}_{+},\mathbb{X})\) for all \(f\in L^{p}(\mathbb{R}_{+},\mathbb{X})\) , and

  • \(\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})\) is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.

Communicated by Jerome A. Goldstein.