Semigroup Forum

, Volume 87, Issue 1, pp 18–34 | Cite as

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

  • Nguyen Van Minh
  • Gaston M. N’guérékata
  • Ciprian Preda
RESEARCH ARTICLE

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.

Keywords

Semilinear evolution equations Exponential stability 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Nguyen Van Minh
    • 1
  • Gaston M. N’guérékata
    • 2
  • Ciprian Preda
    • 3
  1. 1.Department of Mathematics and PhilosophyColumbus State UniversityColumbusUSA
  2. 2.Department of MathematicsMorgan State UniversityBaltimoreUSA
  3. 3.Faculty of Economics and Business AdministrationWest University of TimisoaraTimisoaraRomania

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