On the asymptotic behavior of the solutions of semilinear nonautonomous equations
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On the asymptotic behavior of the solutions of semilinear nonautonomous equations Nguyen Van Minh · Gaston M. N’guérékata · Ciprian Preda
Received: 13 October 2012 / Accepted: 22 November 2012 / Published online: 4 January 2013 © Springer Science+Business Media New York 2013
Abstract We consider nonautonomous semilinear evolution equations of the form dx = A(t)x + f (t, x). dt 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 t U (t, τ )f τ, x(τ ) dτ, t ≥ s. x(t) = U (t, s)x + 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
Communicated by Jerome A. Goldstein. N. Van Minh Department of Mathematics and Philosophy, Columbus State University, Columbus, GA 31907, USA e-mail: [email protected] G.M. N’guérékata () Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA e-mail: Gaston.N’[email protected] C. Preda Faculty of Economics and Business Administration, West University of Timisoara, J.H. Pestalozzi Street, No. 16, 300115 Timisoara, Romania e-mail: [email protected]
On the asymptotic behavior of the solutions
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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+ . t Considering the Green’s operator (Gf )(t) = 0 X(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 − Ggq ≤ Kf − gp ,
for all f, g ∈ Lp (R+ , X),
then the above mild solution will have an exponential decay. Keywords Semilinear evolution equations · Exponential stability
1 Introduction In the last few decades, we can note an increasing research interest in the asymptotic behavior of the solutions of the linear differential equations dx = A(t)x, dt
t ∈ [0, ∞), x ∈ X,
(1)
where A(t) is in general an unbounded linear operator on a Banach space X, for every fixed t. In the case that A(t) is a matrix continuous function, O. Perron [19] first observed a connection between the asymptotic behavior of the solutions of the above equation d and the properties of the differential operator dt − A(t) as an operator on the space n n Cb (R+ , R ) of R -valued, bounded and continuous functions on the half-line R+ . This
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