REGULARITY AND STABILITY FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Khalil EZZINBI1 and Aziz OUHINOU2

In this work, we study the asymptotic behavior of solutions to some partial functional differential equations as ( (1)

dx = Ax(t) + Lxt , t ≥ 0, dt x0 = ϕ ∈ B.

where A generats a strongly continuous semigroup on a Banach space X, B is a normed space consists of functions mapping (−∞, 0] to X characterized by some axioms introduced by J.Hale and J.Kato, xt ∈ B defined by xt (θ) = x(t + θ) for θ ∈ (−∞, 0], and L is a bounded linear operator from the phase space B into X. We establish a new representation for the generator A of the solution semigroup to equation (1). Furthermore, we give spectral properties of A. Using this results, we discuss stability criteria of the solutions to equation (1).

References [1] M. Adimy, H. Bouzahir and K. Ezzinbi, Local Existence and stability for some partial functional differential equations with infinite delay, Nonlinear Analysis (2002) 323-348. [2] T. Naito, J. S. Shin, S. Murakami, The generator of the solution semigroup for the general linear functional differential equation,Bull.Univ. Electro-comm. Vol. , N. 2, 29-38,(1998).

1

Universit´e Cadi-Ayyad, Facult´e des Sciences Semlalia, D´epartement de Math´ematiques, B.P 2390 Marrakech Maroc.(e-mail: [email protected]) 2 Universit´e Cadi-Ayyad, Facult´e des Sciences Semlalia, D´epartement de Math´ematiques, B.P 2390 Marrakech Maroc.(e-mail: [email protected]) 1