Local existence and stability for some partial functional differential equations with infinite delay☆

In this work, we are concerned with the local existence and stability for the following partial functional differential equations with infinite delay$\text{d}\text{x}\text{d}\text{t}\text{(t)=Ax(t)+F(x}{}_{\mathrm{t}}\text{),}\text{t≥0,}\text{x}{}_0\text{=ϕ∈}\text{B}\text{,}$where A is a closed linear operator on a Banach space (E,|.|), the phase space $\text{B}$ is a linear space of functions mapping (−∞,0] into E satisfying some axioms which are described in the sequel, F is an E-valued appropriate function, and for every t≥0, the function $\text{x}{}_{\mathrm{t}}\text{∈}\text{B}$ is defined by$\text{x}{}_{\mathrm{t}}\text{(θ)=x(t+θ)}\text{for}\text{θ∈(−∞,0].}$

In the literature devoted to equations with finite delay, the state space is much of time the space of all continuous functions on [−r,0], r>0, endowed with the uniform norm topology. When the delay is infinite, the selection of the state space $\text{B}$ plays an important role in the study of both qualitative and quantitative theory. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato [12], Kappel and Schappacher [18], and Schumacher [31]. For a detailed discussion on this topic, we refer the reader to the book by Hino et al. [17].

In recent years, the theory of partial functional differential equations with delay has attracted widespread attention. The development was initiated for equations with finite delay and about existence and stability by Travis and Webb [35], [36], and Webb [37], [38]. For later development, we cite only a paper by Arino and Sanchez [5] and a recent book by Wu [40]. In the standard framework for Eq. (1), one assumes that the operator A is the infinitesimal generator of a C₀-semigroup of bounded linear operators in E or, equivalently$\text{∥(λ−ω)}{}^{\mathrm{n}}\text{(λI−A)}{}^{\mathrm{-n}}\text{∥≤β}\text{for}\text{all}\text{n∈}\text{N}\text{,}$where $\text{L}\text{(E)}$ is the space of bounded linear operators from E into E. In that case, the classical semigroups theory ensures the well posedness of Problem (1). More recently, it has been shown in [1], [2] that the density condition is not necessary (in a certain sense) to deal with partial functional differential equations with finite or infinite delay. In the applications, it is sometimes convenient to take initial functions with more restrictions. There are many examples in concrete situations where evolution equations are not densely defined. Only hypothesis (ii) holds. One can refer for this to [8] for more details. Non-density occurs, in many situations, from restrictions made on the space where the equation is considered (for example, periodic continuous functions, Hölder continuous functions) or from boundary conditions (e.g., the space $\text{C}{}^1$ with null value on the boundary is non-dense in the space of continuous functions).

Concerning the case of infinite delay, an extensive theory is developed for Eq. (1) with A=0. We refer the reader to Hale and Kato [12], Corduneanu and Lakshmikantham [7], Hale [10], [11], Shin [32], Hino et al. [17], and Lakshmikantham et al. [20]. The extension to the case when A is the infinitesimal generator of a C₀-semigroup (T(t))_t≥0 was later on studied by Henriquez in his three consecutive papers [13], [14], [15]. Following an axiomatic approach, he developed several fundamental results on the existence of solutions, regularity, existence of periodic solutions and stability. Henriquez proved his results by using the following variation-of-constants formula:$\text{x(t,ϕ)=}\text{T(t)ϕ(0)+}\text{∫}\text{0}\text{t}\text{T(t−s)F(s,x}{}_{\mathrm{s}}\text{)}\text{d}\text{s}\text{for}\text{t≥0,}\text{ϕ(t)}\text{for}\text{−∞<t≤0.}$In [22], Milota obtained some results on existence and stability. In [29], Ruan and Wu developed a general theory of existence, comparison, invariance and monotonicity and provide some applications to reaction diffusion systems with general distributed delays. Recently, in [1], we treated Eq. (1) when A is non-densely defined and satisfies the Hille–Yosida condition. We obtained some results about the global existence under the assumption that F is globally Lipschitz continuous.

As in [1], we study Eq. (1) without assuming necessarily that A is densely defined. We state the local existence and regularity of solutions under a locally Lipschitz condition on F. We extend the results obtained in [2] to the case of infinite delay. In the case of global existence, we give some properties of the solution map. Mainly, we discuss the stability of the trivial solutions. The method used here is based on the integrated semigroup theory.

Let us now briefly discuss about the advantage of using the integrated semigroups. In the case when the operator F of Eq. (1) is equal to zero, the problem can still be handled by using the classical semigroup theory because A generates a strongly continuous semigroup in the space $\text{D(A)}\text{.}$ But, when F≠0 it is necessary to impose additional restrictions, the simplest of which is that F takes values in $\text{D(A)}$. It is the integrated semigroups theory that allows the range of the operator F to be a subset of E not necessarily contained in $\text{D(A)}$.

The organization of this work is as follows: in Section 2, we collect some useful results on Hille–Yosida operators and integrated semigroups. In Section 3, we study the local existence and the regularity of solutions of Eq. (1). In Section 4, we state some properties of the solution operator associated to Eq. (1). We investigate the stability near an equilibrium and we prove that if the linearized semigroup around an equilibrium is exponentially stable, then the equilibrium of Eq. (1) is also exponentially stable. The last section is devoted to an application to a reaction diffusion equation with infinite delay.