Existence, Regularity and Compactness Properties in the α -Norm for Some Partial Functional Integrodifferential Equations with Finite Delay

In this work, we study in the α -norm, the existence, the continuity dependence, regularity and compactness of solutions for some partial functional integro-differential equations by using the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer and Pritchard (J Diff Equ 50:234–259, 1983). The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results


Introduction
The main purpose of this work is to study the existence, regularity and compactness properties of a class for partial functional integrodifferential equations of retarded type with deviating arguments in terms involving spatial partial derivatives in the form where −A is the infinitesimal generator of an analytic semigroup (T (t)) t≥0 on a Banach space X. B(t) is a closed linear operator with domain D(B(t)) ⊃ D(A) time-independent. For 0 < α < 1, A α is the fractional power of A which will be precised in the sequel. The domain D(A α ), endowed with the norm x α = A α x is a Banach space. C α is the Banach space C([−r, 0], D(A α )) of continuous functions from [−r, 0] to D(A α ) endowed with the following norm F : R + × C α → X is a continuous function and as usual, the history function u t ∈ C α is defined by As a model for this class one may take the following Lotka-Voltera equation ∂ ∂ x u(t + θ, x))dθ for t ≥ 0 and x ∈ [0, π], u(t, 0) = u(t, π) = 0 for t ≥ 0, u(θ, x) = u 0 (θ, x) for θ ∈ [−r, 0] and x ∈ [0, π], and h : R + −→ R are appropriate functions. The theory of partial functional integrodifferential equations and its applications are an active areas of research, see for instance [1,10,17,20,21] and the references therein. In the particulary case where α = 0, that is we have the following integrodifferential Eq.  [1], Hannsgen [11], Smart [18], Miller [12,13], and Miller and Wheeler [14,15]. In the case where the non-linear part involves spatial derivative, the above obtained results become invalid. To overcome this difficulty, we shall restrict our problem in a Banach space Y α ⊂ X, to obtain our main results for Eq. (1.1).
Considering the case where B = 0, Travis et al. in [19] obtained results on the existence, stability, regularity and compactness of Eq. (1.1). To achieve their goal, the authors assumed that −A is the infinitesimal generator of a compact analytic semigroup and F is only continuous with respect to a fractional power of A in the second variable. The present paper is motivated by the paper of Travis et al. in [19].
The paper is organized as follows. In "Fractional Power of Closed Operators and Resolvent Operator for Integrodifferential Equations" section, we recall some fundamental properties of the resolvent operator and fractional powers of closed operators. The global existence, uniqueness and continuous dependence with respect to the initials data are studied in "GLobal Existence, Uniqueness and Continuous Dependence with Respect to the the Initials Data" section. In "Local Existence, Blowing Up Phenomena and the Compactness of the Flow" section, we study the local existence and bowing up phenomena. In "Regularity of the Mild Solutions" section we prove under some conditions, the regularity of the mild solutions. And finally we illustrate our main results in "Application" section by examining an example.

Fractional Power of Closed Operators and Resolvent Operator for Integrodifferential Equations
In this section, we first introduce some notations and definitions which we will use in the whole work.
We shall write Y for D(A) endowed with the graph norm x Y = x + Ax , Y α for D(A α ) and L(Y α , X) will denote the space of bounded linear operators from Y α to X and for Y 0 = X, we write L(X) with norm . L(X) . We also frequently use the Laplace transform of f which is denoted by f * . If we assume that −A generates an analytic semigroup and, without loss of generality that 0 ∈ (A), then one can define the fractional power A α for 0 < α < 1, as a closed linear operator on its domain Y α with its inverse A −α given by where is the Gamma function We have the following known results.
Theorem 2.1 [16] The following properties are true.
Now, we collect the definition and basic results about the theory of resolvent operator for integrodifferential equations in Banach spaces. Definition 2.2 [7] A family of bounded linear operators (R(t)) t≥0 in X is called resolvent operator for the following equation It what follows, we assume the hypothesis taken from [8] which implies the existence of an analytic resolvent operator (R(t)) t≥0 .
We give in next, the definition of the so-called strict and mild solutions. Consider the following nonhomogeneous equation Definition 2.5 [7] A continuous function u : [0, b] → X is called a strict solution of the Eq.

GLobal Existence, Uniqueness and Continuous Dependence with Respect to the Initials Data
This section is asserted to the results of global existence and continuous dependence with respect to the initials data. We give the definitions of the so-called mild and strict solutions of Eq. (1.1).

Definition 3.2 A continuous function
Now to obtain our first result, we take the following assumption.
(H1) There exists a constant L F > 0 such that
Proof Let a > 0. For ϕ ∈ C α , we define the set ∧ by We consider the operator defined on ∧ by We claim that (∧) ⊂ ∧ . In fact for y ∈ ∧ we have ( y)(0) = ϕ(0) and by continuity of F and R(t)x for x ∈ X, we deduce that y ∈ ∧. In order to obtain our result, we apply the strict contraction principle. In fact, let u, v ∈ ∧ and t ∈ [0, a]. Then Using the α-norm, we have Then is a strict contraction on ∧ and it has an unique fixed point y which is the unique mild solution of Eq.
Notice that the solution of Eq. (3.2) is given by Letz be the function defined byz Consider now again the set ∧ defined by provided with the induced topological norm. We define the operator a on ∧ by We have ( a z)(a) = y(a) and a z is continuous. Then it follows that a ∧ ⊂ ∧. Morever, for u, v ∈ ∧, one has Then we deduce that a has an unique fixed point in ∧ which extends the solution y in [a, 2a]. Proceeding inductively, y is uniquely and continuously extended to [na, (n + 1)a] for all n ≥ 1 and this ends the proof.
Now we show the continuous dependence of the mild solutions with respect to the initial data.

Theorem 3.4 Assume that (V1)-(V3) and (H1) hold. Then the mild solution u(., ϕ) of equa-
Moreover there exist a real number δ and a scalar function P such that for t ≥ 0 and ϕ 1 , ϕ 2 ∈ C α we have Proof We use the Gamma formula where k > 0 (see [9], p. 265). The continuity is obvious on what the map t → u t (., ϕ) is Let δ a real number be such that We define the function P by Fixt > 0 and let E = sup 0≤s≤t e −δs w s . If 0 ≤ τ ≤t, then from (3.4), we have Therefore, (3.5) and (3.6) imply that Then from (3.7) and (3.8) we deduce that for 0 ≤ t ≤t Then the result follows.

Local Existence, Blowing up Phenomena and the Compactness of the Flow
In this section, we establish the local existence and blowing up phenomena under assumption that F is continuous. We also study the compactness of the flow. To achieve our goal, we need the following Lemma used in [3] in the case of the usual norm on X (α = 0). It will be seen in the case of α-norm. We take the following assumption.
Proof Let a > 0 and x ∈ X. Then We deduce that R(t + h)x satisfies the equation of the form Then by Theorem 2.4, it follows that which yields that Taking the α-norm, we obtain that Let p such that 1/q + 1/ p = 1, so p < 1/β. Then it follows that And the proof is complete.
The local existence result is given by the following Theorem. Proof Let ϕ ∈ . For any real ζ ∈ J and p > 0, we define the following sets For φ ∈ H p , we choose ζ and p such that (t, φ + ϕ) ∈ I ζ × H p and H p ⊆ . By continuity of F, there exists N 1 ≥ 0 such that F(t, φ + ϕ) ≤ N 1 for (t, φ) in I ζ × H p . We consider ϕ ∈ C [−r, ζ ]; Y α be the function defined byφ(t) = R(t)ϕ(0) for t ∈ I ζ andφ 0 = ϕ. Suppose thatp < p and choose 0 < b < ζ such that Notice that finding a fixed point of S in K 0 is equivalent to find a mild solution of Eq. (1.1) in K 0 . Furthermore, S is a mapping from K 0 to K 0 , since if η ∈ K 0 we have Sη 0 = 0 and Then In fact, let β be such that 0 < α ≤ β < 1. The above estimate show that A β Sη (t) : η ∈ K 0 is bounded in X. Since A α−β is compact operator, we infer that A α−β A β Sη (t) : η ∈ K 0 is compact in X, hence Sη (t) : η ∈ K 0 is compact in Y α . Next, we show that Sη (t) : η ∈ K 0 is equicontinuous. The equicontinuity of Sη (t) : η ∈ K 0 at t = 0 follows from the above estimation of Sη (t). Now let 0 < t 0 < t ≤ b with t 0 be fixed. Then we have (4.2) Using Theorem 4.1, it follow that As the set Sη (t 0 ) : η ∈ K 0 is compact in Y α , we have that We obtain the same results by taking t 0 be fixed with 0 < t < t 0 ≤ b. Then we claim that uniformly in η ∈ K 0 which means that Sη (t) : η ∈ K 0 is equicontinuous. Then by Ascoli-Arzela Theorem, Sη : η ∈ K 0 is relatively compact in K 0 . Finally, we prove that S is continuous. Since F is continuous, given ε > 0, there exists δ > 0, such that sup 0≤s≤b η(s) −η(s) α < δ implies that F(s, η s +φ s ) − F(s,η(s) +φ s ) < ε.
This yields the continuity of S and using Schauder's Fixed Point Theorem we deduce that S has a fixed point. Then the proof of the Theorem is complete.
The following result gives the blowing up phenomena of the mild solution in finite times.  b 1 > b and a function u ., u

By continuity of A α R(t), we claim that A α R(t + h) − R(t) ϕ(0) is uniformly continuous on each compact set. Moreover, Theorem 4.1 implies that
Then using Theorem 4.1, we obtain that We claim that the set is relatively compact. In fact, let (t n ) n≥0 be a sequence of [0, b ϕ ). Then there exist a subsequence (t n k ) k and a real number t 0 such that t n k → t 0 . Using Dominate Convergence Theorem we deduce that

This implies that
is relatively compact. Now using Banach Steinhauss's theorem we deduce that one can show a similar results by using the same reasoning. This implies that u(., ϕ) is uniformly continuous. Therefore lim t→b − ϕ u(t, ϕ) exists in Y α . And consequently, u(., ϕ) can be extended to b ϕ which contradicts the maximality of [0, b ϕ ).
Next result gives the global existence of the mild solutions under weak conditions of F. To achieve our goal, we introduce a following necessary result which is a consequence of Lemma 7.1.1 given in [2](p. 197, Exo 4).

Theorem 4.5 Assume that (V1)-(V3), (H0) and (H2) hold and F is a completly continuous function on R + × C α . Moreover suppose that there exist continuous non-negative functions
Then taking the α-norm, we obtain Now we claim that the function is non-decreasing. In fact, let 0 ≤ t 1 ≤ t 2 . Then which yields the result. Then it follow from (4.3) that Then using Lemma 4.4, we deduce that u(., ϕ) is bounded in [0, b ϕ ). Consequently lim t→b − ϕ u(t, ϕ) α < ∞, which contradicts our hypothesis. Then the mild solution is global.
We focus now to the compactness of the flow defined by the mild solutions.
Proof We use Ascoli-Arzela's theorem. Let E = ϕ γ : γ ∈ be a bounded subset of C α and let t > r be fixed, but arbitrary. We will prove that U (t)E is compact. It follows from (H1) and inequality (3.3) that there exists N 5 such that For each γ ∈ , we define f γ ∈ C α by f γ = u t (., ϕ γ ). We show now that for fixed θ ∈ [−r, 0], the set f γ (θ ) : γ ∈ is precompact in Y α . For any γ ∈ , we have , u s (., ϕ))ds.

Then it follows that
Using the compactness of the set , u s (., ϕ γ ))ds : γ ∈ and the continuity of t → R(t)x for x ∈ X the right side of the above inequality can be made sufficiently small for h > 0 small enough. Then we conclude that f γ : γ ∈ is equicontinuous. Consequently, by Ascoli-Arzela's Theorem we conclude that the set {U (t)ϕ : ϕ ∈ E} is compact, which means that the operator U (t) is compact for t > r .

Regularity of the Mild Solutions
This section is devoted to the regularity of the mild solution. We define the set C 1 α by C 1 α = C 1 ([−r, 0]; Y α ) as the set of continuously differentiable functions from [−r, 0] into Y α . We assume the following hypothesis.
(H3) F is continuously differentiable and the partial derivatives D t F and D ϕ F are locally Lipschitz in the classical sense with respect to the second argument. Proof Let a > 0. Take ϕ ∈ C 1 α be such that ϕ(0) ∈ Y andφ(0) = −Aϕ(0) + F(0, ϕ) and let u be the mild solution of Eq. (1.1) which is defined on [0, +∞[. Consider the following equation Then it follows Consequently, the map t −→ w t and t −→ t 0 R(t − s)F(s, w s )ds are continuously differentiable and the following formula holds This implies that On the other hand, from equality (2.2), we have We rewrite w as following Then it follows that We deduce, for t ∈ [0, a], that F(τ, w τ ))v τ dτ ds. (5.2) The set H = {u s , w s : s ∈ [0, a]} is compact in C α . Since the partial derivatives of F are locally Lipschitz with respect to the second argument, it is well-known that they are globally Lipschitz on H . Then we deduce that If we choose a such that . Now we will prove that u = w in [0, +∞). Assume that there exists . By continuity, one has u(t) = w(t) for t ≤ t 1 and there exists ε > 0 such that u(t) − w(t) > 0 for t ∈ (t 1 , t 1 + ε). Then it follows that, for t ∈ (t 1 , t 1 + ε) Now choosing ε such that which gives a contradiction. Consequently, u(t) = w(t) for t ≥ 0. We conclude that t → u t from [0, +∞) to Y α and t → F(t, u t ) from [0, +∞) × C α to X are continuously differentiables. Thus, we claim that u is a strict solution of Eq. (1.1) on [0, +∞).
We assume the following assumptions.
Considering D = C ∞ 0 ([0, π]), we see that the condition (V1)-(V3) and (H0) are verified. Hence the homogeneous linear equation of equation (6.1) has an analytic compact resolvent operator (R(t)) t≥0 . The function F is continuous in the first variable from the fact that g is continuous in the first variable. Moreover from Lemma 6.1 and the continuity of g, we deduce that F is continuous with respect to the second argument. This yields the continuity of F in R + × C 1/2 . In addition, by assumption (H5) we deduce that Then F is a continuous globally Lipschitz function with respect to the second argument. We obtain the following important result. Proposition 6.2 Suppose that the assumptions (H4)-(H5) hold. Then the Eq. (6.2) has a mild solution wich is defined for t ≥ 0.