INTEGRODIFFERENTIAL EQUATIONS WITH ANALYTIC SEMIGROUPS

In this paper we study a class of integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness, regularity and continuation of solutions to these integrodifferential equations.


Introduction
In this paper we are concerned with the following integrodifferential equation in a Banach space X: du(t) dt + Au(t) = f (t, u(t)) + K(u)(t), t > t 0 , (1.1) where In (1.1), we assume that −A generates an analytic semigroup, S(t), t ≥ 0 on X, the function a is real-valued and locally integrable on [0, ∞), and the nonlinear maps f and g are defined on [0, ∞)×X into X.We first establish that under Assumption F0, stated below, there exists a unique local mild solution to (1.1).Then under Assumption F, stated below, we study the regularity of the mild solution to (1.1) and show under additional condition of Hölder continuity on a that the mild solution to (1.1) is in fact the classical solution.Further, we analyze the continuation of the solutions to (1.1) under different conditions.Finally, at the end we give an example of a class of parabolic integrodifferential equations as an application of the results obtained for the abstract integrodifferential equation (1.1).Equation (1.1) represents an abstract formulation of certain classes of parabolic integrodifferential equations.These type of equations model the physical phenomena involving certain type of memory effects.For instance, Nohel [12] has considered a nonlinear Volterra equation of the type (1.1), in which g(t, u(t)) = Bu(t), where −B is a nonlinear accretive operator.For more details on such formulations and corresponding techniques used to study such problems, we refer to Bahuguna and Pani [2], Barbu [4,5,6], Crandall, Londen and Nohel [7].
Heard and Rankin [9] have considered the following integrodifferential equation in a Banach space X: where the linear operator −A(t) for each t ≥ 0 is the infinitesimal generator of an analytic semigroup in X, the nonlinear map g, defined on [0, ∞) × D(A(0)) into X, is such that g(t, .) is Lipschitz continuous on the domain D(A(0)) of A(0) into X with respect to the graph norm of A(0), the nonlinear map f , defined on [0, ∞) × X α into X, satisfies the condition that there exist constants L > 0, 0 < η, γ ≤ 1 and 0 for all (t, x), (t, y) ∈ [0, ∞) × X α .Here X α for 0 ≤ α ≤ 1 is the Banach space D(A α ) endowed with the norm u α = A α u .
Webb [15] has also considered (1.3) and has assumed that f maps R × X 1 into X α and for each t ∈ R there exists a positive constant C(t) such that (1.5) for all x, y ∈ X 1 .
The existence result is proved by first solving the following integrodifferential equation uniquely: where v(t) is chosen from a closed, bounded, convex subset S of an appropriate Banach space.Existence of a unique u(t) is established by proving that the mapping K(v) = u v is a strict contraction from S into S.This is possible because of the extra smoothness assumption (1.5) on f .Since Heard and Rankin [9] assumed a weaker condition (1.4), they require an estimate of the following type on the map K: and they use Schauder's fixed point theorem to establish the existence.
When (1.4) is replaced by the stronger assumption (2.1), stated below, the methods used by Heard and Rankin [9] do not automatically imply that the solution is unique.They are able to prove uniqueness in the case in which X is a Hilbert space.Furthermore, the nonlinear map g is assumed to be defined from [t 0 , T ) × W into X where W is an open subset of X 1 and satisfies the local Lipschitz condition: for all t, s ∈ [t 0 , T ) and x, y

Preliminaries
Let X denote a Banach space and let J denote the closure of the interval [t 0 , T ) t 0 < T ≤ ∞.Let −A be the infinitesimal generator of an analytic semigroup S(t), t ≥ 0 in X.We note that if −A is the infinitesimal generator of an analytic semigroup then −(A + αI) is invertible and generates a bounded analytic semigroup for α > 0 large enough.This allows us to reduce the general case in which −A is the infinitesimal generator of an analytic semigroup to the case in which the semigroup is bounded and the generator is invertible.Hence for convenience, we suppose that where ρ(−A) is the resolvent set of −A.It follows that for 0 ≤ α ≤ 1, A α can be defined as a closed linear invertible operator with its domain D(A α ) being dense in X.
We denote by X α the Banach space D(A α ) equipped with norm which is equivalent to the graph norm of A α .We have and the embedding is continuous.By a classical solution to (1.1) on J, we mean a function u ∈ C(J; X)∩C 1 (J\{t 0 }; X) satisfying (1.1) on J.By a local classical solution to (1.1) on J we mean that there exist a T 0 , t 0 < T 0 < T , and a function u defined from To establish the existence of a unique classical solution to (1.1) in later sections, we shall require the following assumption on the maps f and g.
By a mild solution to (1.1) on J we mean a continuous function u defined from J into X satisfying the following integral equation (2.2) We say that (1.1) has a local mild solution if there exist a T 0 , 0 < T 0 < T and a continuous function u defined from J 0 = [t 0 , T 0 ] into X such that u is a mild solution to (1.1) on J 0 .
To establish the existence of a unique local mild solution, we only need the following assumptions on f and g.
3) for all (s, u) and (s, v) in V .

Local Existence of Mild Solutions
As pointed out earlier, we may suppose without loss of generality that the analytic semigroup generated by −A is bounded and that −A is invertible.Furthermore, we assume that 0 < T < ∞ to establish local existence.With these simplifications we have the following theorem.Theorem 3.1: Suppose that the operator −A generates the analytic semigroup S(t) with S(t) ≤ M , t ≥ 0 and that 0 ∈ ρ(−A).If the maps f and g satisfy Assumption F0 and the real valued map a is integrable on J, then (1.1) has a unique local mild solution for every u 0 ∈ X α .
Proof: We shall use the notions and notations introduced in the preceding section.We fix a point (t 0 , u 0 ) in the open subset U of [0, ∞) × X α and choose t 1 > t 0 and δ > 0 such that (2.3), with some constant L 0 > 0 holds for the functions f and g on the set and where C α is a positive constant depending on α satisfying and Let Y = C([t 0 , t 1 ]; X) be endowed with the supremum norm Then Y is a Banach space.We define a map on Y by F y = ỹ where ỹ is given by Then for y ∈ S we have where the last two inequalities follow from (3.2) and (3.3).Thus, we have that F : S −→ S. Now we show that F is a strict contraction on S which will ensure the existence of a unique continuous function satisfying equation (2.2).Let y and z be in S; then Using Assumption F0 on f and g and (3.4), (3.5), we get using (3.3) in the last inequality.Thus F is a strict contraction map from S into S and therefore by the Banach contraction principle there exists a unique fixed point y of F in S, i.e., there is a unique y ∈ S such that

.11)
Let u = A −α y.Then for t ∈ [t 0 , t 1 ], we have Hence u is a unique local mild solution to (1.1).

Regularity of Mild Solutions
In this section we establish the regularity of the mild solutions to (1.1).Again, let J denote the closure of the interval [t 0 , T ), t 0 < T ≤ ∞.In addition to the hypotheses mentioned in the earlier sections, we assume the following on the kernel a: (H) There exist constants C 0 ≥ 0 and 0 < β ≤ 1 such that for all t, s ∈ J.
Theorem 4.1:Suppose that −A generates the analytic semigroup S(t) such that S(t) ≤ M for t ≥ 0, and 0 ∈ ρ(−A).Further, suppose that the maps f and g satisfy Assumption F and the kernel a satisfies (H).Then (1.1) has a unique local classical solution for each u 0 ∈ X α .
Proof: From Theorem 3.1, it follows that there exist T 0 , t 0 < T 0 < T and a function u such that u is a unique mild solution to (1.1) on J 0 = [t 0 , T 0 ) given by where For simplification, we set Then (4.3) can be rewritten as Since u(t) is continuous on J 0 and the maps f and g satisfy Assumption F, it follows that f and g are continuous, and therefore bounded on J 0 .Let We show that f and g are locally Hölder continuous on J 0 .For this, we first show that v(t) is locally Hölder continuous on J 0 .From Theorem 2.6.13 in Pazy [13], it follows that for every 0 < β < 1 − α and every 0 < h < 1, we have Next, we have where M 1 depends on t and blows up as t decreases to t 0 .Furthermore where M 2 is independent of t.Also, we have where M 3 is also independent of t.From the estimates (4.9)-(4.11), it follows that there exists a constant C 1 such that for every t 0 > t 0 , we have for all t 0 < t 0 < t, s < T 0 .Now, Assumption F together with (4.12) implies that there exist constants C 2 , C 3 ≥ 0 and 0 < γ, η < 1 such that for all t 0 < t 0 < t, s < T 0 , we have Let Now we show that h(t) is locally Hölder continuous on J 0 .For s ≤ t, we have for some constants C 4 ≥ 0 and 0 < δ < 1.Consider the following initial value problem By Corollary 4.3.3 in Pazy [13], (4.16) has a unique solution v ∈ C 1 ((t 0 , T 0 ]; X) given by For t > t 0 , each term on the right hand side belongs to D(A) and hence belongs to D(A α ).Applying A α to both sides of (4.17) and using the uniqueness of v(t), we have that A α v(t) = u(t).Thus, it follows that u is the classical solution to (1.1) on J 0 .

Global Existence
In order to establish the global existence of classical solutions to (1.1), we need the following lemma.Lemma 5.1:Let φ(t, s) ≥ 0 be continuous on 0 ≤ s ≤ t ≤ T < ∞.If there are positive constants A, B and α such that ) which holds for every α, β > 0. Integrating (5.1) n − 1 times using (5.2) and replacing t − s by T , we get Let n be large enough so that nα > 1.We majorize (t − σ) nα−1 by T nα−1 to obtain then the initial value problem (1.1) has a unique classical solution u on [t 0 , ∞) for every u 0 ∈ X α .Proof: From Theorem 4.1 it follows that there exist a T 0 , t 0 < T 0 and a unique classical solution u on for t ∈ J 0 for some positive constant C, then the solution u(t) may be continued further on the right of T 0 .Therefore it suffices to prove that if a classical solution u to (1.1) exists on [t 0 , T ], t 0 < T < ∞ then u(t) α is bounded as t ↑ T .Since u(t) is a classical solution as well it is a mild solution.Therefore we have Making use of the fact that S(t) commutes with A and that for t ≥ t 0 in (5.8), after applying A α and taking norms on both sides, we get a(s − τ )g(τ, u(τ ))dτ ds.(5.9) For the last term in (5.9), we have the estimate Incorporating the estimate of (5.10) in (5.9), we get After a slight modification in (5.11), we get (5.12)

Applications
Let Ω ⊂ R n be a bounded domain with smooth boundary ∂Ω.Consider the linear partial differential operator x ∈ Ω, t > t 0 , u(x, t 0 ) = u 0 (x) x ∈ Ω u(x, t) = 0, x ∈ Ω, t ∈ [t 0 , T ), t 0 < T ≤ ∞, where D j stands for any j-th order derivative.We assume that f and g are continuously differentiable functions of all their variables, except possibly in x.
The parabolic integrodifferential equation (6.3) can be reformulated as the following abstract integrodifferential equation in X = L p (Ω): du(t) dt + A p u(t) = F (t, u(t)) + F (t, u)(x) = f (x, t, u(x, t), Du(x, t), ..., D 2m−1 u(x, t)) (6.5) G(t, u)(x) = g(x, t, u(x, t), Du(x, t), ..., D 2m−1 u(x, t)) (6.6)where we assume the usual sufficient Caratheodory and growth conditions on the functions f and g for the Nemyckii operators in (6.5) and (6.6) to be well defined.Here we assume that λ is large enough so that A p is invertible.It follows that −A p is the infinitesimal generator of an analytic semigroup on X.Also, from imbedding theorems it follows that X α is continuously imbedded in C 2m−1 ( Ω) for 1 − 1/2m < α < 1 and p large enough.It can be verified that the Assumption F is satisfied by F and G.Under suitable assumptions on the kernel a, Theorem 4.1 assures the existence of a unique global classical solution to (6.4) for p large enough which in turn guarantees the existence of a unique global classical solution to (6.3).