Existence and uniqueness of a solution to a partial integro-differential equation by the method of lines

In this work we consider a partial integro-differential equation. We reformulate it a functional integro-differential equation in a suitable Hilbert space. We apply the method of lines to establish the existence and uniqueness of a strong solution. w(x,t) @x2 + q(x) @w(x,t) @t + f1(x,t,w(x,t), @w(x,t) @t ) + R t 0 k(t,s) h @ 2 w(x,s) @x2 + f2(x,s,w(x,s), @w(x,s) @s ) i ds, (x,t) 2 (0,1) × (0,T), 0 < T < 1, w(x,0) = g1(x), @w @t (x,0) = g2(x), x 2 (0,1),

In the next section we transform (1.1) as a Cauchy problem for the following functional integro-differential equation in the product Hilbert space H := H 1 0 (0, 1)× L 2 (0, 1), where A : D(A) ⊂ H → H is shown to be the infinitesimal generator of a contraction semigroup in H and the nonlinear function F : [0, T ] × C 0 → H.Here the space C t := C([−T, t]; H), t ∈ [0, T ], is the Banach space of all continuous functions from [−T, t] into H endowed with the norm . H being the norm in H given by The space C 0 is called the "history space" or the phase space.We also show that F verifies a Lipschitz condition under certain assumptions on the functions f 1 and f 2 .
Our aim is to apply Rothe's method to (1.2) involving delays to establish the existence and uniqueness of a strong solution which in turn will guarantee the wellposedness of (1.1).The delay differential equations arise in the study of various population dynamical models [9].The method of lines is a powerful tool for proving the existence and uniqueness of solutions to evolution equations.This method is oriented towards the numerical approximations.For instance, we refer to Rektorys [11] for a rich illustration of the method applied to various interesting physical problems.Until today, the application of method of lines includes only those nonlinear differential and Volterra integro-differential equations (VIDEs) in which bounded, though nonlinear, operators appear inside the integrals, see Kacur [6,7], Rektorys [11], Bahuguna and Raghavendra [2] and Bahuguna, Pani and Raghavendra [5].
In the present study we extend the application of the method of lines to a class of nonlinear VIDEs in which differential operators occur inside the integrals and hence are unbounded.Motivation for considering such problems arises from the theory of wave propagation under the influence of damping, see Bahuguna [1], and Bahuguna and Shukla [4] and references cited therein.
We assume that q : [0, 1] → C is measurable and satisfies the following conditions.(C1) There exist constants γ ≥ 0 and δ > 0 such that where Re q(x) and Im q(x) denote the real and imaginary parts of q(x).
(C2) For every 0 We define Under the conditions (C1) and (C2) on q, it follows that the closure A, of A 0 is given by where , is a bounded linear operator on L 2 (0, 1), is the infinitesimal generator of a C 0 -semigroup of contractions in H (cf. Engel and Nagel [8], page 381).Here, for any operator L, L represents the closure of L.
With these operators introduced and for g 1 ∈ H 1 0 (0, 1) and g 2 ∈ L 2 (0, 1), we rewrite (1.1) as where and, for i = 1, 2, F i : [0, T ] × H → H are given by (2.9) we may rewrite (2.5) as (1.2) where F : [0, T ] × C 0 → H, given by and φ ∈ C 0 is given by We list here the properties of the linear operator A, the nonlinear map F and the kernel k.
(P1) The operator A : D(A) ⊂ H → H is the infinitesimal generator of a C 0 semigroup S(t) of contractions in H.
(P2) The function F : [0, T ] × C 0 → H satisfies the Lipschitz condition, i.e., there exists a positive constant L F such that We have the following main result.

Discretization Scheme and A Priori Estimates
To apply Rothe's method, We use the following procedure.For any positive integer n we consider a partition t n j defined by t n j = jh; h = T /n, j = 0, 1, 2, . . ., n. Set u n 0 = φ(0) for all n ∈ N.For j = 1, 2, . . ., n, we define u n j ∈ D(A) the unique solutions of each of the equations Where 1 ≤ i ≤ j.Now, the existence of a unique u n j ∈ D(A) satisfying (3.1) follows from the mdissipativity of A and by Theorems 1.4.2 and 1.4.3 in Pazy [10].In order to ensure the existence of a unique solution u n j ∈ D(A) of (3.1) we rewrite it as as (I − hA) −1 exists for all h > 0. The existence of unique u n j ∈ D(A) satisfying (3.1) is ensured.Definition 3.1.We define the Rothe sequence {U n } ⊂ C([−T, T ]; H) given by We prove the convergence of the sequence {U n } to the unique solution of the problem as n → ∞ using some a priori estimates on {U n }.For convenience, we shall denote by C a generic constant, i.e., KC, e KC , etc., will be replace by C where K is a positive constant independent of j, h and n.
We shall use later the following lemma due to Sloan and Thomee [12].EJQTDE, 2008 No. 4, p. 4 Lemma 3.2.Let {w n } be a sequence of nonnegative real numbers satisfying where {α n } is a nondecreasing sequence of nonnegative real numbers and β n ≥ 0. Then Furthermore, we also require the following lemma for later use.
Lemma 3.3.Let C > 0, h > 0 and let {α j } n j=1 be a sequence of nonnegative real numbers satisfying Then Proof.From (3.2) Putting in (3.2) By repeating the above process This completes the proof of the lemma.Proof.From (3.1) for j = 1 we get [10] implies that 1) for j − 1 from (3.1) for j, we get Applying Theorem 1.4.2 of [10] again, we get Now using Lipschitz continuity of the function F Then (3.6) becomes Au n i H + Ch. (3.8) From (3.1), for 2 ≤ i ≤ j, we have Au n p H . (3.9) Again using Lipchitz continuity of F Then (3.9) becomes To use Lemma 3.3 in (3.13), we take α j = max 1≤p≤j δu n p H and the fact that Again we apply Lemma 3.2 to get the required estimate.This completes the proof of the lemma.
Remark 3.5.Each of the functions {U n } is Lipschitz continuous with uniform Lipschitz constant, i.e., Definition 3.6.We define the sequence {F n } and {K n } of step functions [0, T ] into H by Lemma 3.7.Under the given assumptions we have (c) for t ∈ (t n j−1 , t n j ], AX n (t) = Au n j and d − u n dt (t) = 1 h (u n j − u n j−1 ).Therefore, This completes the proof of the lemma.
In the next lemma we prove the local uniform convergence of the Rothe sequence.
and the compactness of (I + A) −1 imply that U n = (I + A) −1 V n has a convergent subsequence.For convenience, we again denote this convergent subsequence by {U np }.Thus, U np (t) → u(t) as p → ∞.Also, X np (t) → u(t) as p → ∞.

Lemma 3 . 4 .
There exists a constant C independent of j, h and n such that δu n j H ≤ C. EJQTDE, 2008 No. 4, p. 5

H
≤ C, and k : [0, T ] → R Lipschitz continuous imply that the last two terms on the right hand side tend to zero strongly and uniformly on [0, T ] as EJQTDE, 2008 No. 4, p. 9 Lemma 3.8.There exist a subsequence {U np } of {U n } and a function u: [0, T ] → D(A) such that U np → u in C([0, T ]; H),and AU np (t) Proof.Since {U n (t)} and {AX n (t)} are uniformly bounded in the Hilbert space H, there exist weakly convergent subsequences {U np (t)} and {AX np (t)} (we take the same indices without loss of generality otherwise we first take the subsequence EJQTDE, 2008 No. 4, p. 8 {U np (t)} of {U n (t)} and then take the subsequence {U np n (t)} and {AX np n (t)} of {U np (t)} and {AX np (t)}, respectively).
[2]the maximal dissipativity of A, it follows that u(t) ∈ D(A) and AX np (t) Au(t).Since U np is Lipschitz continuous with uniform Lipschitz constant, it follows that {U np } is equi-continuous in C([0, T ]; H) and {U np (t)} is relatively compact in H. Hence by Ascoli-Arzela theorem, U np → u as p → ∞ in C([0, T ]; H).To show the weak continuity of Au(t) in t, let {t p } ⊂ [0, T ] such that t p → t as p → ∞, t ∈ [0, T ].Then u(t p ) → u(t) and since Au(t p ) H ≤ C, there exists a subsequence {Au(t kp )} ⊂ {Au(t p )} such that Au(t pm ) z(t) as m → ∞.Since u(t pm ) → u(t) and Au(t pm ) z(t) as m → ∞, it follows as above that u(t) ∈ D(A) and Au(t) = z(t).Hence Au(t) is weakly continuous.This completes the proof of the lemma.Lemma 3.9.Au(t) is Bochner integrable on [0, T ].For a proof of this lemma we refer to Bahuguna and Raghavendra[2].Lemma 3.10.Let {K n (t)} be the sequence of functions defined by (3.15) and where φ : [0, T ] → H is Bochner integrable.We haveK np (t) K(Au)(t),uniformly on [0, T ] as p → ∞.Proof.We first show that K np (t) − K(AX np )(t) → 0 uniformly on [0, T ] as p → ∞.For t ∈ (tnp j−1 , t np j ], we have K np (t) − K(AX np )(t) = h