Nonlinear Partial Functional Differential Equations: Existence and Stability

Nonlinear Partial Functional Differential Equations: Existence and Stability Tom´ as Caraballo Dpto. Ecuaciones Diferenciales y An´alisis Num´erico. Universidad de Sevilla. Apdo. Correos 1160. 41080-SEVILLA. Spain. e-mail: [email protected] Dedicated to Pepi, my wife. Abstract Existence and uniqueness of solutions for a class of nonlinear functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied. The stability results derived are also applied to ordinary differential equations with hereditary characteristics. Keywords: Partial differential equation; partial functional differential equation; exponential stability. AMS 2000 Classifications: 35R10. 1 Introduction The study of functional differential equations is motivated by the fact that when one wants to model some evolution phenomena arising in Physics, Biology, Engineering, etc., some hereditary characteristics such as aftereffect, time lag and time delay can appear in the variables. Typical examples arise from the researches of materials with termal memory, biochemical reactions, population models, etc. (see, for instance, Hale and Lunel [8], Ruess [16]-[17], Webb [20], Wu [23] and the references therein). On the other hand, one important and interesting problem in the analysis of functional differential equations is the stability, the theory of which has been greatly developed over the last years. As is well known, in the case without any hereditary features, Lyapunov’s technique is available to obtain sufficient conditions for the stability of solutions of (partial) differential equations. However, in the case of differential equations with hereditary properties, for instance, even in the case of constant time delays, Lyapunov’s method becomes difficult to apply effectively 1 as Krasovskii [12] pointed out. The main reason is that it is much more difficult (or even impossible in some cases) to construct proper Lyapunov functions (or functionals) for functional differential equations than for those without any hereditary characteristics. As a consequence, a comparison technique has been developed by various authors such as Krasovskii [12] and Mao [14] (among others). Let us illustrate this point in more detail. Consider the following one-dimensional delay differential equation dx(t) dt =f(t, x(t), x(t−h)), t > 0,(1) where h > 0, or equivalently, dx(t) dt =f(t, x(t), x(t)) + [f(t, x(t), x(t−h)) −f(t, x(t), x(t))] (2) Clearly if h > 0 is small enough, the perturbation term f(t, x(t), x(t−h)) −f(t, x(t), x(t)) could be expected to be so small that the perturbed equation (2) would behave asymptotically as equation dx(t) dt =f(t, x(t), x(t)).(3) In particular, it can be proved that, under some circumstances, exponential stability is transferred from the nondelay equation (3) to the delay one (1) if the constant time lag h > 0 appearing in the problem is sufficiently small (see, e.g. [14]). So, in order to find out whether the functional equation (1) is exponentially stable, one can check the exponential stability of the equation (2) and then compute whether the time lag h > 0 is sufficiently small. Nevertheless, it is worth pointing out that this kind of results can be somewhat restrictive for many practical applications. In fact, the situation turns out to be rather complicated when one considers the general functional differential equations, even the usual delay differential systems. To this respect, we should mention that in a wide variety of problems, the history of the phenomenon has a decisive influence on the future behaviour of the system, and, in some cases, not only a short period of the past has to be taken into account, but a large one. So it seems rather unnatural to look for results which hold only for small values of the deviating arguments. In this work, we shall carry out an investigation in this direction. One of the main aims of this paper is to give sufficient conditions (which, in particular, may contain the corresponding results in finite dimension) in order to transfer the exponential stability of partial differential equations to partial functional differential equations. The problem we are referring to is devoted to the consideration of an infinite dimensional version of (1) in which fhas the following form: f(t, x, y) = A(t, x) + f1(t, y), with the family of (non-linear) operators A(t, ·) satisfying some kinds of coercivity conditions (see Section 3) as well as f1satisfying Lipschitz continuous ones. We would also like to mention 2 that, in some sense, a suitable coercivity condition implies the (exponential) stability of solutions in nondelay cases. In addition to this, we will be able to assure exponential stability for a great number of finite dimensional functional differential equations where the results in [14] only guarantee this kind of stability for constant and sufficiently small delays. In Section 2, we begin with some preliminary results. Section 3 is devoted to establish some results on the existence and uniqueness of solutions for a class of partial functional differential equations in a variational context. Some results on exponential stability are studied in Section 4. Finally, several examples are given in Section 5 to illustrate the theory derived in the preceding sections. 2 Preliminaries First of all, we introduce the framework in which our analysis is going to be carried out. Let V be a separable Banach space and Hbe a real separable Hilbert one such that V ,→H≡H0,→V0, where V0is the dual of Vand the injections are continuous and dense. We denote by k · k ,|·| and k·k∗the norms in V,Hand V0respectively; by h·,·i the duality product between V0, V , and by (·,·) the scalar product in H. Let us take h≥0, p≥1 and T > 0, let CH=C(−h, 0; H) be the space of all continuous functions from [−h, 0] into Hwith sup-norm kψkCH= sup−h≤s≤0|ψ(s)|,ψ∈CH, similarly, let CV=C(−h, 0; V), Lp V=Lp(−h, 0; V) and Lp H=Lp(−h, 0; H). Given a function x(·)∈ Lp(−h, T;V)∩C(−h, T;H), we associate with an Lp V∩CH-valued function xt, t ≥0, by setting xt(s) = x(t+s), s∈[−h, 0]. The first purpose of this paper is to establish existence and uniqueness results for a class of nonlinear partial functional differential equations of the form    dx(t) dt =A(t, x(t))+f(t, xt), t ∈[0, T], x(t) = ψ(t), t ∈[−h, 0], (4) where, in general, the operators are assumed to be nonlinear. In fact, we are interested in the case in which A(t, ·) : V→V0is a family of nonlinear monotone and coercive operators and f(t, ·) : X→His Lipschitz continuous, where Xwill denote CH, CV, Lp Hor Lp V. On the one hand, it is worth pointing out that, in many applications, Ausually denotes a partial differential operator (linear or nonlinear), while fuses to be a first order partial differential one. On the other, we want to mention that the problem of existence of solutions to (4) and its stability have been previously analyzed by Travis and Webb in [18] for linear autonomous operator A generating a strongly continuous semigroup, and in [19] when Ais the generator of an analytic semigroup by using the tools of the semigroup theory of operators. Also Webb [21] considers a similar problem for autonomous and accretive operator Aby the same theory of (linear or not) semigroups of operators. Moreover, there exits a wide literature on the existence of different 3 classes of solutions (strong, mild, integral, etc.) to functional differential equations even in the more general context of differential inclusions. It is well worth reading the work by Ruess [16] (see also [17]), where we can find a description of some of the different techniques used to handle with this question, in addition to a large list of references concerning these methods. Among others, let us mention, for the univalued case, the method of lines (cf. Kartsatos [9]), the Galerkin approximations (cf. Kartsatos and Parrott [10]), the Kato approximants (cf. Kartsatos and Parrott [11]), etc. Extensions to the multivalued framework can be found in several works included in the references in [16]. However, on the one hand, the case involving unbounded (linear or not) operators f(i.e. when fis a partial differential operator of first order), despite of its importance in applications, has only been treated in some specific and particular situations (see, for instance, Fitzgibbon [7], Aizicovici [1], Crandall et al. [5]) and not always in a systematic way; on the other hand, what is missed in the literature is a general treatment of this problem from a variational point of view (see Artola [2]-[3] for some particular linear and nonlinear partial differential equations with delays). Consequently, in this paper, we shall first establish some existence results by a variational type of argument similar to that one carried out by Lions [13] for a case without delays, but subject to necessary changes to make our scheme go through when f(t, ·) : Lp H→H. Then we will treat the more general case with f(t, ·) : Lp V→Hby using a Galerkin approximation technique. 3 Existence and uniqueness of solutions Let A(t, ·) : V→V0be a family of (nonlinear) operators defined a.e.t. (for almost every t) and p≥2. Assume the following hypotheses: Coercivity: ∃α > 0, λ, ν ∈Rsuch that: −2hA(t, x), xi+λ|x|2+ν≥αkxkp,∀x∈V , a.e.t.; (5) Monotonicity: −2hA(t, x)−A(t, y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e.t.; (6) Boundedness: ∃γ > 0 : kA(t, x)k∗≤γkxkp−1,∀x∈V , a.e.t.; (7) Hemicontinuity: θ∈R→ hA(t, x +θy), zi ∈ Ris continuous ∀x, y, z ∈V , a.e.t.; (8) Measurability: t∈(0, T)→A(t, x)∈V0is Lebesgue-measurable ∀x∈V , a.e.t. (9) 4 Let f(t, ·) : L2 H→Hbe a family of nonlinear operators defined a.e.t., and satisfy the following conditions: ∃cf≥0 : sup 0≤t≤T |f(t, 0)| ≤ cf<+∞; (10) ∃k1=k1(h)>0 : |f(t, η)−f(t, ξ)| ≤ k1kη−ξkCH,∀η, ξ ∈CH,a.e.t.; (11) t∈(0, T)7→ f(t, η)∈His Lebesgue-measurable ∀η∈L2 H.(12) Given an initial value ψ∈Lp(−h, 0; V)∩C(−h, 0; H),the first objective in this Section is, under the conditions described above, to find a unique function x(·)∈Lp(−h, T;V)∩C(−h, T;H) such that (x(t) = ψ(0) + Rt 0[A(s, x(s))+f(s, xs)]dt, t ∈[0, T], x(t) = ψ(t), t ∈[−h, 0],(13) where the first equality is understood in V0. We will refer to this solution as the variational solution to (4). Remark 1 (1) First, we notice that, although the results can be proved for p > 1, the interesting situations in the applications appear when p≥2. Because of this, we content ourselves with the analysis of the case p≥2. (2) We have fixed the initial data in the space Lp(−h, 0; V)∩C(−h, 0; H)just only to ensure that the solution belongs to Lp(−h, T ;V)∩C(−h, T;H).However, we can of course take ψ∈Lp Vand a value x0∈Hinstead of ψ(0) in the problem. In this case, the argument we will use provides a solution in the space Lp(−h, T;V)∩C(0, T;H). (3) Although it is possible to extend the results derived here to more general sytems involving coefficients such that f(t, x(t), xt), we restrict ourselves to this more simple situation in order to avoid unnecessary technicalities. Now we shall prove that there exists at most one solution of (13). This result will be deduced mainly from (6) and the energy equality. Theorem 2 Assume the preceding hypotheses hold. Then, there exists at most one solution of (13) in Lp(−h, T;V)∩C(−h, T;H). Proof. Suppose that x, y ∈Lp(−h, T;V)∩C(−h, T;H) are two solutions of (13). Then, taking into account (6), we obtain |x(t)−y(t)|2= 2 Rt 0hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids +2 Rt 0(f(s, xs)−f(s, ys), x(s)−y(s))ds ≤λRt 0|x(s)−y(s)|2ds +2 Rt 0|f(s, xs)−f(s, ys)||x(s)−y(s)|ds. 5 Now, it follows from (11) that for any t∈[0, T] sup 0≤s≤t |x(s)−y(s)|2≤(|λ|+ 1) Zt 0 |x(s)−y(s)|2ds (14) +k2 1Zt 0 ||x(s)−y(s)||2 CHds (15) On the other hand, since x(s) = y(s) for s≤0, we easily get Rt 0||x(s)−y(s)||2 CHds =Rt 0sup −h≤r≤0 |xs(r)−ys(r)|2ds =Rt 0sup −h≤r≤0 |x(s+r)−y(s+r)|2ds ≤Rt 0sup 0≤r≤s |x(r)−y(r)|2ds. (16) Thus, it follows from (14)–(16) sup 0≤s≤t |x(s)−y(s)|2≤2h|λ|+1+k2 1iZt 0 sup 0≤r≤s |x(r)−y(r)|2ds, ∀t∈[0, T], and Gronwall’s lemma obviously implies uniqueness. Remark 3 Observe that if we assume the following monotonicity hypothesis For all ξ, η ∈Lp(−h, T;V)with ξ0=η0it holds −2hA(t, ξ(t))+f(t, ξt)−A(t, η(t)) −f(t, ηt), ξ(t)−η(t)i +λ|ξ(t)−η(t)|2≥0, t ∈[0, T],(17) instead of (6), uniqueness is also easily deduced. Indeed, notice that in this case, (17) implies |x(t)−y(t)|2≤λZt 0 |x(s)−y(s)|2ds ∀t∈[0, T], for arbitrary two solutions x, y of the problem. Now, before proving our first existence result, we shall state a theorem on existence and uniqueness of solutions of evolution equations. Theorem 4 Assume (5)–(9) hold with λ= 0.Then, given f1∈Lp0(0, T;V0)(with 1 p+1 p0= 1) and x0∈H, there exists a unique function x∈Lp(0, T;V)∩C(0, T;H)such that x(t) = x0+Zt 0 [A(s, x(s))+f1(s)]ds, for all t∈[0, T]. In addition to this, the following energy equality holds: |x(t)|2=|x0|2+ 2 Zt 0 hA(s, x(s))+f1(s), x(s)ids, t ∈[0, T]. 6 Proof. See Lions [13] (Theorem 1.2, page 162). Theorem 5 Assume that (5)–(9), (11) and (12) hold. Then, for each initial datum ψ∈ Lp(−h, 0; V)∩C(−h, 0; H), there exists a unique solution of the problem (13) in Lp(−h, T;V)∩ C(−h, T;H). Proof. Uniqueness follows from Theorem 2. For the existence, we consider the equations (x1(t) = ψ(0) + Rt 0A(s, x1(s)) −λ 2x1(s)ds, t ∈[0, T], x1(t) = ψ(t), t ∈[−h, 0],(18)        xn+1(t) = ψ(0) + Rt 0A(s, xn+1(s)) −λ 2xn+1(s)ds +λ 2Rt 0xn(s)ds +Rt 0f(s, xn s)ds, t ∈[0, T],∀n≥1, xn+1(t) = ψ(t), t ∈[−h, 0],∀n≥1. (19) By virtue of (5)–(9), the family A1(t, .) : V→V0defined as A1(t, x) = A(t, x)−(λ/2)x , satisfies assumptions in Theorem 4. Consequently, (18) has a unique solution x1∈Lp(−h, T;V)∩ C(−h, T;H).We note that, from (11) it follows that the mapping t∈(0, T)7→ f(t, x1 t)∈H belongs to L2(0, T;H).Consequently, bearing these remarks in mind, we can use Theorem 4 and get that there exists a unique function x2∈Lp(−h, T;V)∩C(−h, T;H) , which is the solution of (19) for n= 1 . By recurrence, we obtain a sequence of solutions of (18)–(19), {xn}n≥1⊂ Lp(−h, T;V)∩C(−h, T;H).Now, we want to prove that the sequence {xn}converges to a function xin Lp(−h, T;V)∩C(−h, T;H) , which will be the solution of (13). For this end, we shall first prove the following lemmas. Lemma 6 {xn}n≥1is a Cauchy sequence in C(−h, T;H). Proof. Indeed, it follows for n≥2 |xn+1(t)−xn(t)|2= 2 Rt 0hA(xn+1)−A(xn), xn+1 −xnids −λRt 0|xn+1 −xn|2ds +λRt 0(xn+1 −xn, xn−xn−1)ds +2 Rt 0(f(xn s)−f(xn−1 s), xn+1 −xn)ds, (20) where, for short we denote, xn:= xn(s), A(xn):=A(s, xn(s)) and f(xn):=f(s, xn s).Now, it is easy to deduce from (6) sup 0≤θ≤t |xn+1(θ)−xn(θ)|2≤ |λ|Rt 0|xn+1 −xn||xn−xn−1|ds +2 Rt 0|f(xn s)−f(xn−1 s)||xn+1 −xn|ds. (21) 7 Firstly, |λ|Rt 0|xn+1 −xn||xn−xn−1|ds ≤1 4sup 0≤θ≤t |xn+1(θ)−xn(θ)|2 +λ2TRt 0sup 0≤θ≤s |xn(θ)−xn−1(θ)|2ds, (22) and noticing that xn(s) = xn−1(s) for −h≤s≤0, 2Rt 0|f(xn s)−f(xn−1 s)||xn+1 −xn|ds ≤1 4TRt 0|xn+1 −xn|2ds + 4k2 1TRt 0||xn s−xn−1 s||2 CHds ≤1 4sup 0≤θ≤t |xn+1(θ)−xn(θ)|2+ 4k2 1TRt 0sup 0≤θ≤s |xn+1(θ)−xn(θ)|2ds. (23) If we set ϕn(t) = sup0≤θ≤t|xn+1(θ)−xn(θ)|2,it then follows from (21)–(23) that there exists a positive constant k > 0 such that ϕn(t)≤kZt 0 ϕn−1(s)ds . (24) By iteration from (24), we get ϕn(t)≤kn−2Tn−1 (n−2)! ϕ2(T),∀n≥2,∀t∈[0, T].(25) Therefore, sup 0≤θ≤T |xn+1(θ)−xn(θ)|2≤kn−2Tn−2 (n−2)! ϕ2(T),∀n≥2.(26) Obviously, since xn+1(θ) = xn(θ) for θ∈[−h, 0], (26) implies that {xn}is a Cauchy sequence in C(−h, T;H). Lemma 7 The sequence {xn}is bounded in Lp(−h, T;V). Proof. Indeed, for n≥2 we immediately obtain |xn(T)|2= 2 RT 0hA(xn), xnids −λRT 0|xn|2ds +|ψ(0)|2+ 2 RT 0(f(xn−1), xn)ds +λRT 0(xn−1, xn)ds. (27) Therefore, −2RT 0hA(xn), xnids +λRT 0|xn|2ds ≤ |ψ(0)|2+ 2 RT 0|f(xn−1)||xn|ds +|λ|RT 0|xn−1||xn|ds. (28) Since {xn}converges in C(−h, T;H), it will be bounded in this space. Now, it is not difficult to check that there exists a positive constant k0>0 such that the right-hand side of (28) is 8 bounded by this constant. For instance, we will estimate one of those terms. Firstly, we observe that RT 0||xn−1 s||2 CH≤RT 0sup −h≤θ≤s |xn−1(θ)|2ds Next, (10) and (11) imply 2RT 0|f(xn−1)||xn|ds ≤RT 0|f(xn−1)|2+|xn|2ds ≤Tk1||xn−1||C(−h,T;H)+cf2+T||xn||2 C(0,T;H),(29) which, in addition to (28) and (5), leads to αZT 0 kxn(s)kpds ≤ −2ZT 0 hA(xn), xnids +λZT 0 |xn|2ds +νT ≤k0, and the lemma is proved. Now in order to complete the proof of the theorem, we shall prove that the limit of the sequence {xn}is a solution to (13). Firstly, observe that Lemma 6 implies that there exists x∈C(−h, T;H) such that xn→x in C(−h, T;H). Now, thanks to (11), we have f(xn)→f(x) (in L∞(0, T;H)). On the other hand, by virtue of Lemma 7, {xn}has a subsequence which converges weakly in Lp(−h, T;V). But, since xn→xin C(−h, T;H), we can assure that xn→xweakly in Lp(−h, T;V) (in the sequel, we will denote this by xn* x in Lp(−h, T;V)). Nevertheless, it follows from (7) that {A(xn)}is bounded in Lp0(0, T ;V0) (with p0such that (1/p) + (1/p0) = 1), since ZT 0 kA(t, xn(t))kp/(p−1) ∗dt ≤γZT 0 kxn(t)kpdt ≤γk0/α. Therefore, from each subsequence of {A(xn)}, we can get another subsequence weakly convergent in Lp0(0, T;V0). Now, as it is easy to see that all the limits of different subsequences coincide, we finally get that A(xn)* v in Lp0(0, T;V0). In conclusion, we have proved: xn→xin C(0, T;H),(30) f(xn)→f(x) in L∞(0, T;H),(31) xn* x in Lp(−h, T;V),(32) A(xn)* v in Lp0(0, T;V0).(33) Finally, by the virtue of (30)-(33), we can take limits in (19) and obtain x(t) = ψ(0) + Zt 0 v(s)ds +Zt 0 f(s, xs)ds. (34) 9 4 STABILITY OF SOLUTIONS In this section we shall show that under suitable conditions exponential stability can be transferred from equations without time lags to those with time lag ones. Since we are mainly interested in exponential stability problems, we will assume there exists x∈L2(−h, T;V)∩ C(−h, T;H),∀T > 0,which is the variational solution of the following problem:    dx(t) dt =A(t, x(t))+f(t, xt)t > 0, x(t) = ψ(t), t ∈[−h, 0]. (61) In other words, x(t) satisfies the following integral equation (in V0): (x(t) = ψ(0) + Rt 0[A(s, x(s))+f(s, xs)]ds, t ≥0, x(t) = ψ(t), t ∈[−h, 0].(62) In particular, we suppose in this section that all conditions in Section 3 hold so that there exists a unique solution to the functional differential equation (61). First of all, we investigate the case without hereditary characteristics. In other words, consider Eq. (61) with h= 0 and thus k1>0 in (11) does not depend on h. Then Eq. (61) reduces to      dx(t) dt =A(t, x(t))+f(t, x(t)), t ≥0, x(0) = x0. (63) If it is possible to know the existence of a Lyapunov function, we could prove exponential stability of solutions. Indeed, assume there exist v∈C2(H;R+) and positive constants ci,1≤i≤4 , such that v0(x)∈Vfor all x∈Vand c1|x|2≤v(x)≤c2|x|2,Lv(x)≤ −c3v(x),|v0(x)| ≤ c4|x|, for all x∈V, where Lis the associated Lyapunov operator defined as Lv(x) = hA(t, x) + f(t, x), v0(x)i,∀x∈V. We can get for the function ec3tv(x), and x∈H ec3tv(x(t)) = v(x(0)) + c3Rt 0ec3sv(x(s))ds +Rt 0ec3shA(s, x(s))+f(s, x(s)), v0(x(s))ids. Observing that Lv(x)≤ −c3v(x) , we have v(x(t)) ≤e−c3tv(x(0)) ,∀t≥0, and from the assumptions on v, we easily deduce that |x(t)|2≤c2 c1 e−c3t|x(0)|2,∀t≥0. 16 Consequently, what we have proved is that every solution to (61) converges exponentially to zero, even if zero is not a stationary solution. In particular, if f(t, 0) = 0 for all t≥0,our result implies that the trivial solution to (61) is globally exponentially stable. Although, as we have mentioned before, the construction of Lyapunov functions is not, in general, a trivial problem, there exists a condition which makes v(x) = |x|2become a natural Lyapunov function. We are refering to the following hypothesis (H): there exists a positive constant γ > 0 such that 2hA(t, x) + f(t, x), xi ≤ −γ|x|2,∀x∈V (observe that on this occasion Lv(x) = 2hA(t, x) + f(t, x), xi ≤ −γ|x|2). Remark 10 In what follows, we assume that f(t, 0) = 0,for all t≥0since we are interested in analysing the stability of the trivial solution to our problem. To this respect, it is worth mentioning that in a variety of practical situations, the following assumption (H)0(which seems easier to check) implies (H): (H)0: There exists a positive constant bα > 0such that −2hA(t, x), xi ≥ bα|x|2,∀x∈Vand −bα+ 2k1<0, where k1is the nonnegative constant in (11). Indeed, note that 2hA(t, x) + f(t, x), xi≤−bα|x|2+ 2(f(t, x), x) ≤(−bα+ 2k1)|x|2 and denoting γ=bα−2k1, assumption (H) follows. Now, we shall show that the same hypotheses as above (mainly (11) and (H)0) imply exponential stability of the trivial solution to the functional differential equation (61). However, it is particularly worth pointing out that on this occasion the constants k1is generally dependent on the time lag constant h > 0. This fact simply means that, in order to obtain exponential stability, the time lag must be sufficiently small. However, as will be shown by the examples in the final section (see also Theorem ??), on some occasions such as the time delay case, the constant k1could be independent on h > 0 so that the stability holds true for any h > 0. For our ends, let us firstly study some stability criteria for the functional differential equation (61) by using a Razumikhin type argument. Theorem 11 Assume that operators Aand fare continuous with respect to time t, and that there exists a positive constant λ > 0such that for all t≥0, 2hA(t, φ(0)) + f(t, φ), φ(0)i<−λ|φ(0)|2(64) 17 provided φ={φ(s) : −h≤s≤0} ∈ C(−h, 0; V)satisfying kφk2 CH≤eλh|φ(0)|2.(65) Then, there exists a positive constant K≥1such that for all ψ∈C(−h, 0; V)for which the solution to (61) corresponding to the initial value ψ, denoted by x(t, ψ),belongs to C(−h, T;V) for all T > 0, satisfies |x(t, ψ)|2≤K||ψ||2 CH·e−λt,for all t≥0.(66) Proof. Suppose the assertion does not hold. Then, for any K≥1 there exists an initial data ψsuch that (66) is not true. Let us fix a K≥1 and take the corresponding ψ. Thus, we can affirm that there exists a ρ≥0 such that eλt|x(t;ψ)|2≤eλρ|x(ρ;ψ)|2=K||ψ||2 CH,(67) for all 0 ≤t≤ρ, and there is a sequence {tk}k≥1in R+such that tk↓ρ, as k→ ∞, and eλtk|x(tk;ψ)|2> eλρ|x(ρ;ψ)|2.(68) On the other hand, by virtue of (67) we deduce |x(ρ+θ;ψ)|2≤eλ(ρ−t)|x(ρ;ψ)|2≤eλh|x(ρ;ψ)|2, for all −h≤θ≤0, which, in view of the assumptions (64)-(65), immediately implies that 2hA(ρ, x(ρ))+f(ρ, xρ), x(ρ;ψ)i<−λ|x(ρ)|2. By the continuity of the solution and the functions Aand f, we see that for some sufficiently small h > 0, 2hA(t, x(t))+f(t, xt), x(t;ψ)i<−λ|x(t)|2, for all t∈[ρ, ρ +h]. Thus, for all sufficiently small h > 0, eλ(ρ+h)|x(ρ+h;ψ)|2−eλρ|x(ρ;ψ)|2 =Rρ+h ρeλt λ|x(t;ψ)|2+ 2 hA(t, x(t;ψ))+f(t, xt), x(t;ψ)idt ≤0. However, this contradicts (68), so the result (66) must be true. Theorem 12 Suppose the assumptions in Section 3 for existence and uniqueness of solutions to (13) hold. In addition to (11) and (H)0, assume that operators Aand fare continuous with respecto to time t. Then, there exist λ > 0and K≥1such that for all ψ∈C(−h, 0; V)for which the solution to (13) belongs to C(−h, T ;V)for all T > 0, i.e. x(·;ψ)∈C(−h, T;V),it follows |x(t;ψ)|2≤K· ||ψ||2 CH·e−λt,for all t≥0. 18 Proof. As 0 <bα−2k1, we can take a sufficiently small positive λsuch that 0 <bα−2k1eλh. Now if t≥0 and φ={φ(s) : −h≤s≤0} ∈ C([−h, 0]; V) satisfies kφkCH≤eλh|φ(0)|, then Assumption (11) and condition (H)0imply 2hA(t, φ(0)) + f(t, φ), φ(0)i ≤ −bα+ 2k1eλh|φ(0)|2.(69) Therefore, in view of Theorem 11 the proof is complete. Remark 13 In the finite dimensional case, that is, when V=H=Rn, Theorem 12 guarantees exponential stability for the solutions to (61) if (H)0is fulfilled, where (H)0can be rewritten now as (H)00 There exists a positive constant bαsuch that −2xTA(t, x)≥bα|x|2,∀x∈Rnand −bα+ 2k1<0, where xTdenotes the transpose of x. Notice that this is the same condition deduced by Hale and Lunel [8], so our assumption becomes a natural extension of the finite-dimensional results. Futhermore, under this condition the results in Mao [14] only ensure exponential stability if the delay function there is t−hwith hsufficiently small. Remark 14 Observe that the stability result just proved requires some regularity conditions on the problem and some additional continuity assumptions on the operators. Nevertheless, for some particular cases, better results could be obtained. As an example, we shall prove below an exponential stability result for the case of a partial differential equation with variable delay. To this respect, let us consider F:H→Hglobally Lipschitz with constant bk1and F(0) = 0, ω:R+→[0, h] continuously differentiable with ω0(t)≤0 for all t≥0,and define f(t, φ) = F(φ(−ω(t)),for φ∈C(−h, 0; H).This situation corresponds to the problem (dx(t) dt =A(t, x(t))+F(x(t−ω(t)), t > 0, x(t) = ψ(t), t ∈[−h, 0].(70) Theorem 15 In the preceding situation, assume that −2hA(t, x), xi ≥ bα|x|2,∀x∈Vwith −bα+ 2bk1<0. Then, there exist λ > 0, K ≥1such that for every ψ∈CH, the solution to (70), denoted again x(t, ψ),satisfies |x(t;ψ)|2≤K· ||ψ||2 CH·e−λt,for all t≥0. 19 Proof. Let us denote ρ(t) = t−ω(t), t ≥0.Then, ρis a continuously differentiable function with ρ0(t)≥1.This immediately implies that ρ−1(t)≤t+ρ−1(0) = t+k. Now, we can choose λ > 0 such that λ−bα+bk1(1 + eλk)<0,and, by setting x(t) = x(t;ψ),it follows that eλt |x(t)|2≤ |ψ(0)|2+Zt 0 eλs |x(s)|2ds + 2 Zt 0 eλs hA(s, x(s)), x(s)ids + 2 Zt 0 eλs (F(x(ρ(s)), x(s)) ds ≤ ||ψ||2 CH+ (λ−bα)Zt 0 eλs |x(s)|2ds + 2bk1Zt 0 eλs |x(ρ(s))| |x(s)|ds ≤ ||ψ||2 CH+ (λ−bα+bk1)Zt 0 eλs |x(s)|2ds +bk1Zt 0 eλs |x(ρ(s))|2ds, and evaluating the last term by using the change of variables u=ρ(s) in the integral, it holds Zt 0 eλs |x(ρ(s))|2ds ≤Zt −h eλu+λk |x(u)|2du ≤Z0 −h eλu+λk |ψ(u)|2du +Zt 0 eλu+λk |x(u)|2du ≤heλk||ψ||2 CH+eλk Zt 0 eλu |x(u)|2du. Thus, we have eλt |x(t)|2≤(1 + bk1heλk)||ψ||2 CH+ (λ−bα+bk1+bk1eλk)Zt 0 eλs |x(s)|2ds, and therefore the proof is complete. 5 Examples Now, we are going to apply the results proved in the previous sections to obtain stability of some functional differential equations. First of all, we consider a general situation concerning the semilinear case previously studied by several authors (see Travis and Webb [18],[19], Martin and Smith [15], Wu [23], among others). Example 1. In our variational setting, consider a linear operator A∈ L(V, V 0) satisfying the coercivity condition (5) with ν= 0.If we set D(A):={v∈V:Av ∈H},then the operator Arestricted to this set D(A) becomes a closed and densely defined operator. Moreover, (see 20 Dautray and Lions [6] for the details) Ais the generator of a strongly continuous semigroup of linear operators in H, denoted by S(t),which satisfies: |S(t)| ≤ eλt/2,∀t≥0. Conversely, if S(t) is a strongly continuous semigroup satisfying, for some ω∈R, |S(t)| ≤ eωt,∀t≥0, then, it is not difficult to prove that its generator Asatisfies hAx, xi ≤ ω|x|2,∀x∈D(A). Now, consider the semilinear problem    dx(t) dt =Ax(t) + f(xt), t ≥0, x(t) = ψ(t), t ∈[−h, 0]. (71) where we assume that fis continuous, satisfies the Lipschitz condition (11) and f(0) = 0. Then, by applying the results in [18], we obtain that there exists a unique mild solution to (71) which, in addition, is a strong solution (whose existence is also ensured by our theory). On the other hand, if ω < 0 and k1<−ω, the results in [18] guarantee that the null solution to (71) is exponentially asymptotically stable. As we can easily see, the same result follows from our theory, since on this occasion, (H)0holds with bα=−2ω, and therefore, −bα+ 2k1<0 is equivalent to ω+k1<0. In conclusion, our theory gives a natural extension of the results previously obtained in the semilinear case. Example 2. Consider the following semilinear heat equation with finite time lags r1,r2 (r > r1, r2≥0) and with µ > 0, α1≥0 d dtX(t, x) = µ∂2 ∂x2X(t, x) + α1Z0 −r1 X(t+u, x)h(u)du +α(X(t))X(t−r2, x), t ≥0, X(t, 0) = X(t, π) = 0, t ≥0, X(s, x) = φ(s, x), φ(·, x)∈C:= C(−r, 0; R), φ(s, ·)∈L2(0, π), s ∈[−r, 0], x ∈[0, π],kφkC<∞, where α:R→R,h: [−r1,0] →Rare two bounded, Lipschitz continuous function with |α(x)| ≤ K,|h(u)| ≤ M,x∈R,u∈[−r1,0], M, K > 0. Define V=H1 0[0, π], H=L2[0, π] with the corresponding boundary conditions above. Let A=∂2 ∂x2with the domain D(A) = nu∈L2(0, π),∂u ∂x,∂2u ∂x2∈L2(0, π), u(0) = u(π) = 0o, so it is easy to deduce 2hAu, ui ≤ −2µkuk2, u ∈V. 21 On the other hand, it is clear that α1Z0 −r1 X(t+u, ·)h(u)du 2 ≤2(α1r1M)2+K2kXtk2 C. By a straightforward computation and applying Theorems 11, 12 to the above equation, if µ > 2(α1r1M)2+K2and r2>0 is arbitrary, then the null solution is exponentially stable. Example 3. Let us now exhibit a final nonlinear example. Let D= [0,1],2< p < +∞, r > 0 and consider the following nonlinear partial functional differential equation          d dtu(t, x) = ∂ ∂x  ∂u(t, x) ∂x  p−2∂u(t, x) ∂x !−a(x)u(t, x) + f(t, ut(x)), t > 0, x ∈D, x(t, x) = ψ(t, x), t ∈[−r, 0], x ∈D, x(t, 0) = x(t, 1) = 0, t > 0, (72) where a∈L∞(D), a(x)≥ea > 0,for all x∈D, and f(t, φ) = g(φ(−ω(t))),where ω:R→[0, r] is a measurable function and g:R→Ris a Lipschitz function with constant L > 0 and satisfying g(0) = 0.If we set H=L2(D), V =W1,p 0(D) and consider the operator Adefined as hAu, vi=−ZD" ∂u(x) ∂x  p−2∂u(x) ∂x ∂v(x) ∂x +a(x)u(x)v(x)#dx, ∀u, v ∈V. Then it is not difficult to check that assumptions (5)-(9) hold with λ=−2ea, α = 2.Consequently, condition (H)0holds provided −ea+L < 0.So, we get exponential stability. 6 Conclusions and final remarks We have proved some results on the existence and uniqueness of solutions to a nonlinear partial functional differential equation with finite delay by using a variational approach. Then, by means of a Razumikhin type argument, we have established a stability result in the case of additional regularity of the problem. However, nothing has been said in this context concerning the existence of solutions in the case of infinite delays. Also, a study of the multivalued version of our problem could be analyzed and some more stability results should be obtained. Finally, a comparison of these results and the ones previously proved by the evolution operator approach seems to be an interesting task which should be done. We plan to investigate these questions in some subsequent papers. 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