Stability of solutions to some abstract evolution equations with delay

The global existence and stability of the solution to the delay differential equation (*)$\dot{u} = A(t)u + G(t,u(t-\tau)) + f(t)$, $t\ge 0$, $u(t) = v(t)$, $-\tau \le t\le 0$, are studied. Here $A(t):\mathcal{H}\to \mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $\mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $\mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $\Re \lambda \le \gamma(t)$, where $\gamma(t)$ is not necessarily negative and $\|G(t,u)\| \le \alpha(t)\|u\|^p$, $p>1$, $t\ge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $\infty$, under the non-classical assumption that $\gamma(t)$ can take positive values, are proposed and justified.


Introduction
Consider the following delay differential equatioṅ Reference to equation (1) means reference to both equations (1a) and (1b). Here, A(t) : H → H is a closed, densely defined, linear operator in a Hilbert space H for any fixed t ≥ 0, Re u, A(t)u ≤ γ(t) u 2 , u ∈ Dom(A) ⊂ H, G(t, u) is a nonlinear operator in H for any fixed t ≥ 0, and f (t) is a function on R + = [0, ∞) with values in H, Here, ·, · and · denote the inner product and the norm in H, respectively. The functions γ(t) and α(t) are continuous on [0, ∞) and real-valued. Functional differential equations have been studied extensively in the literature (see, e.g., [1]- [5] and references therein). The usual assumption to derive the global existence and the stability of the solution to equation (1) is: γ(t) ≤ γ 0 < 0, ∀t ≥ 0. If A(t) is a square matrix, then the condition γ(t) ≤ γ 0 < 0 implies that all the eigenvalues of A(t) lie in the half-plane Re λ ≤ γ 0 < 0. In [6], [7], and [9], stability of solution to abstract differential equation (1) when τ = 0, i.e., without delay, was studied for the cases 0 < γ(t) ց 0 and 0 > γ(t) ր 0. The main tool for the study of the stability in [6], [7], and [9] under these non-classical assumptions is some nonlinear inequalities. These inequalities were also used in the study of the Dynamical Systems Method (DSM) for solving operator equations in [8]. In [4] the stability of the solution to equation (1) with τ = 0, i.e., without delay, was studied for the case when γ(t) can take positive and negative values. In [4] the nonlinear inequalities were not used, in contrast to [6], [7], and [9]. Using a nonlinear inequality with delay, the global existence and stability of equation (1) were studied in [10] for the case when f (t) = 0 and G(t, u) is of the form B(t)F (t, u) under the non-classical assumption that (2) holds but the inequality γ(t) ≤ γ 0 < 0 does not hold for any γ 0 < 0. In this paper, we are interested in having the stability results for the solution to equation (1) without using nonlinear inequalities similar to those used in [10].
A common approach to obtain the global existence of the solution to equation (1) is to estimate u(t) for t ≥ 0 and use the local existence of the solution to extend the existence of the solution to [0, ∞). An estimate of u(t) for t ≥ 0 can be derived from a nonlinear inequality (see inequality (7) below) which is obtained from equation (1) as follows. Take the inner product of both sides of equation (1a) with u to get Denote g(t) := u(t) , take the real part of equation (5) and use the triangle inequality, the Cauchy-Schwarz inequality, and inequalities (2)-(4) to geṫ The derivativeġ(t) in (6) is understood as the right derivative at t if g(t) = 0. From inequality (6) and equation (1b) one getṡ If g(t) > 0, ∀t ≥ 0, then it is clear that (7a) follows from (6). If g(t) = 0 for some t > 0, then these t form a set of isolated points by the uniqueness theorem and therefore equations (6) and (7a) are equivalent. This equivalence is proved differently in Lemma 2.1 below. By studying the global existence and boundedness of a solution g(t) to inequality (7) and using the local existence of the solution u(t) to equation (1), one can obtain the global existence and boundedness of u(t).
In [10] the stability of the solution to the following differential equation with delay was studied:u It was assumed in [10] that A(t) and B(t) are linear operators in a Hilbert space H and F (t, u) is a nonlinear operator in H for any fixed t ≥ 0. It was also assumed in [10] that relation (2) holds, B(t) ≤ b(t), and F (t, u) ≤ α(t, u ). It is clear that equation (8) is a special case of equation (1) when the function f (t) vanishes and G(t, u) = B(t)F (t, u).
Using the assumptions on A(t), B(t), and F (t, u) and an assumption on the existence of a local solution to problem (8), the following result was proved in [10]: If there exists a function µ(t) > 0, defined for all t ≥ −τ , such that then the solution to (8) exists for all t ≥ 0 and Using this result the global existence and the boundedness of the solution to equation (8) were obtained for some classes of functions b(t), γ(t), and α(t, y) (see [10]).
Although the mentioned result in [10] is quite general, it requires to find a function µ(t) which solves inequality (9). In general, it is not easy to find such function µ(t) if the functions b(t), γ(t), and α(t, y) are not simple.
In this paper we are interested in having stability results for equation (1) under nonclassical assumptions on the operator A(t), namely that the function γ(t) in (2) may change signs. In particular, we want find sufficient conditions on the functions γ(t), α(t), and β(t) (see (2)-(4)) which yield the global existence and the boundedness of the solution to equation (1).
The main results of this paper are Theorem 2.5, Theorem 2.7, and Corollary 2.8. In Theorems 2.5 and 2.7 sufficient conditions on γ(t), α(t), and β(t) for the solution of equation (1) to exist globally, to be bounded, and to decay to zero as t → ∞, are given. A direct consequence of Theorem 2.5 for the case f = 0 is formulated in Corollary 2.8. The novelties of our results compared to those in [10] include: Our results do not require one to find the function µ(t) > 0 which solves the nonlinear inequality (9).
Thus, these results are applicable when the functions γ(t), α(t), and β(t) are quite general. Moreover, our results cover the case when f = 0 which was not considered in [10].
Throughout this paper, we suppose that the following assumption holds: has a unique local solution for all a ≥ 0 and v a ∈ C([a − τ, a]; H). It is known that Assumption A holds if A(t) is a generator of a C 0 semigroup and the functions α(t) and β(t) are continuous and bounded on [0, ∞) (see, e.g., [1]).

Main results
Let us first show that inequalities (6) and (7) are equivalent.
Consider the following delay differential equatioṅ We have the following comparison lemma.
We claim that T = T n where [−τ, T n ) is the maximal interval of the existence of h n (t).
To prove this claim, assume the contrary. Then T < T n and h n (T ) = g(T ), by the continuity of g(t) and h n (t) and the definition of T . Since g(t) < h n (t), ∀t ∈ (0, T ), and g(T ) = h n (T ), one concludes thatġ (T ) ≥ḣ n (T ).
Lemma 2.2 says that for any solution g(t) ≥ 0 to inequality (7) one has 0 ≤ g(t) ≤ h(t). Thus, the global existence and boundedness of h(t) imply the global existence and boundedness of g(t). Therefore, to study the global existence and boundedness of g(t), we will study the global existence and boundedness of the solution h(t) to equation (18).
Let us consider equation (18). Since the functions h(t), α(t), and f (t) are nonnegative on R + , it follows from (18a) thatḣ This inequality is equivalent to Integrate this inequality from t − τ to t to get Thus, This and inequality (18a) implẏ It follows from equation (18a) that Integrate this equation from 0 to τ to get This and the relation ν(0) = 1 imply
The following lemma gives a sufficient condition for the solution to equation (18) to exist globally.

Lemma 2.3. Let h(t) be a solution to equation (18). Assume that
where h τ is defined in (29). Then h(t) exists globally and satisfies the estimate Remark 2.4. It follows from inequality (30) that the right-hand side of (32) is well-defined for all t ≥ τ . Inequality (30) is equivalent to Moreover, inequality (31) is equivalent to Thus, for the existence of ω satisfying both (30) and (31), it suffices to assume that Given the function γ(t) and the numbers p, τ , h τ and ν(τ ), inequality (33) holds true if α(t) and β(t)/α(t) are sufficiently small. In this case, one can choose In the case when f (t) is absent from equation (1), i.e., β(t) = 0, then inequality (33) becomes Proof of Lemma 2.3. Since h(t) is the solution to (18), it satisfies inequality (28a). From (28a) and (31), one getṡ Here, the inequality a p + b p ≤ (a + b) p , a ≥ 0, b ≥ 0, p > 1, was used. Inequality (35) can be written as Thus, Integrate inequality (37) from τ to t to get This implies It follows from (30) that the right-hand side of inequality (39) is positive. Thus, from (39) one gets Inequality (32) follows from (40). Lemma 2.3 is proved.
It follows from (41) in Theorem 2.5 that if the right-hand side of (41) is positive, then one can obtain the global existence and boundedness of u(t) by choosing f (t) so that β(t) := f (t) is sufficiently small. This raises the question: Given β(t) := f (t) , is this possible to 'control' α(t) so that the global existence and boundedness of u(t) are still guaranteed? We will address this question in the following results. Lemma 2.6. Let h(t) be the solution to equation (18) and Assume that Then Proof. From equation (54) with t = τ and the assumption q > 1, one gets It follows from the continuity of ζ(t) and h(t) that there exists θ > 0 such that Let T > 0 be the largest real value such that We claim that T = ∞. Assume the contrary. Then T is finite and by the definition of T we have Since h(t) is the solution to (18), it satisfies inequality (28a). It follows from inequalities (28a), (59), and (55) thaṫ From (61) one gets d dt Integrate this inequality from τ to t to get This and (54) imply Therefore, This contradicts to equation (60). The contradiction implies that T = ∞, i.e., inequality (56) holds. Lemma 2.6 is proved.