Blow-up and superexponential growth in superlinear Volterra equations

This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and nonexplosive solutions for a class of equations with nonsingular kernels under weak hypotheses on the nonlinearity. In this superlinear setting we must be content with estimates of the form $\lim_{t\to\tau}A(x(t),t) = 1$, where $\tau$ is the blow-up time if solutions are explosive or $\tau = \infty$ if solutions are global. Our estimates improve on the sharpness of results in the literature and we also recover well-known blow-up criteria via new methods.


Introduction
This paper concerns the blow-up and asymptotic behaviour of positive solutions to initial value problems of the form x ′ (t) = t 0 w(t − s)f (x(s)) ds, t ≥ 0; x(0) > 0. (1) We assume that the nonlinearity, f , obeys f ∈ C((0, ∞); (0, ∞)), f is asymptotically increasing, lim The positivity and monotonicity hypotheses in (2) are natural when studying growing solutions to (1). Moreover, f (x)/x → ∞ as x → ∞ is necessary for the existence of a solution to (1) which blows up in finite-time. Sufficient conditions for the existence and uniqueness of local solutions are readily available [5]. It is well-known that the behaviour of the kernel near zero is crucial in the analysis of blow-up problems of the type studied in this paper [2]. Hence we assume that w(0) > 0, w ∈ C(R + ; R + ).
(3) There is a rich and active literature on blow-up problems in Volterra integral equations (VIEs) (see the survey articles [18,19] and the recent papers [7,8]). Much of this interest stems from the connections between VIEs and PDEs of parabolic-type in which the source term has a highly localised spacial dependence [6,15,16]. In this context, a blow-up solution represents the scenario in which the energy entering the system via the source term outweighs the ability of the medium to dissipate this energy and a literal explosion occurs in the physical system . In many cases, the leading order behaviour in such models is governed by a nonlinear VIE of the form nonlinearities, in fact their particular interest is n-th order equations [3]. In this special case, they improve upon Mydlarczyk's results by proving that lim t→τ B τ (x(t), t) = 1, τ ∈ {T, ∞}, for an appropriately chosen function B τ . To the best of our knowledge, this is the most complete result available in the extant literature. We first outline our results for the case H ≡ 0. Under (2) and (3), we identify a decreasing function F B such that where T is the blow-up time. Similarly, in the nonexplosive case, we identify an increasing function F U such that under the additional assumption that w ∈ L 1 (R + ; R + ). The functions F B and F U depend only on f and hence can be estimated from the problem data. Furthermore, our assumptions on the nonlinearity are nonparametric and allow a good deal of generality while still yielding strong conclusions. Interestingly, in spite of the dependence of these growth rates on w, the presence of a blow-up is completely independent of the value of w(0) and the structure of the kernel under (3).
If H ∈ C 1 ([0, ∞); [0, ∞)), then (5) is unchanged. However, in the nonexplosive case, H can impact the growth rate of solutions. When H is sufficiently small the growth rate from (6) is preserved and we characterise these rate preserving perturbations.
The outline of the paper is as follows: in Section 2 we give precise blow-up criteria for equation (1), explain how they can be recovered from previous work, and outline the novelty of our methods. Section 3 details the asymptotic growth rates of solutions to (1) when H ≡ 0 and Section 4 extends these results to the case when H is nontrivial. We provide some simple examples to illustrate the application of our results in Section 5. All proofs are deferred to the closing sections of the paper; Section 6 contains proofs of preliminary results and lemmas while Section 7 contains the proofs of our main results.

Blow-up Conditions
Definition 1. A solution to (1) blows up in finite-time if there exists T > 0 such that x ∈ C([0, T ); [0, ∞)) but lim t→T − |x(t)| = ∞; the minimal such T is the blow-up time.
The following result characterises the finite-time blow-up of solutions to (1).
Under (2), the negation of (7) is of course and, by Theorem 2, condition (8) guarantees that solutions to (1) are global; we record condition (8) for future reference. Theorem 2 is a special case of the following result.
Theorem 3 (Brunner and Yang [2, Theorem 3.9]). Suppose ψ > 0, h(t) ≥ 0 for t ≥ 0, w(t) = t β−1 w 1 (t) ≥ 0 for t ≥ 0, β > 0, and w 1 is bounded on every compact interval with inf s∈[0,δ] w 1 (s) > 0 for some δ > 0. Suppose that G : R + × R + → R + is continuous (uniformly in its second argument), increasing in its second argument, and satisfies lim u→∞ G(0, u)/u = ∞. Solutions to blow-up in finite-time if and only if there exists a t * > 0 such that To recover Theorem 2 from Theorem 3, set h ≡ 0, β = 1, and G(s, u) = f (u). Thus (10) holds if min u∈[0,∞) f (u) t * 0 W (s) ds − u > 0 and we can choose t * > 0 sufficiently large to satisfy this condition since ∞ 0 W (s) ds = ∞. In our case, condition (11) reduces to the finiteness of the integral i.e. the conclusions of Theorems 2 and 3 are consistent. While the conclusion of Theorem 2 is known, unlike Theorem 3, its proof yields considerable insight into the rate at which solutions to (1) grow. The proof of Theorem 3 proceeds by integrating (9) to obtain an integral equation of the form The integral equation above is discretised along a sequence (t n ) n≥1 upon which the solution to (9) grows geometrically, i.e. u(t n ) = R n for each n ≥ 1 and some R > 1. In all cases, lim n→∞ t n+1 − t n = 0 and moreover, if there is a global solution, h n = t n+1 − t n tends to zero so fast that ∞ n=1 h n < ∞, contradicting the existence of a global solution. Conversely, in the presence of a blow-up solution, (h n ) n≥1 is proven not to be summable using similar difference inequalities. Hence lim n→∞ t n = ∞, contradicting the assumption that the solution explodes in finite-time. In both cases, the summability of the sequence (h n ) n≥1 hinges on (11). Naturally, some rough rates of growth are implicit in the construction described above, but it is difficult to see how one could obtain sharp estimates on rates of asymptotic growth of solutions from this approach, even for the simpler equation (1).
In contrast, we exploit the enhanced differential structure of (1) and employ comparison equations of the form to establish sharp blow-up conditions. The fact that comparison equations such as (13) yield sharp blow-up criteria suggests that these bounded delay equations are promising candidates for investigating the more subtle issue of asymptotic behaviour. Under mild continuity assumptions, Solutions of (1) and (13) will grow extremely rapidly when f (x)/x → ∞ as x → ∞ so we conjecture that the delayed term in (14) is negligible asymptotically. Following this line of reasoning, we expect the second order ODE z ′′ (t) = f (z(t)) to give a good asymptotic approximation to solutions of (1); this approximation is at the heart of our analysis and the definitions which follow are the product of our efforts to systematically exploit this idea.
If φ, f ∈ C((0, ∞); (0, ∞)) obey φ ∼ f and φ preserves superexponential growth, then so does f . The following lemma (whose proof is elementary and thus omitted) records several important classes of nonlinear functions which preserve superexponential growth and frequently arise in applications.

Growth Rates of Solutions
In order to compute rates of growth of solutions, define the functions and for each x > 0. (16) F B characterises the rate of growth to infinity of solutions which blow-up in finite time, while F U captures rates of growth of unbounded but nonexplosive solutions. In order to compute growth rates, we ask that the nonlinearity preserves superexponential growth, in the sense of Definition 5. As discussed in Section 3, preservation of superexponential growth is a relatively mild hypothesis satisfied by broad classes of nonlinearities commonly found in applications (see Proposition 1).
Theorem 6. Suppose (2) and (3) hold. If (7) holds and f preserves superexponential growth, then solutions to (1) blow-up in finite-time and obey where T denotes the blow-up time.
When studying growth rates of non-explosive solutions, we further ask that If w does not have finite L 1 -norm, then it can contribute to faster growth in the convolution term when the solution is global; assuming (17) rules this out and allows us to prove the following analogue of Theorem 6 for non-explosive solutions.

Extensions to Perturbed Equations
We now consider the case when a nonautonomous forcing term is added to (1), i.e.
and demonstrate that the results of Section 2 are preserved under "small" perturbations. We do not require h to be nonnegative and hence solutions to (21) are no longer necessarily monotone; due to the nature of our comparison arguments this relaxation does not present any additional difficulties. Suppose that the forcing term, h, obeys Results regarding the finite-time blow-up of solutions require no additional hypotheses. However, for results regarding rates of growth we ask that the nonlinearity obeys in order to simplify and shorten the proofs. Our first result regarding solutions to the forced Volterra equation (21) shows that the blow-up condition and rate of explosion are unchanged by forcing terms obeying (22).
where T denotes the blow-up time.
Previously we assumed that f preserves superexponential growth when proving results regarding the rate of growth of solutions; henceforth we replace this hypothesis with the assumption that x → f (x)/x is eventually increasing.
(25) By Proposition 1, f preserves superexponential growth when (25) holds. As we show presently, the stronger hypothesis (25) allows us to characterise the perturbation terms which preserve the rate of growth when h ≡ 0, i.e. the asymptotic relation (18) still holds, in the non-explosive case. Our next result also shows that our blow-up conditions remain necessary if the nonautonomous forcing term is sufficiently small in an appropriate sense. x ∈ C([0, ∞); (0, ∞)). If we further suppose x → f (x)/x is eventually increasing, (23) holds, and w obeys (17), then the following are equivalent: It is evidently of interest to study the case when lim sup t→∞ F U (H(t))/t > 2w(0) and we conjecture that the perturbation likely dominates the dynamics of the system in this case. The results of [1] provide a road map as to how this issue could be addressed.

Examples
Since our results are insensitive to the structure of the memory, the examples which follow do not require a functional form for w (so long as continuity and integrability assumptions hold). For example, with ω > 0 arbitrary, the following kernels would be admissible: where Γ denotes the Gamma function.
Example 11. Suppose f (x) = (1 + x) β for all x > 0 and for some β > 1. Choose any w obeying (3). Note that this choice of f obeys (2) and also preserves superexponential growth; to see this check any of (i. − iii.) in Proposition 1. We first check condition (7) to determine whether or not solutions to (1) blow-up in finite-time. First note that For η > 0 arbitrary and N > 0 sufficiently large, we have for each η > 0 and β > 1.
Therefore, by Theorem 2, solutions to (1) blow-up for every w obeying (3). It can be shown that Thus, by Theorem 6, solutions to (1) obey Furthermore, solutions to (21) will still obey (26) for any perturbation h obeying (22). In this example one may "invert" the asymptotic relation (26) to obtain the leading order behaviour of the solution at blow-up. In other words, (26) can be improved to Example 12. Suppose f (x) = (x + e) log(x + e) for x > 0 and let w obey (3). Once again, it is straightforward to verify that f satisfies (2) and preserves superexponential growth.
Moreover, x → f (x)/x = (x + e) log(x + e)/x is eventually increasing. We first check condition (7) to see if solutions to (1) blow-up in finite-time. Direct computation shows that Therefore, by Theorem 2, solutions to (1) are global if w obeys (3). Furthermore, , as x → ∞ and thus, by Theorem 7, solutions to (1) obey Equation (27) is of course equivalent to saying that log(x(t)) ∼ w(0)t 2 /4 as t → ∞. Now we consider the effect of forcing terms on the asymptotic growth rate captured by (27). Firstly suppose h obeys (22) and H(t) ∼ t α as t → ∞, for some α > 0. Then Hence, by Theorem 10, solutions to (21) still obey (27) for any perturbation tending to infinity no faster than a power.

Preliminary Results and Lemmas
We first characterise the behaviour of solutions of two auxiliary equations, namely and for some C > 0 and δ > 0. We often use solutions to equations of the form (28) and (29) as comparison solutions for the more complex Volterra equations (1) and (21). The hypotheses on the nonlinearity are as before and the initial function, denoted by ψ, is assumed positive throughout, i.e. ψ ∈ C([−δ, 0]; (0, ∞)).
(30) The functionF given byF appears frequently and inherits useful properties from f , as noted in the following corollary.
for each t > 2δ. Using the positivity of z and (2) yields the lower bound for each t > 2δ. The estimate above can (equivalently) be written as for each t > 2δ.
We immediately have the following useful lemma which we record for future use.
Our final lemma identifies the growth rate of solutions to (29). The corresponding results for (1) and (21) consist of carefully constructing comparison solutions using equations of the form of (29) and then invoking this lemma.
It follows from (2) that z(t) 0 f (u) du → ∞ as t → ∞ and hence z ′ (t) → ∞ as t → ∞ by integration of (38). Now use L'Hôpital's rule to show that It follows that for each ǫ > 0 there exists T * (ǫ) > 0 such that Suppose t > T * (ǫ) and integrate the inequality above to yield By making the substitution y = z(u) it is straightforward to show that Let t → ∞ and then ǫ → 0 + in the inequalities above to complete the proof.

Dividing across byw(δ)
t 0 f (x(u)) du in (41) and taking the limsup thus yields lim sup Letting δ → 0 + in the limit above shows that lim sup Similarly, we can obtain the following lower estimate on the derivative where w(δ) = inf u∈[0,δ] w(u) > 0. Following the same steps as above quickly reveals that We claim that (42) implies Using (42), (43) is equivalent to Letting I(t) = t 0 f (x(s)) ds, the limit above is in turn equivalent to However, since I(t) → ∞ as t → T − and where the final equality follows from (42). Thus (42) implies (43), as claimed. By (43), for each ǫ ∈ (0, 1), there existsT (ǫ) ∈ (0, T ) such that Let t and T L be such thatT < t < T L < T and integrate the inequalities above from t to T L ; this yields forT < t < T L < T . Make the substitution y = x(u) in the integral to obtain forT < t < T L < T . Now let T L → T − and divide across by T − t to show that Let ǫ → 0 + in the inequalities above to complete the proof.
Proof of Theorem 9. We first show that (7) is a sufficient condition for the finite-time blowup of solutions to (21). Suppose T = ∞. For an arbitrary τ > 0, we have the trivial lower estimate x ′ (t) > t t−τ w(t − s)f (x(s)) ds + h(t) for t ≥ τ. By (2) there exists an increasing, positive function φ asymptotic to f and a finite, positive constant C such that where H τ (t) = t τ h(s) ds. Define the lower comparison solution y by Of course, y also obeys the delayed integro-differential equation Note that x(t) ≥ x(0) for each t > 0 due to (22), and y(t) < x(t) for each t ∈ [0, τ ] by construction. Use the continuity of h to choose τ > 0 sufficiently small that τ 0 h(s) ds ≤ x(0)/4 and suppose T B > τ is the minimal time such that y(T B ) = x(T B ). Thus a contradiction. Therefore y(t) < x(t) for each t ≥ 0. The proof of necessity in Theorem 2 now shows that T = ∞ produces a contradiction and hence that T ∈ (0, ∞), as required.
Integration of the inequalities above rules out the finite-time explosion of x, a contradiction. Therefore lim t→T − I(t) = ∞ and it follows from L'Hôpital's rule that Combining the limit above with (47) and (48) quickly establishes that Next let δ → 0 + in the inequalities above to show that We are now in the same position as at equation (42) in the proof of Theorem 6. Repeat verbatim the arguments which follow equation (42) to complete the proof.
We state without proof several elementary lemmas required for the proof of Theorem 10.