Global vs Blow-Up Solutions and Optimal Threshold for Hyperbolic ODEs with Possibly Singular Nonlinearities

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbations or large initial values, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold.


Introduction
In this paper, we study the following ordinary differential equation: where λ > 0, α ≥ 0, β > 0, u 0 ∈ R and v 0 ∈ R.Under appropriate assumptions on f and g, we discuss the global existence and blow-up of the solutions to (1).For α ≥ 0 and the initial value (u 0 , v 0 ) ∈ R 2 , we denote by λ * (u 0 , v 0 ) (α) the dynamical threshold for the existence of a global solution of (1).Namely, the solution of (1) exists globally in time for 0 < λ < λ * (u 0 , v 0 ) (α) and blows up for λ > λ * (u 0 , v 0 ) (α).At the same time, let us denote by λ * the stationary threshold, see Theorem 1 below, for the existence of solutions to βφ = λg(φ). ( From the point of view of applications to Micro Electro Mechanical Systems, the value λ * plays an important role as it is connected to the so-called "pullin instability", see [1,2] and references therein.
So far, a huge literature has been devoted to (1) and related problems from the theoretical point of view as well as from the point of view of applications, see for instance [12,9,6] and references therein.
In [5], Flores studies the following problem He proves that 0 < λ (0, 0) (α) < λ * for α ≥ 0 and that lim α→+∞ λ (0, 0) (α) = λ * .In [7], Haraux considers the following for c > 0, α > 0, β > 0 and studies the existence of sign-changing solutions, number of the zeros and the decay estimates for solutions depending on the value of the parameters c, α, β.Moreover, the results are generalized to for c > 0, α > 0, β > 0 for a continuous function f (t) with some decaying properties.In [11], Souplet studies the following backward equation and proves the existence of unbounded global solutions, unbounded oscillatory solutions, as well as their blow-up rate and asymptotic behaviour.In [3,10], the authors consider They investigate for which value of parameters one has existence of solutions, derive their asymptotic behaviour and classify the ground state solutions according to the value of parameters.
Here we aim at extending the result in [5] to more general nonlinearities f and g.Henceforth, we consider (1) and (2) under Assumptions 1 and 2 unless otherwise stated.We consider first the stationary problem (2) for which we obtain the bifurcation diagram of the solution set {(λ, φ)}.
In order to investigate the dynamical behaviour, we find the solution (λ, φ 2 ) with the following properties: Theorem 2 There exists a unique solution (λ, φ 2 (λ)) = (λ, φ 2 ) of (2) satisfying the following: Next, on the one hand we consider the time-dependent equation ( 1) and derive the conditions for the existence of global bounded solutions.
The main result of this paper is concerned with establishing an optimal dynamical threshold, indeed we have The function λ (0, 0) (α) is continuous and monotone increasing with respect to α ≥ 0 and satisfies: This paper is organized as follows: In Section 2, we consider the stationary problem.Thanks to the assumptions on g, we get at most two positive solutions.In Section 3, we settle preliminary lemmas, involving energy and dynamical system, in order to investigate the dynamical behaviour.On the one hand, in Section 4, we establish the existence of a global solution and periodic orbit under some appropriate conditions.On the other hand, for large values of the perturbation parameter or large initial values, we prove that the solution blows up.In Section 5, we discuss qualitative properties of the orbit such as openness, monotonicity and continuity.In Section 6, we prove our main result, namely Theorem 6.

The stationary problem
Here we study the solution set of the function equation ( 2) and obtain the upper bound λ * for the existence of solutions.We regard φ = φ(λ) as a function of the parameter λ.
Proof of Theorem 1.Since solutions are positive, we consider in (2) φ ≥ 0. By Assumption 2, is well-defined for u ≥ 0 and the following holds From F (0) = F (b) = 0 and F ′ (0) > 0, there exists p > 0 such that F ′ (p) = 0. Next we show the uniqueness of such p.Set G(u) := g(u) − ug ′ (u), and consider the sign of G(u).We have Then βF (u) = λ is equivalent to H λ (u) = 0 and the sign of H λ (u) is given by the value of λ.
Proof.Noting that H λ (φ 1 ) = 0, we apply the mean value theorem to obtain the conclusion, as Proof of Theorem 2. Let us define We claim that I(u 0 ) = 0 for some p < u 0 < b.As a consequence, we find a solution (λ, φ 2 ) = (βu 0 /g(u 0 ), u 0 ) of ( 2) by Theorem 1 and Let us prove the claim.First we have .
has a singularity at u = 1.However, Assumption 2 is satisfied with b = 1.In particular, in the case of p = 2 we have

A dynamical system
Consider the following energy functional Then, E λ (u(t), u t (t)) turns out to be the Lyapunov function for (1).In fact, we have which yields Hence, by defining the energy inequality holds by (4).Every local solution satisfies ( 5) as long as it exists.To extend the solution globally in time, we consider some properties of J λ (u) in the following two lemmas: The following hold: ) is monotone decreasing in λ.It follows from Theorem 2 that J λ (φ 2 (λ)) = 0, which yields the conclusions.✷ In the next section, we consider dynamical properties of the solution of ( 1), which can be written in the following form Now under Assumptions 1 and 2, we obtain a local solution.Next, we establish the existence of a global solution exploiting Lemmas 2 and 3.For this purpose, we consider the stability of the equilibrium point.At the equilibrium point (u, u t ) = (φ i (λ), 0) for i = 1, 2, the linearized equation is given by and the eigenvalues µ ± i of coefficient matrix are given respectively by Along with Corollary 1, we have the following two lemmas: and it is a centre for αf ′ (0) = 0.

The time-dependent problem
In this Section, for small parameters and small initial values, we establish the existence of a global solution by means of the Lyapunov function method.For the dissipative case, that is α > 0, we show that the global solution converges to the stationary solution.For the conservative case, namely α = 0, we consider the periodic orbit starting at (0, 0).Finally, we show that the solution becomes unbounded for large parameters as well as for large initial values.
In other words, the solution exists globally in time for the initial value (u 0 , v 0 ) with φ 1 < u 0 < φ 2 and v 0 = 0.
Proof of Theorem 4. Let us divide the proof into four cases: Integrating (1) and thanks to Corollary 2, we have which implies that u(t) reaches b for finite T ∞ < ∞.Then by ( 4) and Assumption 2.
II.The case α = 0 and I = R.
Assume by contradiction that T ∞ = ∞.By ( 8), there exists sufficiently large T > 0 such that for all t > T , where τ is given in Assumption 2 and T can be taken as T = 2ξ −1 R.Then, integrating inequality (9) twice, with respect to u, over [u(T ), u] we get as g ′ (u) ≥ 0 for u ≥ 0. Thus we have for all t > T .Since u ′ (t) > 0 holds for all t > T by (8), we have Take T 1 ∈ (T, +∞) such that holds for all t > T 1 .For instance, we can take T 1 as follows Then we have for all t > T 1 .Integrating this inequality over [T 1 , t], we have which implies that lim t→T 2 u(t) = +∞, where contradicting the maximality of T ∞ and hence necessarily T ∞ < +∞.Thanks to (4), we also have which implies that both u and u t blow up to +∞ as t → T ∞ .
If v 0 = 0, we have Hence for v 0 ≥ 0, we have u t (t) > 0 for sufficiently small t > 0. If there exists T 3 ∈ (0, T ∞ ) such that u t (t) > 0 for all t ∈ (0, T 3 ) and u t (T 3 ) = 0, then we have u tt (T 3 ) > 0 similarly to (12), which contradicts the positivity of u t .Hence, if necessary, we retake the initial value as u 0 = u(T 4 ) and v 0 = v(T 4 ) for some T 4 > 0 so that u 0 > 0, v 0 > 0, u(t) > 0 and u t (t) > 0 hold for all t ∈ (0, T ∞ ).We estimate u t (t).First if u tt (0) > 0 holds, we have u t (t) ≥ v 0 for sufficiently small t > 0. On the other hand, if u tt (t) ≤ 0 holds for some t ≥ 0, we have and then .

Thus we have
for all t ∈ (0, T ∞ ), which yields for all t ∈ (0, T ∞ ), which brings back to the same situation of case I above.
IV.The case α = 0 and I = R.By estimates carried out in the case III, we have that ( 13) and ( 14) hold.Hence, (u, u t ) is unbounded in R 2 for t ∈ (0, T ∞ ), where T ∞ ≤ +∞.In the case of T ∞ < +∞, both u and u t blow up to +∞ as t → T ∞ similarly to (11).Next let us consider the case T ∞ = +∞ and let us prove that u t (t) → +∞, as t → +∞.Now suppose that there exists a constant for all t ≥ T .Then, buying the line of ( 10) we obtain the following differential inequality for all t ≥ T .Thus for sufficiently large t > T , we have which yields u t (t) → +∞, as t → +∞ by integration and contradicting the boundedness of u t .Hence, both u and u t blow up to +∞ as t → T ∞ .✷ Proof of Theorem 5.By hypothesis, φ 2 < u(t) < b holds for sufficiently small t > 0. If v 0 = 0, we have by Corollary 1 and Remark 3. Hence we may assume that φ 2 < u 0 < u(t) < b and u t (t) > 0 holds for all t ∈ (0, T ∞ ).Now set ξ = −H λ (u 0 ) > 0 and proceed as in the proof of Theorem 4. ✷ Let us denote by γ(t; λ, α) the orbit of solution of (1) starting at (0, 0).We note that from Lemma 4, one has αf ′ (0) = 0 if and only if (φ 1 (λ), 0) is a centre.

Properties of the dissipative orbit
In this Section, we study the properties of the orbit starting at the origin for the dissipative case, that is, α > 0.Moreover, we also assume that f (v) = v.Then, (φ 1 (λ), 0) is a hyperbolic sink for all λ ∈ (λ, λ * ).The argument proceeds in the same way as in Section 3 of [5].
Let us first define a few sets which will be used in the sequel: where γ(t; λ, α) is the orbit of the solution (u, v) = (u, u t ) of (1) starting at (0, 0).In what follows we will also use for convenience the following equivalent notations for any fixed α, λ > 0.
Proof.Since we have Φ 2 (0) − Φ 1 (0) = 0 and (dΦ 2 /dU We are interested in the set L(λ) of all points P α defined by L(λ) consists of the points where the unstable manifold intersects the positive U -axis.We shall show that K(λ) is a non-empty interval and that L(λ) is an unbounded interval.For this purpose, let us define two lines parallel to the V -axis as follows: Thanks to the continuity of the intersection point with respect to α proved in [4], M Φ (λ) is an interval in M (λ).
Hence it enters Then, we can find T > 0 such that φ 2 (λ) < u(T ) < b, v(T ) > 0 and v t (T ) = 0 by Proposition 5. Hence the solution blows up by Theorem 5.