Blow-up result for a plate equation with fractional damping and nonlinear source terms

Abstract: In this work, we consider a plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in finite time. The background of the problem comes from the modeling of the downward displacement of suspension bridge using a thin rectangular plate. The result in the article shows that in the present of fractional damping and a nonlinear source such as the earthquake shocks, the suspension bridge is bound to collapse in finite time.


Preliminaries
Throughout the paper, C i , i = 1, 2, 3, .. or c are generic positive constants that may change within lines and (, ) 2 and . 2 denote respectively the inner product and norm in L 2 (Ω). We recall some useful materials. We consider the Hilbert space ( see [1]) together with the inner product and denote by H(Ω) the dual of H 2 * (Ω).
For completeness, we state without proof a local existence result for problem (1)-(2) (see [19,20] for more on existence).
Then, there exists a weak unique local solution to problem (1) − (2) in the class for some T > 0.
We consider the energy functional E(t) defined by Multiplying (1) by u t and integrating over Ω, using integration by part, definition of fractional derivative (4) and recalling that a 1 ≤ a ≤ a 2 , we obtain for almost all t ∈ [0, T). The result in (14) is for any regular solutions. However, this result remains valid for weak solutions by simple density argument. We define a modify energy functional: for some to be specified later. Differentiating (15) and making use of (1) 1 and (14), we arrive at Also, we define the functional where with where γ = p+1 2 and θ, λ, β are positive constants to be specified later. The differentiation of (18) gives the relation In the next section, we state and prove some useful Lemmas.

Technical lemma
Lemma 3. Suppose E (0) < 0 and p is sufficiently large, then H(t) and H (t) are strictly positive.

Main results
In this section, we show that the solutions of 1-2 blows up in finite time for negative initial energy.

Proof.
We begin by defining the functional G by where σ = p−1 2(p+1) and η > 0 to be specified later. Then differentiating G(t) and using (1) yields Similarly as in the inequalities (23)and (25), we have that and From Lemma 2, we get Substituting (34)-(36) into (33), we obtain Using (31), we obtain From the last inequality, we get Now, we choose δ 5 = BH σ (t) for some B positive to be precise later. Then, (39) becomes From the definition of G(t), we have for some positive constant C such that A combination of (50) and (53) leads to (G(t)) From (50), we see clearly that G (t) ≥ 0. It follows from the definition of G(t) and the assumption on u 0 and u 1 that Hence, G(t) > 0. Integrating (54) over (0, t) yields From (56), we obtain that G(t) blows up in time This completes the proof.

Conclusion
In this paper, we have studied a plate equation supplemented with partially hinged boundary conditions as model for suspension bridge in the presence of fractional damping and non-linear source terms. We showed that the solution blows up in finite time. We saw that, even in the present of a weaker damping, the bridge will collapse in infinite time when the power p of the non-linear source term is sufficiently large. This is a very important factor for engineers to consider when constructing such types of bridges.