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

1. Introduction

In this paper, we consider the following problem

with partially hinged boundary condition where \( \Omega =(0,L)\times(-\ell,\ell)\subset \mathbb{R}^2\) represent a thin rectangular plate as a model of a suspension bridge and \(u=u(x,y,t)\) is the downward displacement of the rectangular plate, see [1,2] for detail description of suspension bridge models. The function \(a=a(x,y,t)\) is bounded, continuous and sign changing. For instance, if \(h:[0,\infty)\rightarrow (-\infty,\infty)\) be any function and \(g:\Omega\rightarrow (-\infty,\infty)\) be a bounded function, then \(a(x,t)=({sign}h)(t)g(x)\) is example of a sign changing function. Furthermore, \(\mu>0, 0 < \nu < \frac{1}{2}\), \(1< p< \infty\) and \(-1< \alpha< 1.\) The notation \(\partial^{1+\alpha}_t\) stand for the Capito's fractional derivative (see [3,4]) of order \(1+\alpha\) with respect to \(t\) defined by where \(I^{\beta} (\beta>0)\) is the fractional derivative defined by

For \(-1< \alpha< 0\), the term \(\partial^{1+\alpha}_t u\) is called the fractional damping while for \(\alpha=-1\) and \(\alpha=0\), it represent respectively the weak and strong damping. We should mention here that the fractional damping plays a dissipative role that is sandwich between the weak and the strong damping (see [5]). Concerning blow up results for plate equations, we mention among others the result of Messaoudi [6], where he studied the Petrovsky equation

where \(a,b>0\) and \({\Omega\subset \mathbb{R}^N, \ N\geq 1}\) is a bounded domain with a smooth boundary \(\partial\Omega\). He established local existence and uniqueness of a weak local solution and that for negative initial energy \((E(0) < 0)\) the local solution blows up in finite time when \(p>m\). In addition, established the existence of global solution when \(m\geq p\). The result in [6] was later improved by Chen and Zhou in [7]. Li et al. [8] considered and established global existence and blow up of solutions. Piskin and Polat [9] considered (6) and investigated the decay of solutions. Alaimia and Tatar [10] studied and proved blow up of the solutions for negative initial energy. For related results with fractional damping, we refer the reader to [11,12,13,14,15,16,17,18,] and references therein. The article is organized as follows: In Section 2, we recall some fundamental materials and useful assumptions on the relaxation function \(g\). In Section 3, we state and prove some technical lemmas. Finally, in section 4, we establish a blow-up result for problem 1.

2. 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 \(L^2(\Omega)\). We recall some useful materials. We consider the Hilbert space ( see [1]) \begin{equation*} H_{\ast}^{2}(\Omega) = \left\{w\in H^{2}(\Omega): w =0 \ {\text{on}} \ \left\{0, L \right\} \times (-\ell,\ell) \right\}, \end{equation*} together with the inner product \begin{equation*} (u,v)_{H_{\ast}^2}=\displaystyle\int_{\Omega}[(\Delta u\Delta v +(1-\nu)(2u_{xy}v_{xy}-u_{xx}v_{yy}-u_{yy}v_{xx})]dxdy, \end{equation*} and denote by \( \mathcal{H}(\Omega)\) the dual of \( H_{\ast}^2(\Omega).\)

Lemma 1. (Embedding, see [19]) Suppose \( 1< p < +\infty \). Then for any \(u \in H_{\ast}^2(\Omega)\), there exists an embedding constant \(S_{p} = S_{p}(\Omega,p)> 0\) such that

where \(S_p=\left(\frac{L}{2\ell}+\frac{\sqrt{2}}{2} \right)(2L\ell)^{\frac{p+2}{2p}} \left(\frac{1}{1-\nu} \right)^{\frac{1}{2}} .\)

The eigenvalue problem

which has been studied in[1], has a unique eigenvalue \(\wedge_1\in (1-\nu,1)\), \(0< \nu< \frac{1}{2}\) and \(\lambda=\wedge_1^2\) is the least eigenvalue. As a consequent, we have the following lemma

Lemma 2. Suppose \(-\wedge_1< a_1\leq a \leq a_2\). Then, the following inequality holds

where \(A_1= \begin{cases} 1+\frac{a_1}{\bigwedge_1},\ a_1 < 0,\\ 1, \ \ \ a_1\geq 0 \end{cases}\) and \(A_2= \begin{cases} 1, \ \ \ a_2< 0,\\ 1+\frac{a_2}{\wedge_1},\ a_2\geq 0 \end{cases}\) which has been proved in [19].

For completeness, we state without proof a local existence result for problem (1)-(2) (see [19,20] for more on existence).

Theorem 1. Let \( (u_0,u_1)\in H_{\ast}^2(\Omega)\times L^2(\Omega)\) be given and assume \(-\wedge_1< a_1\leq a\leq a_2\). Then, there exists a weak unique local solution to problem (1)-(2) in the class

for some \(T>0.\)

Definition 1. A function \(u\) satisfying (11) is called a weak solution of (1) if

a.e \( t\in (0,T)\) and \(\forall w\in H_{\ast}^2(\Omega)\).

We consider the energy functional \(E(t)\) defined by

Multiplying (1) by \(u_t\) and integrating over \(\Omega\), using integration by part, definition of fractional derivative (4) and recalling that \(a_1\leq a\leq a_2\), we obtain for almost all \(t\in [0,T).\) 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 \(\epsilon\) to be specified later. Differentiating (15) and making use of (1) and (14), we arrive at Also, we define the functional where with where \(\gamma=\frac{p+1}{2}\) and \(\theta,\lambda,\beta\) 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.

3. Technical lemma

Lemma 3. Suppose \(E_{\epsilon}(0)< 0\) and \(p\) is sufficiently large, then \(H(t)\) and \(H'(t)\) are strictly positive.

Proof. Differentiating (17) with respect to \(t\) and using (15) yields

Substituting (13),(16) and (20) into (21), we arrive at Using Young's inequality and Lemma 1, we obtain Again, Young's and Cauchy-Schwarz inequalities, we get In a similar way, with the help of lemma 1, we find Substitution of (23)-25 into (22) and using lemma 2, we obtain Adding \(C_1H(t)-C_1H(t)\) to the right hand of (26), for some \(C_1\) to be precise, we arrive Applying (23) to (27), we arrive at Recalling that \(\gamma=\frac{p+1}{2}\) and choosing \(\delta_1=\frac{1}{2}, \delta_2=\delta_3=\frac{\Gamma(-\alpha)\epsilon}{2}\) and \(C_1=\frac{(p+1)\epsilon}{2},\) we get Now, choosing we get that \[\frac{\epsilon}{2}\left[A_1(p-1)-\epsilon S_2^2((p+1) +\mu) \right]> \frac{A_1(p-1)\epsilon}{4}.\] Next, we select \(\beta=1\), we see that for sufficiently large values of \(p\) \[ \frac{(p+1)\epsilon}{2}-\beta>0. \] Finally, we choose \(\theta\) such that the coefficient of the second term is non-negative and the coefficient of the last term is greater than \(\frac{\mu (p+1)^{\alpha}}{2^{\alpha+1}\epsilon^{1-\alpha}\Gamma(-\alpha) }.\) Thus, we arrive at If we choose \(\lambda< -E_{\epsilon}(0),\) then \(H(0)>0.\) Consequently, it follows from (31) that \(H(t)>0\) and \(H'(t)>0.\) This completes the proof.

4. Main results

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

Theorem 2. Assume that \(-\wedge_1< a_1\leq a\leq a_2\), \(-1< \alpha < 0\),\(E(0)< 0\) and \((u_0,u_1)_2\geq 0.\) Then the solutions of 1-2 blows up in finite time for sufficiently large values of \(p\).

Proof. We begin by defining the functional \(G\) by

where \(\sigma=\frac{p-1}{2(p+1)}\) and \(\eta>0\) 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 \begin{eqnarray} \label{p26} G'(t) &\geq& (1-\sigma)H^{-\sigma}(t)H'(t) + \eta\left(1-\frac{\gamma\epsilon}{4\delta_4}\right) e^{-\gamma\epsilon t } \|u_t(t)\|^2_2 - \eta\left( A_1 + \delta_4 \gamma\epsilon S_2^2 \right)e^{-\gamma\epsilon t } \|u(t)\|_{H_{\ast}^2(\Omega)}^2 \notag\\ && + \eta e^{-\gamma\epsilon t }\int_{\Omega}|u|^{p+1}dxdy- \frac{\eta\mu\delta_5}{\Gamma(-\alpha)}e^{-\gamma\epsilon t }\|u(t)\|^2_2 -\frac{ \mu \eta (\gamma\epsilon)^{\alpha}}{4\delta_5}\int_{\Omega}\int_0^t (t-s)^{-(\alpha +1)}e^{-\gamma\epsilon s}u_s^2(s)dxdy. \end{eqnarray} Using (31), we obtain From the last inequality, we get Now, we choose \(\delta_5=B H^{\sigma}(t)\) for some \(B\) positive to be precise later. Then, (39) becomes Adding and subtracting \(H(t)\) on the right hand side of (40) and making use of lemma 2 leads to The term \((u,u_t)_2\) is estimated similarly as in (23) as For the term \(H^{\sigma}(t)\|u(t)\|^2_2,\) we use the definition of \(H(t)\) in (17) and the choice of \(\epsilon\) in (23) to get (see [10] page 141 for detail computations) for some constant \(C_2>0.\) Substituting (42) and (43) into (41) yields Now, we choose are parameters carefully. First, recalling \(\sigma=\frac{p-1}{2(p+1)}\) and selecting \(\epsilon\) so small such that we see that the coefficient of the first term is positive. By choosing \(\eta=\frac{p+3}{4(p+1)}\), \(\delta_4=\delta_6=\frac{1}{2},\) and \(\epsilon\) small enough so that we find that the coefficient of \(\|u(t)\|_{H_{\ast}^2(\Omega)}^2\) is positive. Next, we pick \(\epsilon\) small enough such that to get the coefficient of \(\|u_t(t)\|^2_2\) greater or equal to \(\frac{1}{2}\). We select \(B\) such that to see that the coefficient of \(\int_{\Omega}|u|^{p+1} dxdy\) is greater than \(\frac{p-1}{4(p+1)}\) and the term \[\left(\lambda-\frac{\eta\mu B C_2}{ (p+1)^{\sigma}\Gamma(-\alpha)} \right)>0.\] Thus, for any \(\epsilon\) positive small enough such that we arrive at Using Cauchy-Schwarz and Young's inequalities, we have where \(C_2=C_2(|\Omega|,p)>0, \ C_3=C_3(|\Omega|,p,\sigma)>0\) are constants and \(\frac{1}{r_1}+\frac{1}{r_2}=1.\) We recall that \(\sigma=\frac{p-1}{2(p+1)}\), therefore we select \(r_1=\frac{2(1-\sigma)}{1-2\sigma},\ \ r_2=2(1-\sigma),\) and arrive at We observe that \(\frac{2}{(p+1)(1-2\sigma)}=1,\) so \begin{equation} \nonumber \|u(t)\|_{p+1}^{\frac{2}{1-2\sigma}}=\int_{\Omega}|u|^{p+1} dxdy. \end{equation} From the definition of \(G(t)\), we have for some positive constant \(C\) such that \[C\geq 2^{\frac{1}{1-\sigma}} \max\left\lbrace 1,2 C_3\eta^{\frac{1}{1-\sigma}} ,C_3\eta^{\frac{1}{1-\sigma}}\frac{4(p+1)}{p-1} \right\rbrace.\] A combination of (50) and (53) leads to From (50), we see clearly that \(G'(t)\geq 0\). 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 \begin{equation} \nonumber \left( G(t)\right)^{\frac{-\sigma}{1-\sigma}}\leq \left( G(0)\right)^ {\frac{-\sigma}{1-\sigma}}-\frac{\sigma}{C(1-\sigma)}t \end{equation} which gives From (56), we obtain that \(G(t)\) blows up in time This completes the proof.

5. 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.

acknowledgments

The author thank University of Hafr Al-Batin for its continuous support.

conflictofinterests

The author declares no conflict of interest.