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Blow-up results for a quasilinear von Karman equation of memory type
Boundary Value Problems volume 2019, Article number: 174 (2019)
Abstract
In this paper, we consider the blow-up result of solution for a quasilinear von Karman equation of memory type with nonpositive initial energy as well as positive initial energy. For nonincreasing function \(g>0\) and nondecreasing function f, we prove a finite time blow-up result under suitable condition on the initial data.
1 Introduction
Let \(\rho >0, \alpha >0\), and \(p>2\). Moreover, let us denote by Ω an open bounded set of \(\mathbb{R}^{2} \) with sufficiently smooth boundary Γ. We assume that \(\varGamma _{0} \cup \varGamma _{1}=\varGamma \), \(\varGamma _{0} \cap \varGamma _{1} = \emptyset \), \(\varGamma _{0} \neq \emptyset \), and \(\varGamma _{0} \) and \(\varGamma _{1} \) have positive measure. In this paper we investigate a blow-up result for the following quasilinear von Karman equation of memory type:
where \(\nu =(\nu _{1}, \nu _{2} )\) is the outward unit normal vector on Γ. The relaxation function g is a positive nonincreasing function and f is a nondecreasing function. Here
where
and the constant \(\mu \in (0, \frac{1}{2} )\) represents Poisson’s ratio. The von Karman bracket \([\varpi, \phi ] \) is given by
The authors in [1,2,3,4,5] studied the asymptotic behavior of the solutions to a von Karman system with dissipative effects. The uniform decay rate for the von Karman system with frictional dissipative effect in the boundary has been proved by several authors [6,7,8]. For a von Karman equation with rotational inertia and memory of the form
many authors [9,10,11,12] showed the existence and stability of solutions. Several authors [13,14,15] investigated the general stability for a von Karman system with memory boundary conditions. The stability for a von Karman system with acoustic boundary conditions was treated by [16, 17]. Some authors discussed the energy decay for a von Karman equation with time-varying delay (see [18, 19] and the reference therein).
On the other hand, many authors have considered the global existence, uniform decay rates, and blow-up of solutions for the wave equation with nonlinear damping and source terms:
where \(a, b>0\) and \(p, m>2\). When \(a=0\), Ball [20] showed that the source term \(|u|^{p-2}u\) causes blow-up of solutions with negative initial energy in finite time. For \(m=2\), Levine [21, 22] proved that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [23] extended Levin’s result to the nonlinear damping case. Messaoudi [24] improved the blow-up result of [23] to the solutions with negative initial energy. Messaoudi [25] studied the blow-up property of solutions with negative initial energy for the following viscoelastic wave equation with \(p>m\):
Messaoudi [26] extended the blow-up result of [25] to the solution with positive initial energy. Song [27] proved the finite time blow-up of some solutions whose initial data have arbitrarily positive initial energy for (1.8). Recently, Park et al. [28] showed the blow-up of the solutions for a viscoelastic wave equation with weak damping. Liu and Yu [29] investigated the blow-up of the solutions for the following viscoelastic wave equation with boundary damping and source terms:
For more related works, we refer to [30,31,32,33,34,35,36,37,38] and the references therein.
To our best knowledge, there are no blow-up results of solution for the von Karman equation with memory. Motivated by the previous results, we consider the quasilinear von Karman equation with memory and boundary weak damping. We study a finite time blow-up result under suitable condition on the initial data.
The outline of the paper is the following. In Sect. 2, we give some notations and hypotheses for our work. In Sect. 3, we prove our main result.
2 Preliminary
In this section, we present some material needed in the proof of our result. Throughout this paper we denote
For a Banach space X, \(\Vert \cdot \Vert _{X} \) denotes the norm of X. For simplicity, we denote \(\Vert \cdot \Vert _{L^{2} (\varOmega )} \) by the norm \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{L^{2} (\varGamma _{1})}\) by \(\Vert \cdot \Vert _{ \varGamma _{1}}\), respectively. We define, for all \(1\leq p<\infty \),
Let \(0< \mu <\frac{1}{2}\), we define the bilinear form \(a(\cdot, \cdot )\) as follows:
A simple calculation, based on the integration by parts formula, yields
Thus, for \((y, \kappa ) \in (H^{4} (\varOmega )\cap W)\times W\), it holds
Since \(\varGamma _{0} \neq \emptyset \), we have (see [39]) that \(\sqrt{a(y,y)} \) is equivalent to the \(H^{2} (\varOmega )\) norm on W, that is,
Now we state the assumptions for problem (1.1)–(1.7). We will need the following assumptions.
(H1) Hypotheses on g.
Let \(g: \mathbb{R}^{+} \to \mathbb{R}^{+} \) be a nonincreasing \(C^{1} \) function satisfying
(H2) Hypotheses on f.
Let \(f : {\mathbb{R}} \to {\mathbb{R}} \) be a nondecreasing \(C^{1}\) function with \(f(0)=0 \). There exists an odd and strictly increasing function \(\xi : [-1, 1] \to {\mathbb{R}}\) such that
where \(c_{1}\) and \(c_{2} \) are positive constants, \(m>2\), and \(\xi ^{-1}\) denotes the inverse function of ξ.
We state the well-posedness which can be established by the arguments of [11,12,13, 29, 40].
Theorem 2.1
Suppose that (H1)–(H2) hold and \((y_{0}, y_{1}) \in (H^{4} (\varOmega )\cap W )\times (H^{3} (\varOmega ) \cap V))\). Then, for any \(T>0\), there exists a unique solution of problem (1.1)–(1.7) such that
A direct calculation gives
where
We recall the trace Sobolev embedding
and the embedding inequality
where \(B >0\) is the optimal constant. We define the energy associated with problem (1.1)–(1.7) by
then
So the energy E is a nonincreasing function. Next, we define the functionals
We define
Lemma 2.1
For \(t\geq 0\), we get
where \(e_{0} = \frac{p-2}{2p} ( \frac{C_{1} l}{B^{2}} )^{\frac{p}{p-2}}\) and
Proof
We find that
If \(\frac{d H(\lambda y,\lambda z)}{d\lambda }=0\), then we obtain
It is easy to verify that \(\frac{d^{2} H}{d\lambda ^{2}} |_{\lambda =\lambda _{1}} <0 \), then from (2.3), (2.4), and (2.8)
By the definition of \(e_{0} \), we conclude that \(e_{0}>0\). □
Lemma 2.2
Assume that (H1)–(H2) hold. Suppose that \((y_{0}, y_{1}) \in W\times L^{2}(\varOmega )\) and satisfy
Then, for some \(T>0\), we get \(I(t)<0 \) and
for all \(t\in [0, T)\).
Proof
Using (2.10) and (2.13), we obtain \(E(t)< \epsilon e_{0}\) for all \(t\in [0,T)\). We can also have \(I(t)<0\) for all \(t\in [0,T)\). It can be showed by contradiction. Suppose that there exists some \(t_{0}>0\) such that \(I(t_{0}) =0\) and \(I(t)<0\) for \(0\leq t < t_{0}\). Then
Using Lemma 2.1 and (2.15), we see that
Applying (2.15) and (2.16), we obtain
From \(t \rightarrow \Vert y(t) \Vert _{p,\varGamma _{1}}^{p} >0 \) is continuous, we have \(y(t_{0})|_{\varGamma _{1}} \neq 0\). By (2.12) and \(I(t_{0})=0\), we find that
This is contradiction to \(H(t_{0}) \leq E(t_{0} ) < e_{0} \). From Lemma 2.1, we get (2.14). □
We set
where \(\hat{\epsilon } =\max \{ 0, \epsilon \}\). By (2.10), G is an increasing function. Using (2.9), (2.13), (2.14), and (2.17), we obtain
where \(p_{0}=\frac{\hat{ \epsilon } }{2} + (1-\hat{\epsilon }) \frac{1}{p}\).
Lemma 2.3
Let the conditions of Lemma 2.2 hold. Then the solution y of problem (1.1)–(1.7) satisfies
where \(C_{3}>0\).
Proof
If \(\Vert y(t) \Vert _{p,\varGamma _{1}} \geq 1\), then \(\Vert y(t) \Vert _{p,\varGamma _{1}} ^{s} \leq \Vert y(t) \Vert _{p, \varGamma _{1}}^{p}\).
If \(\Vert y(t) \Vert _{p,\varGamma _{1}} \leq 1\), then
where we used (2.3) and (2.8). Then there exists a positive constant \(C_{4}=\max \{1, \frac{B^{2}}{C_{1}}\} \) such that
By (2.4), (2.9), (2.17), and (2.18),
Using (2.20) and (2.21), we get the desired result (2.19). □
3 A blow-up of solution
To obtain the blow-up result for solutions with nonpositive initial energy as well as positive initial energy, we use a similar method of [26, 29].
Theorem 3.1
Let (H1)–(H2) and the conditions of Lemma 2.2 hold, \(\epsilon < \frac{p-4}{p-2}\) and \(p> \max \{ 4, m \}\). Moreover, we assume that g satisfies
where \(\hat{\epsilon }=\max \{ 0, \epsilon \}\) and
where \(0 < \eta < \min \{2 \theta _{0},2 \theta _{1}, 4 \theta _{2}\}\), \(0<\beta <\eta ^{\frac{1}{p-1}}\), for some \(\delta >0\),
Then the solution of system (1.1)–(1.7) blows up in finite time.
Proof
We suppose that there exists some positive constant \(B_{0}\) such that, for \(t>0\), the solution \(y(t)\) of (1.1)–(1.7) satisfies
Let us define
where \(\varepsilon >0\) shall be taken later and
Using (1.1)–(1.6), (2.2), (2.9), and (2.17), we get
From (2.14), we find that
Moreover, we give
for some \(\delta >0\). Combining (3.9), (3.10), and (3.11), we deduce that
for some δ with \(0<\delta <1+ (\frac{p}{2}-1 )(1- \hat{\epsilon })\). By (3.1), (3.3)–(3.5), estimate (3.12) can be rewritten by
Using a method similar to [30], we now estimate the last term of the right-hand side of (3.13). Setting \(\varGamma _{11} = \{ x\in \varGamma _{1} : |y_{t} (x,t)| \leq 1 \}\) and \(\varGamma _{12} = \{ x\in \varGamma _{1} : |y_{t} (x,t)| > 1 \} \), we obtain
From (2.5) and Young’s inequality, we get
On the other hand, by using (2.6), (2.10), (2.17), and Young’s inequality, we have
Inserting (3.14)–(3.16) into (3.13), we obtain
We choose \(\gamma = ( \tau G^{-\sigma }(t) )^{- \frac{m-1}{m}}\), \(\tau >0\) will be specified later. Using (2.18), (2.19), and (3.8), we see that
where \(C_{5}= \frac{c_{2}^{m} p_{0}^{\sigma (m-1)} C_{3} }{m} \). Substituting (3.18) into (3.17), we have
Adding and subtracting \(\varepsilon \eta G(t) \) on the right-hand side of (3.19) and applying (2.9) and (2.17), we obtain
We fix η such that
then we can choose \(\beta >0\) sufficiently small so that \({ \eta } - {\beta ^{p-1}} >0 \). And then, we select \(\tau >0\) large enough such that \(\frac{ \eta }{p} - \frac{\beta ^{p-1}}{p} - C_{5} \tau ^{1-m} >0 \). Finally, we take \(\varepsilon >0\) with
Condition (3.2) yields
Therefore, we get from (2.3) and (3.20)
where \(C>0\) is a generic constant. Hence we have
By the similar arguments in [31, 32], we see that
Indeed, using Young’s inequality and
we obtain
where \(\frac{1}{\kappa }+\frac{1}{\mu }=1\). By taking \(\kappa =\frac{(1- \sigma )(\rho +2)}{\rho +1}\) and using (3.8), we get \(\kappa >1\) and \(\frac{\mu }{1-\sigma } = \frac{\rho +2}{(1-\sigma )(\rho +2)-( \rho +1)}\). Since G is an increasing function, (2.18) and (3.6), we arrive at
where \(C_{0}\) is the embedding constant. Similarly, by Young’s inequality, we obtain
Like (3.25), we find that
where \(C_{*}\) is the embedding constant. From (2.18), (3.7), (3.24)–(3.27), we see that (3.23) holds. Combining (3.22) and (3.23), we deduce that
By a simple integration of (3.28) over \((0,t)\), we get
Consequently, \(F(t)\) blows up in time \(T^{*} \leq \frac{1-\sigma }{C \sigma F^{\frac{\sigma }{1-\sigma }} (0)} \). Furthermore, we have from (3.23)
This leads to a contradiction with (3.6). Therefore the solution of (1.1)–(1.7) blows up in finite time. □
4 Conclusions
In this paper, we consider the blow-up of solutions for the quasilinear von Karman equation of memory type. In recent years, there has been published much work concerning the wave equation with nonlinear boundary damping. But as far as we know, there was no blow-up result of solutions to the viscoelastic von Karman equation with nonlinear boundary damping and source terms. Therefore, we will prove a finite time blow-up result of solution with positive initial energy as well as non-positive initial energy. Moreover, we generalize the earlier result under a weaker assumption on the damping term.
References
Bradley, M.E., Lasiecka, I.: Global decay rates for the solutions to a von Karman plate without geometric conditions. J. Math. Anal. Appl. 181, 254–276 (1994)
Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics. Springer, New York (2010)
Khanmamedov, A.K.: Global attractors for von Karman equations with nonlinear interior dissipation. J. Math. Anal. Appl. 318, 92–101 (2006)
Lasiecka, I.: Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation. NoDEA Nonlinear Differ. Equ. Appl. 6, 437–472 (1999)
Puel, J., Tucsnak, M.: Boundary stabilization for the von Karman equations. SIAM J. Control Optim. 33, 255–273 (1996)
Chueshov, I., Lasiecka, I.: Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping. J. Differ. Equ. 233, 42–86 (2007)
Favini, A., Horn, M., Lasiecka, I., Tataru, D.: Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation. Differ. Integral Equ. 9(2), 267–294 (1996)
Horn, M.A., Lasiecka, I.: Global stabilization of a dynamic von Karman plate with nonlinear boundary feedback. Appl. Math. Optim. 31, 57–84 (1995)
Kang, J.R.: Energy decay rates for von Karman system with memory and boundary feedback. Appl. Math. Comput. 218, 9085–9094 (2012)
Kang, J.R.: A general stability for a von Karman system with memory. Bound. Value Probl. 2015, 204 (2015)
Raposo, C.A., Santos, M.L.: General decay to a von Karman systems with memory. Nonlinear Anal. 74, 937–945 (2011)
Rivera, J.E.M., Menzala, G.P.: Decay rates of solutions of a von Karman system for viscoelastic plates with memory. Q. Appl. Math. LVII(1), 181–200 (1999)
Rivera, J.E.M., Oquendo, H.P., Santos, M.L.: Asymptotic behavior to a von Karman plate with boundary memory conditions. Nonlinear Anal. 62, 1183–1205 (2005)
Santos, M.L., Soufyane, A.: General decay to a von Karman plate system with memory boundary conditions. Differ. Integral Equ. 24(1–2), 69–81 (2011)
Kang, J.R.: General stability for a von Karman plate system with memory boundary conditions. Bound. Value Probl. 2015, 167 (2015)
Park, J.Y., Park, S.H., Kang, Y.H.: General decay for a von Karman equation of memory type with acoustic boundary conditions. Z. Angew. Math. Phys. 63, 813–823 (2012)
Kang, J.R.: Asymptotic behavior to a von Karman equations of memory type with acoustic boundary conditions. Z. Angew. Math. Phys. 67, 48 (2016)
Park, S.H.: Energy decay for a von Karman equation with time-varying delay. Appl. Math. Lett. 55, 10–17 (2016)
Kim, D.W., Park, J.Y., Kang, Y.H.: Energy decay rate for a von Karman system with a boundary nonlinear delay term. Comput. Math. Appl. 75, 3269–3282 (2018)
Ball, J.: Remarks on blow up and nonexistence theorems for nonlinear evolutions equations. Quart. J. Math. Oxford 28, 473–486 (1977)
Levine, H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form \(Pu_{tt}=Au+F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)
Levine, H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equation. SIAM J. Math. Anal. 5, 138–146 (1974)
Georgiev, V., Todorova, G.: Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109, 295–308 (1994)
Messaoudi, S.A.: Blow up in a nonlinearly damped wave equation. Math. Nachr. 231, 1–7 (2001)
Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003)
Messaoudi, S.A.: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 320, 902–915 (2006)
Song, H.: Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal., Real World Appl. 26, 306–314 (2015)
Park, S.H., Lee, M.J., Kang, J.R.: Blow-up results for viscoelastic wave equations with weak damping. Appl. Math. Lett. 80, 20–26 (2018)
Liu, W.J., Yu, J.: On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal. 74, 2175–2190 (2011)
Ha, T.G.: Blow-up for wave equation with weak boundary damping and source terms. Appl. Math. Lett. 49, 166–172 (2015)
Hao, J., Wei, H.: Blow-up and global existence for solution of quasilinear viscoelastic wave equation with strong damping and source term. Bound. Value Probl. 2017, 65 (2017)
Liu, W.J.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73, 1890–1904 (2010)
Liu, W.J., Sun, Y., Li, G.: On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Topol. Methods Nonlinear Anal. 49(1), 299–323 (2017)
Liu, W.J., Zhuang, H.F.: Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms. Nonlinear Differ. Equ. Appl. 24, 67 (2017)
Messaoudi, S.A., Bonfoh, A., Mukiawa, S., Enyi, C.: The global attractor for a suspension bridge with memory and partially hinged boundary conditions. Z. Angew. Math. Mech. 97(2), 159–172 (2017)
Song, H.: Global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal. 125, 260–269 (2015)
Song, H., Zhong, C.: Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal., Real World Appl. 11, 3877–3883 (2010)
Vitillaro, E.: Global existence for the wave equation with nonlinear boundary damping and source terms. J. Differ. Equ. 186, 259–298 (2002)
Chueshov, I., Lasiecka, J.: Global attractors for von Karman evolutions with nonlinear boundary dissipation. J. Differ. Equ. 198, 196–231 (2004)
Ha, T.G.: Asymptotic stability of the semilinear wave equation with boundary damping and source term. C.R. Acad. Sci. Paris 352, 213–218 (2014)
Acknowledgements
The authors are thankful to the honorable reviewers and editors for their valuable comments and suggestions, which improved the paper.
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Research of MJL was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070738). Research of J-RK was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03028291).
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Lee, M.J., Kang, JR. Blow-up results for a quasilinear von Karman equation of memory type. Bound Value Probl 2019, 174 (2019). https://doi.org/10.1186/s13661-019-1277-y
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DOI: https://doi.org/10.1186/s13661-019-1277-y