Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs
- Runzhang Xu^{1}Email author,
- Xingchang Wang^{2},
- Huichao Xu^{1} and
- Mingyou Zhang^{2}
Received: 26 May 2016
Accepted: 22 November 2016
Published: 1 December 2016
Abstract
This paper proves the global existence of solution for a class of nonlinear wave equations with nonlinear combined power-type nonlinearities of different signs for the initial data at sup-critical energy level.
Keywords
initial boundary value wave equation global existence sup-critical energy1 Introduction
Equation (1) is a class of important mathematical physical models, so there has been a lot of important work, such as [1–6], focused on it. Recently Li and Zhang [7] and Tao et al. [8] considered combined power-type nonlinearities since this kind of nonlinearities is more general than a single source term. The effects of this kind of nonlinearities on the prosperities of solution were well treated. However, the nonlinear terms considered in our paper are more general, we aim to provide some new results in this direction. This paper is a continued study of [9, 10], and [11], so we give a quick introduction here; for detailed background of this problem, we refer the reader to [11] and the references therein. The authors in [9] first considered problem (1)-(3) and obtained the global existence and blow up of solutions for the sub-critical case \(E(0)< d\), where \(E(0)\) is the initial energy and d is the depth of the potential well or the mountain pass level, which will be defined later. In the same paper the critical case \(E(0)=d\) was also considered and the global existence was derived. Later, Yu proved blow up of the solution for the critical case \(E(0)=d\) in [10]. Furthermore, the high energy case \(E(0)>0\) was treated in [11], and the blow up result was also given. Observing the above results for problem (1)-(3), helpful in the potential well method which was introduced by Payne and Sattinger [1], the global existence for the sup-critical case, i.e. \(E(0)>0\), is still not solved. So the present paper solves this problem by introducing a new stable invariant set and, using the method of [12], we are focusing on proving the global existence of the solution for problem (1)-(3) in the sup-critical case \(E(0)>0\).
Throughout the present paper, the following notations are used for a precise statement: \(L^{p}\) denotes the space consisting of all \(L^{p}\)-functions on Ω with norm \(\|u\|_{p}=\|u\|_{L^{p}(\Omega )}\), \(\|u\|_{2}=\|u\|_{L^{2}(\Omega)}\), and the inner product \((u,v)=\int _{\Omega}uv\, dx\).
2 Global existence at sup-critical case \(E(0)>0\)
Next, we give a definition of the weak solution for problem (1)-(3).
Definition 2.1
Weak solution [9]
- (i)
\((u_{t},v)+\int_{0}^{t}(\nabla u, \nabla v)\, d\tau=\int _{0}^{t}(f(u),v)\, d\tau+(u_{1},v) \) for all \(v\in H^{1}_{0}(\Omega)\), \(t\in(0,T_{0})\);
- (ii)
\(u(x,0)=u_{0}(x)\) in \(H_{0}^{1}(\Omega)\), \(u_{t}(x,0)=u_{1}(x)\) in \(L^{2}(\Omega)\);
- (iii)
\(E(t)=E(0)\), \(t\in[0,T)\).
Theorem 2.2
Local existence [9]
The invariance of the stable set \(\mathcal{W}\) under the flow of (1)-(3) plays an essential role while proving the global existence of the weak solution for (1)-(3). In order to obtain the invariance, we need to prove the following lemma at first.
Lemma 2.3
Proof
In the following, we show the invariance of the new stable set \(\mathcal{W}\) under the flow of problem (1)-(3).
Lemma 2.4
Invariance of \(\mathcal{W}\) at sup-critical case \(E(0)>0\)
Let \(u_{0}(x)\in H_{0}^{1}(\Omega)\), \(u_{1}(x)\in L^{2}(\Omega)\), and \(u(x,t)\) be a weak solution of problem (1)-(3) with maximal existence time interval \([0,T_{0})\), \(T_{0}\leq+\infty\). Assume that the initial data satisfy (7). Then all solutions of problem (1)-(3) with \(E(0)>0\) belong to \(\mathcal{W}\), provided \(u_{0}\in \mathcal{W}\).
Proof
Theorem 2.5
Let \(u_{0}(x)\in H_{0}^{1}(\Omega)\), \(u_{1}(x)\in L^{2}(\Omega)\), and let \(u(x,t)\) be a weak solution of problem (1)-(3) with maximal existence time interval \([0,T_{0})\), \(T_{0}\leq+\infty\). Assume that \(E(0)>0\), \(u_{0}\in\mathcal{W}\), and (7) holds, then the solution of problem (1)-(3) exists globally.
Proof
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11471087), the China Postdoctoral Science Foundation (2013M540270), the Heilongjiang Postdoctoral Foundation (LBH-Z13056), the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), the Fundamental Research Funds for the Central Universities. Dr. Xu Runzhang also thanks Prof. Yue Liu for his invitation of visit to UTA. The authors appreciate Prof. Weike Wang for his valuable suggestions.
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Authors’ Affiliations
References
- Payne, LE, Sattinger, DH: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273-303 (1975) MathSciNetView ArticleMATHGoogle Scholar
- Gazzola, F, Squassina, M: Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, 185-207 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Ikehata, R: Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal. 27, 1165-1175 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Cavalcanti, MM, Domingos Cavalcanti, VN, Martinez, P: Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J. Differ. Equ. 203, 119-158 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Cavalcanti, MM, Domingos Cavalcanti, VN, Lasiecka, I: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differ. Equ. 236, 407-459 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Alves, CO, Cavalcanti, MM: On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source. Calc. Var. Partial Differ. Equ. 34(3), 377-411 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Li, K, Zhang, Q: Existence and nonexistence of global solutions for the equation of dislocation of crystals. J. Differ. Equ. 146, 5-21 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Tao, T, Visan, M, Zhang, X: The nonlinear Schrödinger equation with combined power-type nonlinearities. Commun. Partial Differ. Equ. 32, 1281-1343 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Y, Xu, R: Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete Contin. Dyn. Syst., Ser. B 7, 171-189 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Yu, T, Tang, L, Liu, B, Xu, R: Wave equations and reaction-diffusion equations with several nonlinear source terms with critical energy. AIP Conf. Proc. 1479, 2435-2438 (2012) Google Scholar
- Shen, J, Xu, R, Yang, Y, Chen, S, Su, J, Huang, S: Nonlinear wave equations and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level. ANZIAM J. 54, 153-170 (2013) MathSciNetMATHGoogle Scholar
- Yang, Y, Shen, J, Xu, R: Global existence of solutions for 1-D nonlinear wave equation of sixth order at high initial energy level. Bound. Value Probl. 2014, 31 (2014) MathSciNetView ArticleMATHGoogle Scholar