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NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

Published online by Cambridge University Press:  11 June 2013

RUNZHANG XU*
Affiliation:
College of Science, Harbin Engineering University, Harbin 150001, PR China
YANBING YANG
Affiliation:
College of Automation, Harbin Engineering University, Harbin 150001, PR China
SHAOHUA CHEN
Affiliation:
Department of Mathematics, Cape Breton University, Sydney, NS, Canada B1P 6L2
JIA SU
Affiliation:
Science China Press, Beijing 100717, PR China
JIHONG SHEN
Affiliation:
College of Science, Harbin Engineering University, Harbin 150001, PR China
SHAOBIN HUANG
Affiliation:
College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR China
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Abstract

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This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

MSC classification

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Evans, L. C., Partial differential equations, volume 19 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 1998).Google Scholar
Gazzola, F. and Squassina, M., “Global solutions and finite time blow up for damped semilinear wave equations”, Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire 23 (2006) 185207; doi:10.1016/j.anihpc.2005.02.007.CrossRefGoogle Scholar
Gazzola, F. and Weth, T., “Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level”, Differential Integral Equations 18 (2005) 961990; http://projecteuclid.org/euclid.die/1356060117.Google Scholar
Li, K. and Zhang, Q., “Existence and nonexistence of global solutions for the equation of dislocation of crystals”, J. Differential Equations 146 (1998) 521; doi:10.1006/jdeq.1998.3409.Google Scholar
Levine, H. A., “Instability and nonexistence of global solutions to nonlinear wave equation of the form $P{u}_{tt} = - Au+ \mathcal{F} (u)$”, Trans. Amer. Math. Soc. 192 (1974) 121; doi:10.1090/S0002-9947-1974-0344697-2.Google Scholar
Levine, H. A., “Some additional remarks on the nonexistence of global solutions to nonlinear wave equations”, SIAM J. Math. Anal. 5 (1974) 138146; doi:10.1137/0505015.CrossRefGoogle Scholar
Liu, Y. C. and Xu, R. Z., “Wave equations and reaction–diffusion equations with several nonlinear source terms of different sign”, Discrete Contin. Dyn. Syst. Ser. B 7 (2007) 171189; doi:10.3934/dcdsb.2007.7.171.CrossRefGoogle Scholar
Quittner, P., “Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems”, Houston J. Math. 29 (2003) 757799.Google Scholar
Simon, L., “Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems”, Ann. of Math. (2) 118 (1983) 525571; http://www.jstor.org/stable/2006981.CrossRefGoogle Scholar
Tao, T., Visan, M. and Zhang, X., “The nonlinear Schrödinger equation with combined power-type nonlinearities”, Comm. Partial Differential Equations 32 (2007) 12811343; doi:10.1080/03605300701588805.CrossRefGoogle Scholar
Vuillermot, P. A., “Small divisors and the construction of stable manifolds for nonlinear Klein–Gordon equations on ${ R}_{0}^{+ } \times R$”, in: Nonlinear hyperbolic equations and field theory (eds Venkatesha Murthy, M. K. and Spagnolo, S.), (Longman Scientific & Technical, Harlow, 1992), 197213.Google Scholar
Yu, T., Tang, L. Q., Liu, B. W. and Xu, R. Z., “Wave equations and reaction–diffusion equations with several nonlinear source terms with critical energy”, AIP Conf. Proc. 1479 (2012) 24352438; doi:10.1063/1.4756687.Google Scholar
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NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL
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NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL
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