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

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  11 June 2013

RUNZHANG XU ,
YANBING YANG ,
SHAOHUA CHEN ,
JIA SU ,
JIHONG SHEN  and
SHAOBIN HUANG
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
*
✉[email protected]
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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.

Keywords

wave equationreaction–diffusion equationhigh energy levelfinite time blow-upvariational methodcomparison principle

MSC classification

Secondary: 35L05: Wave equation
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 3 , January 2013 , pp. 153 - 170
Copyright
Copyright ©2013 Australian Mathematical Society 

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