A New Critical Behavior for Nonlinear Wave Equations

  • Qi S. Zhang
Article

DOI: 10.1023/A:1010156504128

Cite this article as:
Zhang, Q.S. Journal of Computational Analysis and Applications (2000) 2: 277. doi:10.1023/A:1010156504128
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Abstract

We study the inhomogeneous semilinear wave equations\(\Delta u{\text{ + }}\left| u \right|^p - u_{tt} + w = 0\) on \({\text{M}}^n \times \left( {0,\infty } \right)\)with initial values \(u\left( {x,0} \right) = u_0 \left( x \right)\) and \(u_t \left( {x,0} \right) = v_0 \left( x \right)\),where \({\text{M}}^n \) is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < pp*, the above problem hasno global solution for any nonnegative \(w = w\left( x \right)\) not identicallyzero and for any \(u_0 \) and \(v_0 \); when\(p > p^* \) the problem has a global solution for some\(w = w\left( x \right) > 0\) and some \(u_0 \) and \(v_0 \). If\({\text{M}}^n = {\text{R}}^n \), which is equipped with the Euclideanmetric, then \(p^* = n/\left( {n - 2} \right),n \geqslant 3\). If\(n = 3\) we show that \(p^* = 3\) belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.

inhomogeneous semilinear wave equations blow up 

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Qi S. Zhang
    • 1
  1. 1.Department of MathematicsUniversity of MemphisMemphis

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