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A New Critical Behavior for Nonlinear Wave Equations

Authors:	Qi S Zhang

Institution:	(1) Department of Mathematics, University of Memphis, Memphis, TN, 38152

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 < p p, the above problem hasno global solution for any nonnegative $w = w\left( x \right)$ not identicallyzero and for any and ; when the problem has a global solution for some $w = w\left( x \right) > 0$ and some and . If ${\text{M}}^n = {\text{R}}^n$ , which is equipped with the Euclideanmetric, then $p^ = n/\left( {n - 2} \right),n \geqslant 3$ . If we show that 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.

Keywords:	inhomogeneous semilinear wave equations blow up
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