A New Critical Behavior for Nonlinear Wave Equations

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 $u_0$ 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.