ISSN:
1572-9206
Keywords:
inhomogeneous
;
semilinear wave equations
;
blow up
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1010156504128
Permalink
