We prove a global existence result for the Cauchy problem, in the three-dimensional space, associated with the equation $$ u_{tt} − a(t)\Delta_x u = −u|u|^{p(\lambda)−1} $$ where $a(t)\geq 0$ and behaves as $(t − t_0)^\lambda$ close to some $t_0 > 0$ with $a(t_0) = 0$, and $p(\lambda) = (3\lambda+10)/(3\lambda+2)$ with $3\leq p(\lambda)\leq 5$. This means that we deal with the superconformal, critical nonlinear case. Moreover we assume a small initial energy.
The critical case for a semilinear weakly hyperbolic equation / Lucente, S; Fanelli, Luca. - In: ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 1072-6691. - ELETTRONICO. - 101:(2004), pp. 1-13.
The critical case for a semilinear weakly hyperbolic equation
FANELLI, Luca
2004
Abstract
We prove a global existence result for the Cauchy problem, in the three-dimensional space, associated with the equation $$ u_{tt} − a(t)\Delta_x u = −u|u|^{p(\lambda)−1} $$ where $a(t)\geq 0$ and behaves as $(t − t_0)^\lambda$ close to some $t_0 > 0$ with $a(t_0) = 0$, and $p(\lambda) = (3\lambda+10)/(3\lambda+2)$ with $3\leq p(\lambda)\leq 5$. This means that we deal with the superconformal, critical nonlinear case. Moreover we assume a small initial energy.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
