Electron. J. Diff. Eqns., Vol. 2004(2004), No. 101, pp. 1-13.

The critical case for a semilinear weakly hyperbolic equation

Luca Fanelli, Sandra Lucente

Abstract:
We prove a global existence result for the Cauchy problem, in the three-dimensional space, associated with the equation
$$ u_{tt}-a_\lambda(t) \Delta_x u=-u|u|^{p(\lambda)-1} $$
where $a_\lambda(t)\ge 0$ and behaves as $(t-t_0)^\lambda$ close to some $t_0$ greater than 0 with $a(t_0)=0$, and $p(\lambda)=(3\lambda+10)/(3\lambda+2)$ with $3\le p(\lambda)\le 5$. This means that we deal with the superconformal, critical nonlinear case. Moreover we assume a small initial energy.

Submitted July 22, 2004. Published August 24, 2004.
Math Subject Classifications: 35L70, 35L15, 35L80.
Key Words: Global existence; semilinear wave equations.

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Luca Fanelli
Dipartimento di Matematica
Universita "La Sapienza" di Roma
Piazzale Aldo Moro 2, I-00185 Roma, Italy
email: fanelli@mat.uniroma1.it
Sandra Lucente
Dipartimento di Matematica
Universita degli Studi di Bari
Via E. Orabona 4, I-70125 Bari, Italy
email: lucente@dm.uniba.it

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