RMC - Vol. 9, Num. 2 - J.-P. dias/M. Figueira

Full paper in PDF:
$%J.-P. Dias and M. Figueira, A remark on the blow-up of the solutions of the equation $ut+ f(x)a(u)ux = h(x,u)$ , Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 2, 385–391.%$

A Remark on the Blow-Up of the Solutions of the Equation $ut + f (x)a(u)ux = h(x, u)$

João-Paulo DIAS and Mário FIGUEIRA

CMAF/University of Lisbon
2 Av. Prof. Gama Pinto
1699 Lisboa Codex — Portugal

Received: October 23, 1995

ABSTRACT

We consider the Cauchy problem for the equation $ut+ f(x)a(u)ux = h(x,u)$ where $f$ , $a$ , and $h$ are real $C1$ functions, $f > h > 0$ , $a'> 0$ , $hu> 0$ and $hx < 0$ . Following the ideas of Lax (1964) and Klainerman-Majda (1980), we prove a blow-up result for the solutions with special data corresponding, in certain cases, to the development of a singularity in $ux$ .

1991 Mathematics Subject Classification: 35L65, 35L67.