TY - Type of reference
TI - Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities
AU - Makram Hamouda
AU - Mohamed Ali Hamza
AB - We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider
$$${(Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=}$$$|$$${u_t}$$$|$$${^p+}$$$|$$${u}$$$|$$${^q}$$$, $$${\quad \mbox{in}\ \mathbb{R}^N\times[0,\infty)}$$$,
with small initial data, where $$${m ≥ 0}$$$.
For the problem $$${(Tr)}$$$ with $$${m = 0}$$$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity (|$$${u_t}$$$|$$${^p}$$$ or |$$${u}$$$|$$${^q}$$$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $$${(Tr)}$$$ with $$${m ≥ 0}$$$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $$${m}$$$. As an application of the method developed for the study of the equation $$${(Tr)}$$$ we obtain with a different approach the same blow-up result as in [18] when we consider only one time-derivative nonlinearity, namely we keep only |$$${u_t}$$$|$$${^p}$$$ in the right-hand side of $$${(Tr)}$$$.
DO - 10.21494/ISTE.OP.2021.0698
JF - Advances in Pure and Applied Mathematics
KW - blow-up, generalized Tricomi equation, lifespan, critical curve, nonlinear wave equations, time-derivative nonlinearity, blow-up, generalized Tricomi equation, lifespan, critical curve, nonlinear wave equations, time-derivative nonlinearity,
L1 - http://www.openscience.fr/IMG/pdf/iste_apam21v12nspe_4.pdf
LA - en
PB - ISTE OpenScience
DA - 2021/07/28
SN - 1869-6090
TT - Estimation de l’explosion et de temps d’existence pour l’équation de Tricomi généralisée avec des non-linéarités mixtes
UR - http://www.openscience.fr/Blow-up-and-lifespan-estimate-for-the-generalized-Tricomi-equation-with-mixed
IS - Issue 3 (Special AUS-ICMS 2020)
VL - 12
ER -