TY  - Type of reference 
TI  - Estimation de l’explosion et de temps d’existence pour l’équation de Tricomi généralisée avec des non-linéarités mixtes 
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  - Avancées en Mathématiques Pures et Appliquées 
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  - https://openscience.fr/IMG/pdf/iste_apam21v12nspe_4.pdf 
LA  - fr 
PB  - ISTE OpenScience 
DA  - 2021/07/28 
SN  - 1869-6090 
TT  - Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities 
UR  - https://openscience.fr/Estimation-de-l-explosion-et-de-temps-d-existence-pour-l-equation-de-Tricomi 
IS  - Numéro spécial : AUS-ICMS 2020 
VL  - 12 
ER  -