Titre : Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities
Auteurs : Makram Hamouda, Mohamed Ali Hamza,
Revue : Advances in Pure and Applied Mathematics
Numéro : Issue 3 (Special AUS-ICMS 2020)
Volume : 12
Date : 2021/07/28
DOI : 10.21494/ISTE.OP.2021.0698
ISSN : 1869-6090
Résumé : 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 $$${(Tr)}$$$.
Éditeur : ISTE OpenScience