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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Special Issue: AUS-ICMS 2020   > Article

Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

Estimation de l’explosion et de temps d’existence pour l’équation de Tricomi généralisée avec des non-linéarités mixtes


Makram Hamouda
Imam Abdulrahman Bin Faisal University
KSA

Mohamed Ali Hamza
Imam Abdulrahman Bin Faisal University
KSA



Published on 28 July 2021   DOI : 10.21494/ISTE.OP.2021.0698

Abstract

Résumé

Keywords

Mots-clés

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 $$${(Tr)}$$$.

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 $$${(Tr)}$$$.

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