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

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

Makram Hamouda
Imam Abdulrahman Bin Faisal University
KSA

Mohamed Ali Hamza
Imam Abdulrahman Bin Faisal University
KSA

Publié le 28 juillet 2021   DOI : 10.21494/ISTE.OP.2021.0698

### Keywords

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 ${(Tr)}$.