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

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