Nonexistence result for the generalized Tricomi equation with the scale-invariant damping,mass term and time derivative nonlinearity

Abstract

In this article, we consider the damped wave equation in the scale-invariant case with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$\begin{array}{l}\mathrm{(}\mathit{E}\mathrm{)}{\mathit{u}}_{\mathit{t}\mathit{t}}-{\mathit{t}}^{2\mathit{m}}\mathrm{\Delta }\mathit{u}+{\displaystyle \frac{\mathit{\mu }}{\mathit{t}}}{\mathit{u}}_{\mathit{t}}+{\displaystyle \frac{{\mathit{\nu }}^2}{{\mathit{t}}^2}}\mathit{u}=|{\mathit{u}}_{\mathit{t}}{|}^{\mathit{p}}\mathrm{,}\text{in }{\mathbb{R}}^{\mathit{N}}\times [1\mathrm{,}\infty \mathrm{)}\mathrm{,}\end{array}$$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, ν and $\mathit{\mu }\mathrm{>}0$ , respectively, we prove that blow-up region and the lifespan bound of the solution of $\mathrm{(}\mathit{E}\mathrm{)}$ remain the same as the ones obtained for the case without mass, i.e. $\mathrm{(}\mathit{E}\mathrm{)}$ with $\mathit{\nu }=0$ which constitutes itself a shift of the dimension N by $\frac{\mathit{\mu }}{1+\mathit{m}}$ compared to the problem without damping and mass. Finally, we think that the new bound for p is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions.