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

TL;DR

This paper investigates the blow-up behavior of solutions to a scale-invariant damped wave equation with mass and nonlinear time derivative terms, identifying conditions under which solutions cannot exist globally and proposing a critical exponent.

Contribution

It extends previous blow-up results to include mass and damping effects, establishing that these do not alter the blow-up region or lifespan bounds, and suggests a new candidate for the critical exponent.

Findings

Blow-up region remains unchanged with mass and damping.

Lifespan bounds are similar to the massless case.

Proposes a new critical exponent candidate.

Abstract

In this article, we consider the damped wave equation in the \textit{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: $$ (E) \quad u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t+\frac{\nu^2}{t^2}u=|u_t|^p, \quad \mbox{in}\ \mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, $\nu$ and $\mu>0$, respectively, that the blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained for the case without mass, {\it i.e.} $\nu=0$ in $(E)$. The latter case constitutes, in fact, a shift of the dimension $N$ by $\frac{\mu}{1+m}$ 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.