Blow-up and lifespan estimate for the generalized tricomi equation with the scale-invariant damping and time derivative nonlinearity on exterior domain

TL;DR

This paper investigates the blow-up behavior and lifespan of solutions to a scale-invariant damped wave equation with derivative nonlinearity on exterior domains, using multiplier and test-function techniques.

Contribution

It introduces new analytical methods to estimate blow-up and lifespan for a generalized Tricomi-type equation with scale-invariant damping.

Findings

Solutions blow up in finite time under certain conditions.

Lifespan estimates depend on initial data and nonlinearity exponent.

The approach extends previous methods to exterior domains with time-dependent propagation speed.

Abstract

The article is devoted to investigating the initial boundary value problem for the damped wave equation in the scale-invariant case with time-dependent speed of propagation on the exterior domain. By presenting suitable multipliers and applying the test-function technique, we study the blow-up and the lifespan of the solutions to the problem with derivative-type nonlinearity $ \d u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t=|u_t|^p, \quad \mbox{in}\ \Omega^{c}\times[1,\infty),$ that we associate with appropriate small initial data.