Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

TL;DR

This paper investigates the blow-up and lifespan of solutions to a class of generalized Tricomi equations with derivative nonlinearities, introducing a novel test function and establishing results that extend known wave equation behaviors.

Contribution

It presents a new test function construction for analyzing blow-up in generalized Tricomi equations, extending lifespan estimates and demonstrating independence of blow-up power from the parameter m in two dimensions.

Findings

Blow-up occurs for certain p values with explicit lifespan estimates.

The new test function combines cut-off functions, Bessel functions, and harmonic functions.

Results recover classical wave equation behavior when m=0.

Abstract

We study in this paper the small data Cauchy problem for the semilinear generalized Tricomi equations with a nonlinear term of derivative type $u_{tt}-t^{2m}\Delta u=|u_t|^p$ for $m\ge0$. Blow-up result and lifespan estimate from above are established for $1<p\le 1+\frac{2}{(m+1)(n-1)-m}$. If $m=0$, our results coincide with those of the semilinear wave equation. The novelty consists in the construction of a new test function, by combining cut-off functions, the modified Bessel function and a harmonic function. Interestingly, if $n=2$ the blow-up power is independent of $m$. We also furnish a local existence result, which implies the optimality of lifespan estimate at least in the $1$-dimensional case.