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

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 $m\ge 0$. 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.

arXiv comment29 pages; added a section and an appendix with local existence result and new estimates, corrected typos and minor revision, range of the exponent $m$ restricted to $m\ge 0$

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Cited by in corpus (6)

• Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity
• A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime
• Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques
• On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity
• Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type
• Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds