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arXiv · 2007.16003

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

Ning-An Lai·Nico Michele Schiavone
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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.

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Ning-An Lai, Nico Michele Schiavone. 2020-07-31. Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture. https://doi.org/10.1007/s00209-022-03017-4

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