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A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type. (English) Zbl 1470.35079

Summary: In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation \(\mathcal{T}_\ell u=|\partial_t u|^p\), where \(\mathcal{T_\ell}=\partial_t^2-t^{2\ell}\Delta\). Smooth solutions blow up in finite time for positive Cauchy data when the exponent \(p\) of the nonlinear term is below \(\frac{\mathcal{Q}}{\mathcal{Q}-2}\), where \(\mathcal{Q}=(\ell+1)n+1\) is the quasi-homogeneous dimension of the generalized Tricomi operator \(\mathcal{T}_\ell\). Furthermore, we get also an upper bound estimate for the lifespan.

MSC:

35B44 Blow-up in context of PDEs
35L71 Second-order semilinear hyperbolic equations
35B33 Critical exponents in context of PDEs
35C15 Integral representations of solutions to PDEs

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