A Blow-Up Result for a Generalized Tricomi Equation with Nonlinearity of Derivative Type

Abstract

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.