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.
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Acknowledgements
A. Palmieri is supported by the GNAMPA project ‘Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari dispersive’. S. Lucente is supported by the PRIN 2017 project ‘Qualitative and quantitative aspects of nonlinear PDEs’ and by the GNAMPA project ‘Equazioni di tipo dispersivo: teoria e metodi’. The authors gratefully thank to the referee for the careful reading, the constructive comments which help to improve the paper.
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Lucente, S., Palmieri, A. A Blow-Up Result for a Generalized Tricomi Equation with Nonlinearity of Derivative Type. Milan J. Math. 89, 45–57 (2021). https://doi.org/10.1007/s00032-021-00326-x
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DOI: https://doi.org/10.1007/s00032-021-00326-x