TY - CHAP

T1 - Quasilinear Wave Equations with Decaying Time-Potential

AU - Georgiev, Vladimir

AU - Lucente, Sandra

N1 - Funding Information:
Acknowledgments The authors are partially supported by INDAM, GNAMPA. The first author was partially supported by contract “Problemi stazionari e di evoluzione nelle equazioni di campo non-lineari dispersive” of GNAMPA– Gruppo Nazionale per l’Analisi Matematica 2020, by the project PRIN 2020XB3EFL by the Italian Ministry of Universities and Research, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa. The second author is supported by the PRIN 2017 project Qualitative and quantitative aspects of nonlinear PDEs.
Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

PY - 2022

Y1 - 2022

N2 - An active area of recent research is the study of global existence and blow up for nonlinear wave equations where time depending mass or damping are involved. The interaction between linear and nonlinear terms is a crucial point in determination of global evolution dynamics. When the nonlinear term depends on the derivatives of the solution, the situation is even more delicate. Indeed, even in the constant coefficients case, the null conditions strongly relate the symbol of the linear operator with the form of admissible nonlinear terms which leads to global existence. Some peculiar operators with time-dependent coefficients lead to a wave operator in which the time derivative becomes a covariant time derivative. In this paper we give a blow up result for a class of quasilinear wave equations in which the nonlinear term is a combination of powers of first and second order time derivatives and a time-dependent factor. Then we apply this result to scale invariant damped wave equations with nonlinearity involving the covariant time derivatives.

AB - An active area of recent research is the study of global existence and blow up for nonlinear wave equations where time depending mass or damping are involved. The interaction between linear and nonlinear terms is a crucial point in determination of global evolution dynamics. When the nonlinear term depends on the derivatives of the solution, the situation is even more delicate. Indeed, even in the constant coefficients case, the null conditions strongly relate the symbol of the linear operator with the form of admissible nonlinear terms which leads to global existence. Some peculiar operators with time-dependent coefficients lead to a wave operator in which the time derivative becomes a covariant time derivative. In this paper we give a blow up result for a class of quasilinear wave equations in which the nonlinear term is a combination of powers of first and second order time derivatives and a time-dependent factor. Then we apply this result to scale invariant damped wave equations with nonlinearity involving the covariant time derivatives.

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U2 - 10.1007/978-981-19-6434-3_9

DO - 10.1007/978-981-19-6434-3_9

M3 - Chapter

AN - SCOPUS:85143765331

T3 - Springer INdAM Series

SP - 187

EP - 204

BT - Springer INdAM Series

PB - Springer-Verlag Italia s.r.l.

ER -