Lifespan of solutions to second order Cauchy problems with small Gevrey data

John Paolo O. Soto; Jose Ernie C. Lope; Mark Philip F. Ona; John Paolo O. Soto; Jose Ernie C. Lope; Mark Philip F. Ona

doi:10.3934/math.2024070

AIMS Mathematics

2024, Volume 9, Issue 1: 1434-1442. doi: 10.3934/math.2024070

Previous Article Next Article

Research article Special Issues

Lifespan of solutions to second order Cauchy problems with small Gevrey data

Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines

Received: 21 September 2023 Revised: 16 November 2023 Accepted: 06 December 2023 Published: 11 December 2023
MSC : 35B30, 35G25

Consider the second order nonlinear partial differential equation:

$ \partial_t^2 u = F(u, \partial_x u), \quad (t, x) \in \mathbb{C}\times \mathbb{R}. $

Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter $ \varepsilon $. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that $ F $ is independent of $ u $. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan.
- Cauchy problem,
- lifespan of solutions,
- Gevrey functions,
- small data
Citation: John Paolo O. Soto, Jose Ernie C. Lope, Mark Philip F. Ona. Lifespan of solutions to second order Cauchy problems with small Gevrey data[J]. AIMS Mathematics, 2024, 9(1): 1434-1442. doi: 10.3934/math.2024070

Related Papers:

Abstract

Consider the second order nonlinear partial differential equation:

$ \partial_t^2 u = F(u, \partial_x u), \quad (t, x) \in \mathbb{C}\times \mathbb{R}. $

Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter $ \varepsilon $. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that $ F $ is independent of $ u $. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan.

References

[1]	P. D'Ancona, S. Spagnolo, On the life span of the analytic solutions to quasilinear weakly hyperbolic equations, Indiana Univ. Math. J., 40 (1991), 71–99.
[2]	D. Gourdin, M. Mechab, Problème de Cauchy pour des équations de Kirchhoff généralisées, Commun. Partial Differ. Equ., 23 (1998), 761–776. https://doi.org/10.1080/03605309808821364 doi: 10.1080/03605309808821364
[3]	D. Gourdin, M. Mechab, Problème de Cauchy non linéaire à données initiales lentement oscillantes, Bull. Sci. Math., 125 (2001), 371–394. https://doi.org/10.1016/S0007-4497(01)01082-X doi: 10.1016/S0007-4497(01)01082-X
[4]	H. Kubo, M. Ohta, On systems of semilinear wave equations with unequal propagation speeds in three space dimensions, Funkcial. Ekvac., 48 (2005), 65–98. https://doi.org/10.1619/fesi.48.65 doi: 10.1619/fesi.48.65
[5]	H. Kubo, M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, In: New trends in the theory of hyperbolic equations, Basel: Birkhäuser, 2005,113–211. https://doi.org/10.1007/3-7643-7386-5_2
[6]	H. Kubo, M. Ohta, Blowup for systems of semilinear wave equations in two space dimensions, Hokkaido Math. J., 35 (2006), 697–717. https://doi.org/10.14492/hokmj/1285766425 doi: 10.14492/hokmj/1285766425
[7]	J. E. C. Lope, M. P. F. Ona, Local solvability of a system of equations related to Ricci-flat Kähler metrics, Funkcial. Ekvac., 59 (2016), 141–155. https://doi.org/10.1619/fesi.59.141 doi: 10.1619/fesi.59.141
[8]	J. E. C. Lope, H. Tahara, On the analytic continuation of solutions to nonlinear partial differential equations, J. Math. Pures Appl., 81 (2002), 811–826. https://doi.org/10.1016/S0021-7824(02)01257-6 doi: 10.1016/S0021-7824(02)01257-6
[9]	J. P. O. Soto, J. E. C. Lope, M. P. F. Ona, A constructive approach to obtaining the lifespan of solutions to Cauchy problems with small data, Matimyás Mat., 46 (2023), 22–32.
[10]	M. A. C. Tolentino, D. B. Bacani, H. Tahara, On the lifespan of solutions to nonlinear Cauchy problems with small analytic data, J. Differ. Equ., 260 (2016), 897–922. https://doi.org/10.1016/j.jde.2015.09.013 doi: 10.1016/j.jde.2015.09.013
[11]	C. Wagschal, Le problème de Goursat non-linéaire, J. Math. Pures Appl., 58 (1979), 309–337.
[12]	H. Yamane, Nonlinear Cauchy problems with small analytic data, Proc. Amer. Math. Soc., 134 (2006), 3353–3361.
[13]	H. Yamane, Local existence for nonlinear Cauchy problems with small analytic data, J. Math. Sci. Univ. Tokyo, 18 (2011), 51–65.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)