Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations
DOI:
https://doi.org/10.58997/ejde.2023.06Abstract
In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.
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References
M. Abramowitz, I. A. Stegun; Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, INC., New-York, 1965.
W. Balser; Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New-York, 2000.
W. Balser, M. Loday-Richaud; Summability of solutions of the heat equation with inhomo- geneous thermal conductivity in two variables, Adv. Dyn. Syst. Appl., 4 (2009) (2), 159-177.
M. Canalis-Durand, J. P. Ramis, R. Sch¨afke, Y. Sibuya; Gevrey solutions of singularly per- turbed differential equations, J. Reine Angew. Math., 518 (2009), 95-129.
I. Fukuda; Asymptotic behavior of solutions to the generalized KdV-Burgers equation with a slowly decaying data, J. Math. Anal. Appl., 480 (2019) (2).
J. Gorsky, A. A. Himonas; Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl., 303 (2005) (2), 522-529.
J. Gorsky, A. A. Himonas, C. Holliman, G. Petronilho; The Cauchy problem for a periodic higher order KdV equation in analytic Gevrey spaces, J. Math. Anal. Appl., 405 (2013) (2), 349-361.
, H. Hannah, A. A. Himonas, G. Petronilho; Gevrey regularity in time for generalized KdV type equations, in: Recent Progress on Some Problems in Several Complex Variables and Partial Differential Equations, in: Contemp. Math., 400 (2006), 117-127, Amer. Math. Soc., Providence, RI, 2006.
H. Hannah, A. A. Himonas, G. Petronilho; Gevrey regularity of the periodic gKdV equation, J. Differential Equations, 250 (2011) (5), 2581-2600.
P. Hilton, J. Pedersen; Catalan numbers, their generalization, and their uses, Math. Intelli- gencer, 13 (1991) (2), 64-75.
D. A. Klarner; Correspondences between plane trees and binary sequences, J. Combinatorial Theory, 9 (1970), 401-411.
A. Lastra, S. Malek; On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J. Differential Equations, 259 (2015), 5220-5270.
A. Lastra, S. Malek; On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems, Adv. Differ. Equ., 2015 (2015), 200.
A. Lastra, S. Malek, J. Sanz; On Gevrey solutions of threefold singular nonlinear partial differential equations, J. Differential Equations, 255 (2013), 3205-3232.
A. Lastra, H. Tahara; Maillet type theorem for nonlinear totally characteristic partial differ- ential equations, Math. Ann., 377 (2020), 1603-1641.
M. Loday-Richaud; Divergent Series, Summability and Resurgence II. Simple and Multiple Summability, Lecture Notes in Math., 2154, Springer-Verlag, 2016.
S. Malek; On the summability of formal solutions of nonlinear partial differential equations with shrinkings, J. Dyn. Control Syst., 13 (2007) (1), 1-13.
S. Malek; On Gevrey asymptotic for some nonlinear integro-differential equations, J. Dyn. Control Syst., 16 (2010) (3), 377-406.
S. Malek; On the summability of formal solutions for doubly singular nonlinear partial dif- ferential equations, J. Dyn. Control Syst., 18 (2012) (1), 45-82.
M. Miyake; Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations, J. Math. Soc. Japan, 43 (1991)(2), 305-330.
M. Miyake, A. Shirai; Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain, Ann. Polon. Math., 74 (2000) , 215-228.
M. Miyake, A. Shirai; Structure of formal solutions of nonlinear first order singular partial differential equations in complex domain, Funkcial. Ekvac., 48 (2005), 113-136.
M. Miyake, A. Shirai; Two proofs for the convergence of formal solutions of singular first order nonlinear partial differential equations in complex domain, Surikaiseki Kenkyujo Kokyuroku Bessatsu, Kyoto Unviversity, B37 (2013), 137-151.
M. Nagumo; U¨ ber das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math., 18 (1942), 41-47.
S. Ouchi; Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, 185 (2002) (2), 513-549.
M. E. Pli´s, B. Ziemian; Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables, Ann. Polon. Math., 67 (1997) (1), 31-41.
G. P´olya, G. Szeg¨o; Aufgaben und Lehrs¨atze aus der Analysis, Vol. I, 125, Springer-Verlag, Berlin, G¨ottingen, Heidelberg, 1954.
J.-P. Ramis; D´evissage Gevrey, Ast´erisque, Soc. Math. France, Paris, 59-60 (1978), 173-204.
J.-P. Ramis; Th´eor`emes d’indices Gevrey pour les ´equations diff´erentielles ordinaires, Mem. Amer. Math. Soc., 48, viii+95 , 1984.
P. Remy; Gevrey regularity of the solutions of some inhomogeneous semilinear partial differ- ential equations with variable coefficients, hal-02263353, submitted for publication
P. Remy; Gevrey order and summability of formal series solutions of some classes of inhomo- geneous linear partial differential equations with variable coefficients, J. Dyn. Control Syst., 22 (2016) (4), 693-711.
P. Remy; Gevrey order and summability of formal series solutions of certain classes of in- homogeneous linear integro-differential equations with variable coefficients, J. Dyn. Control Syst., 23 (2017) (4), 853-878.
P. Remy; Gevrey properties and summability of formal power series solutions of some inho- mogeneous linear Cauchy-Goursat problems, J. Dyn. Control Syst., 26 (2020) (1), 69-108.
P. Remy; On the summability of the solutions of the inhomogeneous heat equation with a power-law nonlinearity and variable coefficients, J. Math. Anal. Appl., 494 (2021) (2), 124656.
P. Remy; Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients, Acta Sci. Math. (Szeged), 87 (2021) (1-2), 163-181.
P. Remy; Summability of the formal power series solutions of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity and variable coefficients, Results Math., 76 (2021) (3), 118.
P. Remy; Gevrey regularity of the solutions of the inhomogeneous partial differential equa- tions with a polynomial semilinearity, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. RACSAM, 115 (2021) (3), 145.
P. Remy; Summability of the formal power series solutions of a certain class of inhomogeneous nonlinear partial differential equations with a single level, J. Differential Equations, 313 (2022), 450-502.
A. Shirai; Maillet type theorem for nonlinear partial differential equations and Newton poly- gons, J. Math. Soc. Japan, 53 (2001), 565-587.
A. Shirai; Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac., 45 (2002), 187-208.
A. Shirai; A Maillet type theorem for first order singular nonlinear partial differential equa- tions, Publ. RIMS. Kyoto Univ., 39 (2093), 275-296.
A. Shirai; Maillet type theorem for singular first order nonlinear partial differential equa- tions of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University, 1431 (20025), 94-106.
A. Shirai; Alternative proof for the convergence or formal solutions of singular first order non- linear partial differential equations, Journal of the School of Education, Sugiyama Jogakuen University, 1 (2008), 91-102.
A. Shirai; Gevrey order of formal solutions of singular first order nonlinear partial differen- tial equations of totally characteristic type, Journal of the School of Education, Sugiyama Jogakuen University, 6 (2013), 159-172.
A. Shirai; Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, part II, Opuscula Math., 35 (2015) (5), 689-712.
H. Tahara; Gevrey regularity in time of solutions to nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, 18 (2011), 67-137.
H. Tahara, H. Yamazawa; Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations, 255 (2013) (10), 3592-3637.
W. Walter; An elementary proof of the Cauchy-Kowalevsky theorem, Amer. Math. Monthly, 92 (1985) (2), 115-126.
A. Yonemura; Newton polygons and formal Gevrey classes, Publ. Res. Inst. Math. Sci., 26 (190), 197-204.
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