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On Asymptotic Behavior of Blow-Up Solutions to Higher-Order Differential Equations with General Nonlinearity Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Gevrey Well Posedness of Goursat-Darboux Problems and Asymptotic Solutions

Authors : Jorge Marques, Jaime Carvalho e Silva

Published in: Differential and Difference Equations with Applications

Publisher: Springer International Publishing

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Abstract

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex coefficients and two unknown initial data. In order to prove our results, we first construct an explicit solution of a family of problems with initial data depending on a parameter \(\eta > 0\) and then we obtain an asymptotic representation of a solution as \(\eta \) tends to infinity.

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previous chapter Interval Difference Methods for Solving the Poisson Equation
next chapter Oscillation Criteria for a Difference System with Two Delays
Literature
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Carvalho e Silva, J.: Problème de Goursat-Darboux genéralisé pour un opérateur du troisième ordre. C. R. Acad. Sc. Paris Série I 303(6), 223–226 (1986)
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Carvalho e Silva, J., Leal, C.: The generalized Goursat-Darboux problem for a third order operator. Proc. Am. Math. Soc. 125 (2), 471–475 (1997)
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Carvalho e Silva, J., Srivastava H.M.: Asymptotic representations for a class of multiple hypergeometric functions with a dominant variable. Integr. Transform. Spec. Funct. 11(2), 137–150 (2001)MathSciNetCrossRef
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Gevrey, M.: Sur la nature analytique des solutions des équations aux dérivées partielles. Ann. Ecole Norm. Sup. Paris 35, 129–190 (1918)
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Hadamard, J.: Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, Paris (1932)MATH
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Hasegawa, Y.: On the \(C^{\infty }\) Goursat problem for 2nd order equations with real constant coefficients. Proc. Jpn. Acad. 51(7), 516–519 (1975)MathSciNetCrossRef
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Olver, F.: Introduction to Asymptotics and Special Functions. Academic Press, New-York (1974)MATH
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Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993)CrossRef
Metadata
Title
Gevrey Well Posedness of Goursat-Darboux Problems and Asymptotic Solutions
Authors
Jorge Marques
Jaime Carvalho e Silva
Copyright Year
2018
Book
Differential and Difference Equations with Applications

Print ISBN: 978-3-319-75646-2

Electronic ISBN: 978-3-319-75647-9

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-75647-9_22

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