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November  2014, 13(6): 2351-2358. doi: 10.3934/cpaa.2014.13.2351

Finite speed of propagation for mixed problems in the $WR$ class

Antoine Benoit 1,

1. 

Université de Nantes, Laboratoire de Mathématiques Jean Leray (CNRS UMR6629), 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France

Received  October 2013 Revised  March 2014 Published  July 2014

In this article we are interested in the propagation speed for solutions of hyperbolic boundary value problems in the $WR$ class. Using the Holmgren principle, we show that this speed is finite and we are able to give an explicit expression for the maximal speed. Due to a propagation phenomenon along the boundary that is specific to the $WR$ class, the maximal speed can be larger than the propagation speed for the Cauchy problem. This is consistent with previous examples of the litterature.
Keywords: Mixed problems, Holmgren principle., $WR$ class, hyperbolic, finite speed.
Mathematics Subject Classification: Primary: 35L5.
Citation: Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351
References:
[1]

S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 5 (2002), 1073-1104. doi: 10.1017/S030821050000202X.

[2]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, Oxford Mathematical Monographs, Oxford University Press, 2007.

[3]

A. Chazarain and J. Piriou, Caractérisation des problèmes mixtes hyperboliques bien posés differentiables, (French) [Characterization of well-posed mixed hyperbolic mixed problems.], Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237.

[4]

A. Chazarain and J. Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires,, (French) [Introduction to the theory of linear partial differential equations.], (). 

[5]

J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl., 84 (2005), 786-818. doi: 10.1016/j.matpur.2004.10.005.

[6]

J.-F. Coulombel and O. Guès, Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems, Ann. Inst. Fourier (Grenoble), 60 (2010), 2183-2233.

[7]

M. Ikawa, Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J. Math., 7 (1970), 495-525.

[8]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.

[9]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702. doi: 10.1112/S0024609300007517.

[10]

A. Morando and P. Secchi, Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary, J. Hyperbolic Differ. Equ., 8 (2011), 37-99. doi: 10.1142/S021989161100238X.

[11]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, American Mathematical Society, Providence, RI, 2012.

[12]

M. Sablé-Tougeron, Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension $2$, (French) [Existence for a non-linear elastodynamic Neumann problem in $2$ dimensions], Arch. Rational Mech. Anal., 101 (1988), 261-292. doi: 10.1007/BF00253123.

[13]

T. Shirota, On the propagation speed of hyperbolic operator with mixed boundary conditions, J. Fac. Sci. Hokkaido Univ. Ser. I, 22 (1972), 25-31.

show all references

References:
[1]

S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 5 (2002), 1073-1104. doi: 10.1017/S030821050000202X.

[2]

S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations, Oxford Mathematical Monographs, Oxford University Press, 2007.

[3]

A. Chazarain and J. Piriou, Caractérisation des problèmes mixtes hyperboliques bien posés differentiables, (French) [Characterization of well-posed mixed hyperbolic mixed problems.], Ann. Inst. Fourier (Grenoble), 22 (1972), 193-237.

[4]

A. Chazarain and J. Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires,, (French) [Introduction to the theory of linear partial differential equations.], (). 

[5]

J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl., 84 (2005), 786-818. doi: 10.1016/j.matpur.2004.10.005.

[6]

J.-F. Coulombel and O. Guès, Geometric optics expansions with amplification for hyperbolic boundary value problems: linear problems, Ann. Inst. Fourier (Grenoble), 60 (2010), 2183-2233.

[7]

M. Ikawa, Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J. Math., 7 (1970), 495-525.

[8]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298.

[9]

G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702. doi: 10.1112/S0024609300007517.

[10]

A. Morando and P. Secchi, Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary, J. Hyperbolic Differ. Equ., 8 (2011), 37-99. doi: 10.1142/S021989161100238X.

[11]

J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, American Mathematical Society, Providence, RI, 2012.

[12]

M. Sablé-Tougeron, Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension $2$, (French) [Existence for a non-linear elastodynamic Neumann problem in $2$ dimensions], Arch. Rational Mech. Anal., 101 (1988), 261-292. doi: 10.1007/BF00253123.

[13]

T. Shirota, On the propagation speed of hyperbolic operator with mixed boundary conditions, J. Fac. Sci. Hokkaido Univ. Ser. I, 22 (1972), 25-31.

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