Abstract
In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benmeriem K., Bouzar C.: Generalized Gevrey ultradistributions. N. Y. J. Math. 15, 37–72 (2009)
Bernardi E., Parenti C., Parmeggiani A.: The Cauchy problem for hyperbolic operators with double characteristics in presence of transition. Commun. Partial Differ. Equ. 37(7), 1315–1356 (2012)
Bronšteĭn M.D.: Smoothness of roots of polynomials depending on parameters. Sibirsk. Mat. Zh. 20(3), 493– (1979)
Bronšteĭn M.D.: The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity. Tr. Moskov. Mat. Obshch. 41, 83– (1980)
Cicognani M., Colombini F.: A well-posed Cauchy problem for an evolution equation with coefficients of low regularity. J. Differ. Equ. 254(8), 3573–3595 (2013)
Colombini F., De Giorgi E., Spagnolo S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 6(3), 511–559 (1979)
Colombini F., Del Santo D., Kinoshita T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(2), 327–358 (2002)
Colombini F., Del Santo D., Reissig M.: On the optimal regularity of coefficients in hyperbolic Cauchy problems. Bull. Sci. Math. 127(4), 328–347 (2003)
Colombini F., Jannelli E., Spagnolo S.: Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 10(2), 291–312 (1983)
Colombini F., Jannelli E., Spagnolo S.: Nonuniqueness in hyperbolic Cauchy problems. Ann. Math. (2) 126(3), 495–524 (1987)
Colombini F., Kinoshita T.: On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order. J. Differ. Equ. 186(2), 394–419 (2002)
Colombini F., Orrù N., Pernazza L.: On the regularity of the roots of hyperbolic polynomials. Isr. J. Math. 191(2), 923–944 (2012)
Colombini F., Spagnolo S.: An example of a weakly hyperbolic Cauchy problem not well posed in \({C^{\infty}}\). Acta Math. 148, 243–253 (1982)
D’Ancona P., Spagnolo S.: Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(1), 169–185 (1998)
Gelfand I.M.: Some questions of analysis and differential equations. Am. Math. Soc. Transl. (2). 26, 201–219 (1963)
Garetto C., Ruzhansky M.: On the well-posedness of weakly hyperbolic equations with time-dependent coefficients. J. Differ. Equ. 253(5), 1317–1340 (2012)
Garetto C., Ruzhansky M.: Weakly hyperbolic equations with non-analytic coefficients and lower order terms. Math. Ann. 357(2), 401–440 (2013)
Garetto C., Ruzhansky M.: A note on weakly hyperbolic equations with analytic principal part. J. Math. Anal. Appl. 412(1), 1–14 (2014)
Hörmann G., de Hoop M.V.: Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients. Acta Appl. Math. 67(2), 173–224 (2001)
Hörmann G., de Hoop M.V.: Detection of wave front set perturbations via correlation: foundation for wave-equation tomography. Appl. Anal. 81(6), 1443–1465 (2002)
Hurd A.E., Sattinger D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Am. Math. Soc. 132, 159–174 (1968)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition
Kinoshita T., Spagnolo S.: Hyperbolic equations with non-analytic coefficients. Math. Ann. 336(3), 551–569 (2006)
Lafon F., Oberguggenberger M.: Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case. J. Math. Anal. Appl. 160(1), 93–106 (1991)
Marsan D., Bean C.J.: Multiscaling nature of sonic velocities and lithology in the upper crystalline crust: evidence from the KTB main borehole. Geophys. Res. Lett. 26, 275–278 (1999)
Nishitani T.: Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x. Bull. Sci. Math. (2) 107(2), 113–138 (1983)
Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces. Pseudo-Differential Operators. Theory and Applications, vol. 4. Birkhäuser, Basel (2010)
Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics Series, vol. 259. Longman Scientific & Technical, Harlow (1992)
Parenti C., Parmeggiani A.: On the Cauchy problem for hyperbolic operators with double characteristics. Commun. Partial Differ. Equ. 34(7–9), 837–888 (2009)
Schwartz L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)
Teofanov N.: Modulation spaces, Gelfand–Shilov spaces and pseudodifferential operators. Sampl. Theory Signal Image Process 5(2), 225–242 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Dal Maso
M. Ruzhansky was supported by the EPSRC Leadership Fellowship EP/G007233/1 and by EPSRC Grant EP/K039407/1.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Garetto, C., Ruzhansky, M. Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients. Arch Rational Mech Anal 217, 113–154 (2015). https://doi.org/10.1007/s00205-014-0830-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-014-0830-1