The Cauchy problem for $p$-evolution equations
HTML articles powered by AMS MathViewer
- by Massimo Cicognani and Ferruccio Colombini PDF
- Trans. Amer. Math. Soc. 362 (2010), 4853-4869 Request permission
Abstract:
In this paper we deal with the Cauchy problem for evolution equations with real characteristics. We show that the problem is well-posed in Sobolev spaces assuming a suitable decay of the coefficients as the space variable $x\to \infty$. In some cases, such a decay may also compensate a lack of regularity with respect to the time variable $t$.References
- Massimo Cicognani and Ferruccio Colombini, Sharp regularity of the coefficients in the Cauchy problem for a class of evolution equations, Differential Integral Equations 16 (2003), no. 11, 1321–1344. MR 2016685
- Massimo Cicognani and Ferruccio Colombini, Loss of derivatives in evolution Cauchy problems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 2, 271–280. MR 2273098, DOI 10.1007/s11565-006-0020-7
- Ferruccio Colombini, Ennio De Giorgi, and Sergio Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 511–559 (French). MR 553796
- Ferruccio Colombini and Nicolas Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), no. 3, 657–698. MR 1324638, DOI 10.1215/S0012-7094-95-07721-7
- Ferruccio Colombini and Guy Métivier, The Cauchy problem for wave equations with non Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 2, 177–220 (English, with English and French summaries). MR 2468481, DOI 10.24033/asens.2066
- Wataru Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math. 21 (1984), no. 3, 565–581. MR 759481
- Kunihiko Kajitani and Akio Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math. 119 (1995), no. 5, 459–473. MR 1354247
- Kumano-Go, H.: Pseudo-differential operators. The MIT Press, Cambridge, London, 1982.
- Sigeru Mizohata, On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering, vol. 3, Academic Press, Inc., Orlando, FL; Science Press Beijing, Beijing, 1985. MR 860041
Additional Information
- Massimo Cicognani
- Affiliation: FacoltĂ di Ingegneria II, Via Genova, 181, 47023 Cesena, Italy
- Address at time of publication: Dipartimento di Matematica, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy
- Email: cicognani@dm.unibo.it
- Ferruccio Colombini
- Affiliation: Dipartimento di Matematica, University of Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
- Email: colombini@dm.unipi.it
- Received by editor(s): February 17, 2009
- Published electronically: April 28, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4853-4869
- MSC (2010): Primary 35G10, 35L15
- DOI: https://doi.org/10.1090/S0002-9947-10-05171-8
- MathSciNet review: 2645053