Global time estimates for solutions to equations of dissipative type

Ruzhansky, Michael; Smith, James

Journées équations aux dérivées partielles, (2005), p. 1-29 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé

Global time estimates of $L^{p} - L^{q}$ norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.

Publié le : 2005-01-01

MR 2352779

DOI : https://doi.org/10.5802/jedp.23
Classification: 35A20, 35S30, 58G15, 32D20

@article{JEDP_2005____A12_0,
     author = {Ruzhansky, Michael and Smith, James},
     title = {Global time estimates for solutions to equations of dissipative type},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2005},
     pages = {1-29},
     doi = {10.5802/jedp.23},
     mrnumber = {2352779},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2005____A12_0}
}

Ruzhansky, Michael; Smith, James. Global time estimates for solutions to equations of dissipative type. Journées équations aux dérivées partielles,  (2005), pp. 1-29. doi : 10.5802/jedp.23. http://gdmltest.u-ga.fr/item/JEDP_2005____A12_0/

Bibliographie

[Bre75] Brenner, P., On $L_{p} - L_{p^{'}}$ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251–254. | MR 387819 | Zbl 0321.35052

[Bre77] Brenner, P., $L_{p} - L_{p^{'}}$ -estimates for Fourier integral operators related to hyperbolic equations, Math. Z. 152 (1977), no. 3, 273–286. | MR 430872 | Zbl 0325.35009

[Eva98] Evans, L. C., Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. | MR 1625845 | Zbl 0902.35002

[GKZ94] Gelfand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994. | MR 1264417 | Zbl 0827.14036

[HR03] Hirosawa, F., Reissig, M., From wave to Klein-Gordon type decay rates, Nonlinear hyperbolic equations, spectral theory, and wavelet transformations, 95–155, Oper. Theory Adv. Appl., 145, Birkhäuser, Basel, 2003. | MR 2035136 | Zbl 1052.35110

[Hör97] Hörmander, L., Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin), vol. 26, Springer-Verlag, Berlin, 1997. | MR 1466700 | Zbl 0881.35001

[Kli67] Klinger, A., The Vandermonde matrix, Amer. Math. Monthly 74 (1967), 571–574. | MR 213375 | Zbl 0153.35402

[Lit73] Littman, W., $L^{p} - L^{q}$ -estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, R.I., 1973, pp. 479–481. | MR 358443 | Zbl 0263.44006

[Mat76] Matsumura, A., On the asymptotic behaviour of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., Kyoto Univ. 12 (1976), no. 1, 169–189. | MR 420031 | Zbl 0356.35008

[Pec76] Pecher, H., $L^{p}$ -Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. 150 (1976), no. 2, 159–183. | MR 435604 | Zbl 0318.35054

[Rac92] Racke, R., Lectures on nonlinear evolution equations: Initial value problems, Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992. | MR 1158463 | Zbl 0811.35002

[Ruzh00] Ruzhansky, M., Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys, 55 (2000), 99–170. | MR 1751819 | Zbl 0961.35194

[Str70a] Strichartz, R. S., Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. | MR 256219 | Zbl 0199.17502

[Str70b] Strichartz, R. S., A priori estimates for the wave equation and some applications, J. Funct. Analysis 5 (1970), 218–235. | MR 257581 | Zbl 0189.40701

[Sug94] Sugimoto, M., A priori estimates for higher order hyperbolic equations, Math. Z. 215 (1994), no. 4, 519–531. | MR 1269488 | Zbl 0790.35063

[Sug96] Sugimoto, M., Estimates for hyperbolic equations with non-convex characteristics, Math. Z. 222 (1996), no. 4, 521–531. | MR 1406266 | Zbl 0867.35013

[Sug98] Sugimoto, M., Estimates for hyperbolic equations of space dimension 3, J. Funct. Anal. 160 (1998), no. 2, 382–407. | MR 1665291 | Zbl 0917.35064

[VR03] Volevich, L. R. and Radkevich, E. V., Uniform estimates of solutions of the Cauchy problem for hyperbolic equations with a small parameter multiplying higher derivatives, Diff. Eq. 39 (2003), 521–535. | MR 2132995 | Zbl 02176537

[VR04] Volevich, L. R. and Radkevich, E. V., Stable pencils of hyperbolic polynomials and the Cauchy problem for hyperbolic equations with a small parameter at the highest derivatives, Trans. Moscow Math. Soc. 65 (2004), 63–104. | MR 2193437 | Zbl pre05577564

[vW71] von Wahl, W., $L^{p}$ -decay rates for homogeneous wave-equations, Math. Z. 120 (1971), 93–106. | MR 280885 | Zbl 0212.44201