Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption

Journal of Dynamics and Differential Equations

Volumn 19, Issue 4, 2007, Pages 985-1005

  Laurençot, Philippe ^a   Vázquez, Juan Luis ^b  

^a CNRS   (France)

^b UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

Asymptotic patterns; Hamilton?Jacobi equations; Localization; Nonlinear parabolic equations; P Laplacian equation; Viscosity solutions

Indexed keywords

EID: 36549046605     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1007/s10884-007-9093-y     Document Type: Article

References (37)

1. Andreucci D., Tedeev A.F., Ughi M. (2004). The Cauchy problem for degenerate parabolic equations with source and damping. Ukrainian Math. Bull. 1:1-23
2. Antontsev, S. N. (1996). Quasilinear Parabolic Equations with Non-isotropic Nonlinearities: Space and Time Localization. Energy Methods in Continuum Mechanics (Oviedo, 1994), pp. 1-12, Kluwer Acad. Publ., Dordrecht.
3. Antontsev, S. N., and Diaz, J. I. (1989). On Space or Time Localization of Solutions of Nonlinear Elliptic or Parabolic Equations via Energy Methods. Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), pp. 3-14, Pitman Res. Notes Math. Ser., Vol. 208, Longman Sci. Tech., Harlow.
4. Aronson, D. G. (1986). The porous medium equation. Nonlinear diffusion problems (Montecatini Terme, 1985), pp. 1-46, Lecture Notes in Math., Vol. 1224, Springer, Berlin, 1986.
5. Aronson D.G., Caffarelli L.A. (1983). The initial trace of a solution of the porous medium equation. Trans. Amer. Math. Soc. 280:351-366
6. Bardi, M., and Capuzzo-Dolcetta, I. (1997). Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems Control Found. Appl., Birkhäuser, Boston.
7. Barles G. (1985). Asymptotic behavior of viscosity solutions of first order Hamilton-Jacobi equations. Ricerche Mat. 34:227-260
8. Barles, G. (1994). Solutions de Viscosité des Equations d'Hamilton-Jacobi. Mathématiques & Applications, Vol. 17. Springer-Verlag, Berlin.
9. Barles G., Perthame B. (1988). Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26:1133-1148
10. Bartier, J.-P., and Laurençot, P. Gradient estimates for a degenerate parabolic equation with gradient absorption and applications (Submitted).
11. Benachour S., Karch G., Laurençot P. (2004). Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations. J. Math. Pures Appl. 83:1275-1308
12. N. Asymptot. Anal. 31:229-246
13. x|. Astérisque 236:41-71
14. d, d >2 J. Funct. Anal. 144:301-324
15. Bertsch M., Nanbu T., Peletier L.A. (1982). Decay of solutions of a degenerate nonlinear diffusion equation. Nonlinear Anal. 6:539-554
16. Bertsch M., Kersner R., Peletier L.A. (1982). Sur le comportement de la frontière libre dans une équation en théorie de la filtration. C. R. Acad. Sci. Paris Sér. I 295:63-66
17. Biler P., Guedda M., Karch G. (2004). Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation. J. Evol. Equat. 4:75-97
18. Chaves M., Vázquez J.L. (1996). Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity. Diff. Integral Equat. 9:447-464
19. Chaves M., Vázquez J.L. (1999). Free boundary layer formation in nonlinear heat propagation. Comm. Partial Diff. Equat. 24:1945-1965
20. Crandall M.G., Ishii H., Lions P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27:1-67
21. Diaz J.I., Véron L. (1985) Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc. 290:787-814
22. Gallay T., Laurençot, P. (2007). Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent. Indiana Univ. Math. J. 56:459-479
23. q, large-time behaviour. J. Math. Pures Appl. 84:753-785
24. Húska, J., Poláčik, P., and Safonov, M. V. (2007). Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations (Preprint).
25. Gmira A., Bettioui B. (2002). On the selfsimilar solutions of a diffusion convection equation. NoDEA Nonlinear Diff. Equat. Appl. 9:277-294
26. Ishii H. (1987). A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. Proc. Amer. Math. Soc. 100:247-251
27. Kalashnikov A.S. (1987). Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surv. 42:169-222
28. Kamin S., Vázquez J.L. (1988). Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana 4:339-354
29. Knerr B.F. (1977). The porous medium equation in one dimension. Trans. Amer. Math. Soc. 234:381-415
30. Ladyzhenskaya, O. A., Solonnikov, V. A., and Uraltseva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, R.I.
31. Namah G., Roquejoffre J.-M. (1999). Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations. Comm. Partial Diff. Equat. 24:883-893
32. Roquejoffre J.-M. (2001). Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl. 80:85-104
33. Shi P.-H. (2004). Self-similar singular solution of a p-Laplacian evolution equation with gradient absorption term. J. Partial Diff. Equat. 17:369-383
34. Vázquez, J. L. (1993). Singular Solutions and Asymptotic Behaviour of Nonlinear Parabolic Equations. International Conference on Differential Equations, Vol. 1, 2 (Barcelona, 1991), pp. 234-249, World Sci. Publ., River Edge, NJ.
35. Vázquez, J. L. (2006). Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Ser. Math. Appl., Vol. 33, Oxford University Press, Oxford.
36. Vázquez, J. L. (2007). The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford.
37. Yuan H.-J. (1996). Localization condition for a nonlinear diffusion equation. Chinese J. Contemp. Math. 17:45-58