Regularity and nonuniqueness results for parabolic problems arising in some physical models, having natural growth in the gradient

Journal des Mathematiques Pures et Appliquees

Volumn 90, Issue 3, 2008, Pages 242-269

  Abdellaoui, Boumediene ^a   Dall'Aglio, Andrea ^b   Peral, Ireneo ^c  

^a UNIVERSITY OF TLEMCEN   (Algeria)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

Existence; Nonlinear parabolic problems; Uniqueness and nonuniqueness; Viscous Hamilton Jacobi equations

Indexed keywords

EID: 49749116084     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2008.04.004     Document Type: Article

References (37)

1. Abdellaoui B., Colorado E., and Peral I. Existence and nonexistence results for a class of linear and semilinear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities. J. Eur. Math. Soc. 6 (2004) 119-149
2. Abdellaoui B., Dall'Aglio A., and Peral I. Some remarks on elliptic problems with critical growth in the gradient. J. Differential Equations 222 (2006) 21-62
3. Abdellaoui B., and Peral I. Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential. Ann. Mat. Pura Appl. 182 3 (2003) 247-270
4. Aronson D.G., and Serrin J. Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25 (1967) 81-122
5. Ben-Artzi A., Souplet P., and Weissler F.B. The local theory for the viscous Hamilton-Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. 9 81 (2002) 343-378
6. Benilan P., and Brezis H. Nonlinear problems related to the Thomas-Fermi equation (Dedicated to Philippe Bénilan). J. Evol. Eq. 3 4 (2003) 673-770
7. Berestycki H., Kamin S., and Sivashinsky G. Metastability in a flame front evolution equation. Interfaces Free Bound. 3 4 (2001) 361-392
8. 1 data: Existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect. A 127 6 (1997) 1137-1152
9. Blanchard D., Murat F., and Redwane H. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J. Differential Equations 177 2 (2001) 331-374
10. Blanchard D., and Porretta A. Nonlinear parabolic equations with natural growth terms and measure initial data. Ann. Sc. Norm. Sup. Pisa Cl. 30 3-4 (2001) 583-622
11. Boccardo L., Dall'Aglio A., Gallouët T., and Orsina L. Nonlinear parabolic equations with measure data. J. Funct. Anal. 147 1 (1997) 237-258
12. Boccardo L., and Gallouët T. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 1 (1989) 149-169
13. Boccardo L., Murat F., and Puel J.-P. Existence results for some quasilinear parabolic equations. Nonlin. Anal. T.M.A. 13 (1989) 378-392
14. Brezis H., and Cabré X. Some simple nonlinear PDE's without solutions. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 2 (1998) 223-262
15. A. Dall'Aglio, D. Giachetti, I. Peral, S. Segura de León, Global existence for some slightly super-linear parabolic equations with measure data, Preprint
16. Dall'Aglio A., Giachetti D., and Puel J.-P. Nonlinear parabolic equations with natural growth in general domains. Boll. Un. Mat. Ital. Sez. B 8 (2005) 653-683
17. Dall'Aglio A., Giachetti D., and Segura de León S. Nonlinear parabolic problems with a very general quadratic gradient term. Differential Integral Equations 20 4 (2007) 361-396
18. A. Dall'Aglio, D. Giachetti, S. Segura de León, Nonlinear parabolic problems having a gradient term with natural growth, Preprint
19. Dal Maso G., Murat F., Orsina L., and Prignet A. Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 28 4 (1999) 741-808
20. Droniou J., Porretta A., and Prignet A. Parabolic capacity and soft measures for nonlinear equations. Potential Anal. 19 2 (2003) 99-161
21. Evans L.C. Partial Differential Equations. Graduate Studies in Mathematics vol. 19 (1998), Amer. Math. Soc., Providence
22. Ferone V., Posteraro M.R., and Rakotoson J.M. Nonlinear parabolic problems with critical growth and unbounded data. Indiana Univ. Math. J. 50 3 (2001) 1201-1215
23. q. J. Math. Anal. Appl. 284 (2003) 733-755
24. Grenon N. Existence results for some quasilinear parabolic problems. Ann. Mat. Pura Appl. (4) 165 (1993) 281-313
25. Kardar M., Parisi G., and Zhang Y.C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889-892
26. Korkut L., Pašić M., and Žubrinić D. Some qualitative properties of solutions of quasilinear elliptic equations and applications. J. Differential Equations 170 2 (2001) 247-280
27. Ladyzenskaja O.A., Solonnikov V.A., and Ural'ceva N.N. Linear and Quasi-Linear Equations of Parabolic Type. Translations of Math. Monographs vol. 23 (1968), Amer. Math. Soc., Providence
28. t - Δ u = g (u). Ann. Inst. H. Poincaré Anal. Non Linéaire 15 6 (1998) 687-723
29. Moser J. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964) 101-134
30. ∞ (Q)-estimate and existence of solutions for some nonlinear parabolic equations. Boll. U.M.I. Sez. B 6-B (1992) 631-647
31. Oswald L. Isolated positive singularities for a non linear heat equation. Houston J. Math. 14 4 (1988) 543-572
32. Peral I., and Vázquez J.L. On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term. Arch. Rational Mech. Anal. 129 (1995) 201-224
33. F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Annali di Matematica, DOI 10.1007/s10231-007-0057-y, submitted for publication
34. Pierre M. Parabolic capacity and Sobolev spaces. SIAM J. Math. Anal. 14 3 (1983) 522-533
35. A. Porretta, personal communication
36. p (0, T ; B). Ann. Mat. Pura Appl. 146 (1987) 65-96
37. Stampacchia G. Équations elliptiques du second ordre à coefficients discontinus. Été, 1965. Séminaire de Mathématiques Supérieures vol. 16 (1966), Les Presses de l'Université de Montréal, Montreal