Recent results and open problems on parabolic equations with gradient nonlinearities

Electronic Journal of Differential Equations

Volumn 2001, Issue , 2001, Pages

  Souple, Philippe ^a,b  

^a Université de Picardi INSSET   (France)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Finite time blowup; Global existence; Gradient term; Nonlinear parabolic equations

Indexed keywords

EID: 52549103290     PISSN: 10726691     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (58)

1. F. Andreu, J. M. Mazon, F. Simondon and J. Toledo, Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source, Math. Ann., 314 (1999), 703-728.
2. J. Aguirre and M. Escobedo, On the blow up of solutions for a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh., 123A, 3 (1993), 433-460.
3. L. Alfonsi and F. B. Weissler, Blowup in M.N for a parabolic equation with a damping nonlinear gradient term, in: Diffusion Equations and Their Equilibrium States, III, N. G. Lloyd et al. (eds.), Birkhäuser, Boston-BaselBerlin, 1992, 1-10.
4. S. Angenent and J. Fila, Interior gradient blowup in a semilinear parabolic equation, Differential Integral Equations, 9 (1996), 865-877.
5. J. Ball, Remarks on blowup and nonexistence theorems for nonlinear evolution equations, Quarterly J. Math. Oxford Ser. 28 (1977), 473-486.
6. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences, Springer-Verlag, 1989.
7. J. Bebernes and D. Eberly, Characterization of blow-up for a semilinear heat equation with a convection term, Quart. J. Mech. Appl. Mech., 42 (1989) 447-456.
8. M. Ben-Artzi and H. Koch, Decay of mass for a semilinear parabolic equation, Commun. Partial Differ. Equations, 24 (1999) 869-881.
9. M. Chipot, On a class of nonlinear elliptic equations, Proc. of the Banach Center, 27 (1992), 75-80.
10. M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.
11. M. Chlebik and M. Fila, From critical exponents to blowup rates for parabolic problems, Rend. Mat. Appl., Ser. VII, 19 (1999), 449-470.
12. M. Chlebik, M. Fila and P. Quittner, Blowup of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impulsive Syst., to appear.
13. K. Deng, Stabilizat ion of solutions of a nonlinear parabolic equation with a gradient term, Math. Z., 216 (1994), 147-155.
14. K. Deng and H. A. Levine, The role of critical exponents in blowup theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85-126.
15. M. Escobedo, J. L. Vazquez and E. Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J., 42 (1993), 1413-1440.
16. M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc., Ill (1991), 795-801.
17. [17 M. Fila and P. Quittner, Radial positive solutions for a semilinear elliptic equation with a gradient term, Adv. Math. Sci. Appl., 2 (1993), 39-45.
18. M. Fila and Ph. Souplet, The blowup rate for semilinear parabolic problems on general domains, NoDEA Nonlinear Differential Equations Appl., to appear.
19. A. Friedman and J. B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
20. H. Fujita, On the blowing-up of solutions of the Cauchy problem for ut = Au + u1+a, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124.
21. V. A. Galaktionov and J. L. Vazquez, Regional blowup in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal, 24 (1993), 1254-1276.
22. V. A. Galaktionov and J. L. Vazquez, Blowup for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differ. Equations, 127 (1996), 1-40.
23. V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equations, 109 (1994), 295-308.
24. B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.
25. Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40.
26. M. A. Herrero and J. J. L. Velazquez, Explosion de solutions des équations paraboliques semilinéaires supercritiques, C. R. Acad. Se. Paris, 319, Série I (1994), 141-145.
27. B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differential Integral Equations, 9 (1996), 891-901.
28. S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Commun. Pure Appl. Math., 16 (1963), 327-343.
29. B. Kawohl and L. Peletier, Remarks on blowup and dead cores for nonlinear parabolic equations, Math. Z., 202 (1989), 207-217.
30. A. A. Lacey, Global blowup of a nonlinear heat equation, Proc. Royal Soc. Edinburgh, 104 (1986), 161-167.
31. T. Lee and W. Ni, Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378.
32. H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = -Au + F(u), Arch. Rational Mech. Anal, 51 (1973), 371-386.
33. H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.
34. F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Commun. Pure Appl. Math., 51 (1998), 139-196.
35. L. A. Peletier, J. Serrin and H. Zou, Ground states of a quasilinear equation, Differential Integral Equations, 7 (1994), 1063-1082.
36. P. Quittner, Blowup for semilinear parabolic equations with a gradient term, Math. Methods Appl. Sci., 14 (1991), 413-417.
37. P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolinae, 34 (1993), 105-124.
38. P. Quittner, Global solutions in parabolic blowup problems with perturbations, in: Proceedings of the Third European Conference on Elliptic and Parabolic Problems (Pont-a-Mousson 1997), Pitman Research Notes Series in Mathematics jj 384, Addison Wesley Longman, 1998.
39. P. Quittner and Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, preprint Univ. Versailles # 2000-49.
40. P. Rouchon, Blowup of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data, Z. Angew. Math. Phys., to appear.
41. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blowup in Quasilinear Parabolic Equations, Nauka, Moscou, 1987 ; English translation : Walter de Gruyter, Berlin, New York, 1995.
42. J. Serrin, Y. Yan and H. Zou, A numerical study of the existence and nonexistence of ground states and their bifurcations for the equations of Chipot and Weissler, University of Minnesota preprint n° 93-056 (1993).
43. J. Serrin and H. Zou, Existence and non-existence for ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal. 121 (1992), 101130.
44. S. Snoussi, S. Tayachi and F. B. Weissler, Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient terms, Proc. Royal Soc. Edinburgh, 129A (1999), 1291-1307.
45. Ph. Souplet, Résultats d'explosion en temps fini pour une équation de la chaleur non linéaire, C. R. Acad. Se. Paris, 321, Série I (1995), 721-726.
46. Ph. Souplet, Finite time blowup for a nonlinear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19 (1996), 1317-1333.
47. [47 Ph. Souplet, Geometry of unbounded domains, Poincaré inequalities, and stability in semilinear parabolic equations, Commun. Partial Differ. Equations, 24 (1999), 951-973.
48. Ph. Souplet, Decay of heat semigroups in L°° and applications to nonlinear parabolic problems in unbounded domains, J. Funct. Anal., 173 (2000), 343360.
49. Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, to appear.
50. Ph. Souplet and S. Tayachi, Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math, 87 (2001), in press.
51. Ph. Souplet, S. Tayachi and F. B. Weissler, Exact self-similar blowup of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 48 (1996), 655-682.
52. Ph. Souplet and F. B. Weissler, Self-similar sub-solutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60-74.
53. Ph. Souplet and F. B. Weissler, Poincaré's inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré, Analyse non linéaire, 16 (1999), 337-373.
54. B. Straughan, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998.
55. S. Tayachi, Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term, Differential Integral Equations, 9 (1996), 1107-1117.
56. J. J. L. Velazquez, Blow up for semilinear parabolic equations, in: Research in applied mathematics, Recent advances in partial differential equations, M. A. Herrero and E. Zuazua (eds.), John Wiley & Sons, 1994, 131-145.
57. F. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ. Comenianae, 65 (1996), 53-64.
58. F. B. Weissler, An L°° blowup estimate for a nonlinear heat equation, Commun. Pure Appl. Math., 38 (1985), 291-295.