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Souplet, Philippe

Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. (English) Zbl 0923.35077

J. Differ. Equations 153, No. 2, 374-406 (1999).
The Dirichlet problem for semilinear parabolic equations with nonlocal reaction terms is considered. The nonlinearities are such that blow-up in finite time may occur. For a large class of equations it is shown that solutions have global blow-up and that the rate of blow-up is uniform in compact subsets of the domain. The behavior in a boundary layer is studied, too. Some Fujita-type results are also established for the Cauchy problem.
Reviewer: M.Fila (Bratislava)

Cited in 1 Review
Cited in 160 Documents

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

semilinear parabolic equations; nonlocal reaction terms; finite time blow-up; boundary layer; Fujita-type results
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References:

[1] Bebernes, J.; Bressan, A.; Lacey, A., Total blow-up versus single-point blow-up, J. Diff. Equations, 73, 30-44 (1988) · Zbl 0674.35051
[2] Bricmont, J.; Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity, 7, 539-575 (1994) · Zbl 0857.35018
[3] Cannon, J. R.; Yin, H.-M., A class of non-linear non-classical parabolic equations, J. Diff. Equations, 79, 266-288 (1989) · Zbl 0702.35120
[4] Chadam, J. M.; Peirce, A.; Yin, H.-M., The blow-up property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl., 169, 313-328 (1992) · Zbl 0792.35095
[5] Deng, K., Nonlocal nonlinearity versus global blow-up, Math. Applicata, 124-129 (1995)
[6] Deng, K.; Kwang, M. K.; Levine, H. A., The influence of nonlocal nonlinearities on the long time behaviour of solutions of Burgers equation, Quarterly Appl. Math., 5, 1237-1245 (1992)
[7] Filippas, S.; Kohn, R. V., Refined asymptotics for the blowup of \(u_t Δu u^p \), Comm. Pure Appl. Math., 45, 821-869 (1992) · Zbl 0784.35010
[8] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall New York · Zbl 0144.34903
[9] Friedman, A.; Mcleod, J. B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068
[10] Fujita, H., On the blowing-up of solutions of the Cauchy problem for \(u_t = Δu u^{1+α} \), J. Fac. Sci. Univ. Tokyo, Sect. I, 13, 109-124 (1966) · Zbl 0163.34002
[11] V. A. Galaktionov, H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal.; V. A. Galaktionov, H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. · Zbl 1139.35317
[12] Giga, Y.; Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051
[13] Giga, Y.; Kohn, R. V., Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36, 1-40 (1987) · Zbl 0601.35052
[14] Giga, Y.; Kohn, R. V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42, 845-884 (1989) · Zbl 0703.35020
[15] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0691.35001
[16] Guo, J.-S.; Su, H.-W., The blow-up behaviour of the solution of an integrodifferential equation, Differential Integral Equations, 5, 1237-1245 (1992) · Zbl 0798.35021
[17] Herrero, M. A.; Velázquez, J. J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 10, 131-189 (1993) · Zbl 0813.35007
[18] Herrero, M. A.; Velázquez, J. J.L., Flat blow up in one-dimensional semilinear heat equation, Differential Integral Equations, 5, 973-997 (1992) · Zbl 0767.35036
[19] Herrero, M. A.; Velázquez, J. J.L., Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17, 205-219 (1992) · Zbl 0772.35027
[20] Herrero, M. A.; Velázquez, J. J.L., Generic behaviour of one-dimensional blow up patterns, Annali Scuola Norm. Sup. Pisa, XIX, 381-450 (1992) · Zbl 0798.35081
[21] Martel, Y.; Souplet, P., Estimations optimales en temps petit et près de la frontière pour les solutions de l’équation de la chaleur avec données non compatibles, C.R. Acad. Sci. Paris, 327, 575-580 (1998) · Zbl 0990.35051
[22] Merle, F.; Zaag, H., Stability of blow-up profile for equation of the type \(u_t = Δu u^{p−1}u\), Duke Math. J., 86, 143-195 (1997) · Zbl 0872.35049
[23] Merle, F.; Zaag, H., Optimal estimates for blow-up rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51, 139-196 (1998) · Zbl 0926.35024
[24] F. Merle, H. Zaag, Refined uniform estimates at blow-up and applications for nonlinear heat equations, Geom. Funct. Anal.; F. Merle, H. Zaag, Refined uniform estimates at blow-up and applications for nonlinear heat equations, Geom. Funct. Anal. · Zbl 0926.35024
[25] Merle, F.; Zaag, H., Estimations uniformes à l’explosion pour les équations de la chaleur non linéaires et applications, Séminaire X-EDP, Ecole Polytechnique, 19, 1-8 (1996-97) · Zbl 1069.35505
[26] Pao, C. V., Nonexistence of global solutions for an integrodifferential system in reactor dynamics, SIAM J. Math. Anal., 11, 559-564 (1980) · Zbl 0449.45011
[27] Souplet, P., Blow up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29, 1301-1334 (1998) · Zbl 0909.35073
[28] Souplet, P., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, Technical report (1997)
[29] Souplet, P., Some results on blow-up for nonlocal reaction-diffusion equations, Proceedings of the Third European Conference on Elliptic and Parabolic Problems, Pont-à-Mousson 1997. Proceedings of the Third European Conference on Elliptic and Parabolic Problems, Pont-à-Mousson 1997, Pitman Res. Notes Ser. Math., 384 (1998), Addison-Wesley: Addison-Wesley Longman · Zbl 0919.35058
[30] Souplet, P.; Tayachi, S.; Weissler, F. B., Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 48, 655-682 (1996) · Zbl 0990.35061
[31] Velázquez, J. J.L., Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., 338, 441-464 (1993) · Zbl 0803.35015
[32] Velázquez, J. J.L., Blow up for semilinear parabolic equations, Research in applied mathematics, Recent Advances in Partial Differential Equations (1994), John Wiley, p. 131-145 · Zbl 0798.35072
[33] Wang, L.; Chen, Q., The asymptotic behaviour of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl., 200, 315-321 (1996) · Zbl 0852.35101
[34] Wang, M.; Wang, Y., Properties of positive solutions for non-local reaction- diffusion problems, Math. Methods Appl. Sci., 19, 1141-1156 (1996) · Zbl 0990.35066
[35] Weissler, F. B., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38, 29-40 (1981) · Zbl 0476.35043
[36] Weissler, F. B., An \(L^∞\), Comm. Pure Appl. Math., 38, 291-295 (1985) · Zbl 0592.35071
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