×

Initial blow-up rates and universal bounds for nonlinear heat equations. (English) Zbl 1044.35027

Summary: We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation \[ u_t=\Delta u+ u^p\quad\text{on }(0,T)\times\Omega, \] where \(p> 1\) and \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^N\). Namely, we show that \[ \| u(t)\|_\infty\leq C(p,\Omega, T) t^{-\alpha}\quad\text{on }(0,T/2] \] for some \(\alpha= \alpha(N,p)> 0\). This is proved for a supcritical \(p\) i.e., \(p< (N+ 2)/(N- 2))\) if \(N\leq 4\) (and under a stronger assumption on \(p\) if \(N\geq 5\)). As a consequence, we improve the known results on universal bounds for global solutions. Furthermore, if \(p< (N+ 3)/(N+ 1)\), then we may take \(\alpha= (N +1)/2\) and we show that this value of a is optimal. Interestingly, the rate can be faster than the previously known, maximal initial blow-up rate of the Cauchy problem. Applications to universal blow-up estimates at \(t= T\) are given. The Neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical \(p\) up to dimension \(N= 6\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B45 A priori estimates in context of PDEs
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Birkhaüser, Basel, 1995. · Zbl 0819.35001
[2] Andreucci, D.; Dibenedetto, E., On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. scuola norm. sup. Pisa cl. sci., 18, 4, 363-441, (1991) · Zbl 0762.35052
[3] Andreucci, D.; Herrero, M.; Velázquez, J., Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. inst. Henri Poincaré, 14, 1-53, (1997) · Zbl 0877.35019
[4] Aronson, D.; Benilan, Ph., Régularité des solutions de l’équation des milieux poreux dans \(R\^{}\{N\}\), C. R. acad. sci. Paris, 288, 103-105, (1979) · Zbl 0397.35034
[5] M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, in: Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions, Gauthier-Villars, Paris, 1998, pp. 189-198. · Zbl 0914.35055
[6] Brézis, H.; Cazenave, T., A nonlinear heat equation with singular initial data, J. anal. math., 68, 877-894, (1996)
[7] Cazenave, T.; Lions, P.-L., Solutions globales d’equations de la chaleur semi-linéaires, Commun. partial differential equations, 9, 955-978, (1984) · Zbl 0555.35067
[8] Crandall, M.; Rabinowitz, P.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Commun. partial differential equations, 2, 193-222, (1977) · Zbl 0362.35031
[9] Fila, M.; Souplet, Ph.; Weissler, F., Linear and nonlinear heat equations in Lδq spaces and universal bounds for global solutions, Math. ann., 320, 87-113, (2001) · Zbl 0993.35023
[10] Filippas, S.; Herrero, M.; Velázquez, J., Fast blow-up mechanism for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, Roy. soc. lond. proc. ser. A, 456, 2957-2982, (2000) · Zbl 0988.35032
[11] Friedman, A.; McLeod, J., Blow-up of positive solutions of semilinear heat equations, Indiana univ. math. J., 34, 425-447, (1985) · Zbl 0576.35068
[12] Galaktionov, V.; Vázquez, J.-L., Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. pure appl. math., 50, 1-67, (1997) · Zbl 0874.35057
[13] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041
[14] Giga, Y., A bound for global solutions of semi-linear heat equations, Comm. math. phys., 103, 415-421, (1986) · Zbl 0595.35057
[15] Giga, Y.; Kohn, R., Characterizing blowup using similarity variables, Indiana univ. math. J., 36, 1-40, (1987) · Zbl 0601.35052
[16] Gomes, S., On a singular nonlinear elliptic problem, SIAM J. math. anal., 17, 1359-1369, (1986) · Zbl 0614.35037
[17] Gui, Ch.; Lin, F.-H., Regularity of an elliptic problem with a singular nonlinearity, Proc. roy. soc. Edinburgh sect. A, 123, 1021-1029, (1993) · Zbl 0805.35032
[18] Herrero, M.; Velázquez, J., Explosion de solutions des équations paraboliques semi-linéaires supercritiques, C. R. acad. sci. Paris, 319, 141-145, (1994) · Zbl 0806.35005
[19] Kaplan, S., On the growth of solutions of quasilinear parabolic equations, Comm. pure appl. math., 16, 305-330, (1963) · Zbl 0156.33503
[20] Lazer, A.; McKenna, P., On a singular nonlinear elliptic boundary-value problem, Proc. amer. math. soc., 111, 721-730, (1991) · Zbl 0727.35057
[21] Matos, J.; Souplet, Ph., Universal blow-up rates for a semilinear heat equation and applications, Adv. differential equations, 8, 615-639, (2003) · Zbl 1028.35065
[22] Merle, F.; Zaag, H., Refined uniform estimates at blow-up and applications for nonlinear heat equations, Geom. funct. anal., 8, 1043-1085, (1998) · Zbl 0926.35024
[23] Merle, F.; Zaag, H., A Liouville theorem for vector-valued nonlinear heat equations, Math. ann., 316, 103-137, (2000) · Zbl 0939.35086
[24] Ni, W.-M.; Sacks, P.; Tavantzis, J., On the asymptotic behavior of solutions of certain quasi-linear equations of parabolic type, J. differential equations, 54, 97-120, (1984) · Zbl 0565.35053
[25] Quittner, P., A priori bounds for global solutions of a semilinear parabolic problem, Acta math. univ. comenianae, 68, 195-203, (1999) · Zbl 0940.35112
[26] Quittner, P., Universal bound for global positive solutions of a superlinear parabolic problem, Math. ann., 320, 299-305, (2001) · Zbl 0981.35010
[27] Quittner, P., Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. math., 29, 757-799, (2003) · Zbl 1034.35013
[28] Quittner, P.; Souplet, Ph., Admissible Lp norms for local existence and for continuation in parabolic systems are not the same, Proc. roy. soc. Edinburgh sect. A, 131, 1435-1456, (2001) · Zbl 1006.35048
[29] Sacks, P., Global behavior for a class of nonlinear evolution equations, SIAM J. math. anal., 16, 233-250, (1985) · Zbl 0572.35062
[30] Weissler, F., Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034
[31] Weissler, F., An L∞ blow-up estimate for a nonlinear heat equation, Comm. pure appl. math., 38, 291-295, (1985) · Zbl 0592.35071
[32] Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear anal., 28, 1977-1995, (1997) · Zbl 0874.35061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.