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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
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[1] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Birkhaüser, Basel, 1995.; 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.; 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 \(L_δ^{q\) · 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 \(L_p\) 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 \(L^p\), 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
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