Infinite time blow-up for superlinear parabolic problems with localized reaction

Souplet, Philippe

doi:10.1090/S0002-9939-04-07707-X

Infinite time blow-up for superlinear parabolic problems with localized reaction
HTML articles powered by AMS MathViewer

by Philippe Souplet PDF

Proc. Amer. Math. Soc. 133 (2005), 431-436 Request permission

Abstract:

We consider the nonlocal diffusion equation \[ u_t-u_{xx}=u^p(t,x_0(t)),\] on the space interval $(0,1)$, with Dirichlet boundary conditions. It is known that if the curve $x_0(t)$ remains in a compact subset of $(0,1)$ for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on $x_0$ is sharp: for a large class of functions $x_0(t)$ approaching the boundary as $t\to \infty$, blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.

References

John R. Cannon and Hong-Ming Yin, A class of nonlinear nonclassical parabolic equations, J. Differential Equations 79 (1989), no. 2, 266–288. MR 1000690, DOI 10.1016/0022-0396(89)90103-4
Thierry Cazenave and Pierre-Louis Lions, Solutions globales d’équations de la chaleur semi linéaires, Comm. Partial Differential Equations 9 (1984), no. 10, 955–978 (French). MR 755928, DOI 10.1080/03605308408820353
John M. Chadam, A. Peirce, and Hong-Ming Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992), no. 2, 313–328. MR 1180894, DOI 10.1016/0022-247X(92)90081-N
Weibing Deng, Zhiwen Duan, and Chunhong Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl. 264 (2001), no. 2, 577–597. MR 1876751, DOI 10.1006/jmaa.2001.7696
Marek Fila, Boundedness of global solutions of nonlinear parabolic equations, Elliptic and parabolic problems (Rolduc/Gaeta, 2001) World Sci. Publ., River Edge, NJ, 2002, pp. 88–102. MR 1937530, DOI 10.1142/9789812777201_{0}009
Marek Fila, Philippe Souplet, and Fred B. Weissler, Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions, Math. Ann. 320 (2001), no. 1, 87–113. MR 1835063, DOI 10.1007/PL00004471
Victor A. Galaktionov and Juan L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), no. 1, 1–67. MR 1423231, DOI 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R
Yoshikazu Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), no. 3, 415–421. MR 832917, DOI 10.1007/BF01211756
Zhigui Lin, Chunhong Xie, and Mingxin Wang, The blow-up properties of solutions to a parabolic system with localized nonlinear reactions, Acta Math. Sci. (English Ed.) 18 (1998), no. 4, 413–420. MR 1696579, DOI 10.1016/S0252-9602(17)30596-9
Yvan Martel and Philippe Souplet, Small time boundary behavior of solutions of parabolic equations with noncompatible data, J. Math. Pures Appl. (9) 79 (2000), no. 6, 603–632 (English, with English and French summaries). MR 1770663, DOI 10.1016/S0021-7824(00)00154-9
Ill-Hyeon Nam and Hyeong-Kwan Ju, Asymptotic behavior of blow-up solution of a localized semilinear parabolic equation, Honam Math. J. 19 (1997), no. 1, 61–69. MR 1460353
Wei-Ming Ni, Paul E. Sacks, and John Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), no. 1, 97–120. MR 756548, DOI 10.1016/0022-0396(84)90145-1
Atsuko Okada and Isamu Fukuda, Blow-up of solutions of nonlinear parabolic equations with localized reaction, Free boundary problems: theory and applications, II (Chiba, 1999) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 14, Gakk\B{o}tosho, Tokyo, 2000, pp. 358–366. MR 1794365
M. Pedersen and Zhigui Lin, Coupled diffusion systems with localized nonlinear reactions, Comput. Math. Appl. 42 (2001), no. 6-7, 807–816. MR 1846188, DOI 10.1016/S0898-1221(01)00200-0
Michael Pedersen and Lin Zhigui, The profile near blowup time for solutions of diffusion systems coupled with localized nonlinear reactions, Nonlinear Anal. 50 (2002), no. 7, Ser. A: Theory Methods, 1013–1024. MR 1915320, DOI 10.1016/S0362-546X(01)00798-2
Pavol Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), no. 2, 299–305. MR 1839765, DOI 10.1007/PL00004475
Pavol Quittner, A priori bounds for solutions of parabolic problems and applications, Proceedings of EQUADIFF, 10 (Prague, 2001), 2002, pp. 329–341. MR 1981537, DOI 10.21136/MB.2002.134174
Pierre Rouchon, Boundedness of global solutions of nonlinear diffusion equation with localized reaction term, Differential Integral Equations 16 (2003), no. 9, 1083–1092. MR 1989542
Philippe Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), no. 6, 1301–1334. MR 1638054, DOI 10.1137/S0036141097318900
Philippe Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999), no. 2, 374–406. MR 1683627, DOI 10.1006/jdeq.1998.3535
Liwen Wang and Qingyi Chen, The asymptotic behavior of blowup solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996), no. 2, 315–321. MR 1391152, DOI 10.1006/jmaa.1996.0207