×

Classification of blow-up with nonlinear diffusion and localized reaction. (English) Zbl 1110.35034

The authors study and classify blow-up behaviour of nonnegative solutions of the nonlinear diffusion equation \[ u_t = (u^m)_{xx} + a(x)u^p, \quad \text{in } (x,t)\in \mathbb R\times (0,T) \] with the initial data \[ u(x,0) = u_0(x) \quad \text{in } \mathbb R, \] where \(m > 1\) and \(p >0\), and the coefficient \(a(x)\) is assumed to be compactly supported function. The authors proves that the critical exponent for global existence of the above-mentioned problem is \(p_0 = (m + 1)/2\), while the Fujita exponet is \(p_c = m + 1\). It means that if \(0 < p \leq p_0\) every solution is global in time, if \(p_0 < p \leq p_c\) all solutions blow up and if \(p > p_c\) both global in time solutions and blowing up solutions exists. The blow-up rates, the blow-up sets and the blow-up profiles are found. Moreover, the authors show that reaction happens as in the case of reaction extended to the whole line if \(p > m\), while it concentrates to a point in the form of a nonlinear flux if \(p < m\). If \(p = m\) the asymptotic behaviour is given by a self-similar solution of this problem.

MSC:

35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B45 A priori estimates in context of PDEs
35B33 Critical exponents in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Baras, P.; Cohen, L., Complete blow-up after \(T_{\max}\) for the solution of a semilinear heat equation, J. Funct. Anal., 71, 142-174 (1987) · Zbl 0653.35037
[2] Bimpong-Bota, K.; Ortoleva, P.; Ross, J., Far from equilibrium phenomena at local sites of reaction, J. Chem. Phys., 60, 3124 (1974)
[3] Chadam, J. M.; Yin, H.-M., A diffusion equation with localized chemical reactions, Proc. Edinb. Math. Soc., 37, 101-118 (1994) · Zbl 0790.35045
[4] Cortázar, C.; del Pino, M.; Elgueta, M., On the blow-up set for \(u_t = \Delta u^m + u^m, m > 1\), Indiana Univ. Math. J., 47, 541-561 (1998) · Zbl 0916.35056
[5] Deng, K.; Levine, H. A., The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243, 85-126 (2000) · Zbl 0942.35025
[6] Ferreira, R.; de Pablo, A.; Quirós, F.; Rossi, J. D., The blow-up profile for a fast diffusion equation with a nonlinear boundary condition, Rocky Mountain J. Math., 33, 123-146 (2003) · Zbl 1032.35107
[7] Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068
[8] Galaktionov, V. A., Blow-up for quasilinear heat equations with critical Fujita’s exponents, Proc. Roy. Soc. Edinburgh Sect. A, 124, 517-525 (1994) · Zbl 0808.35053
[9] Galaktionov, V. A., On asymptotic self-similar behaviour for a quasilinear heat equation. Single point blow-up, SIAM J. Math. Anal., 26, 675-693 (1995) · Zbl 0828.35067
[10] Galaktionov, V. A.; Levine, H. A., On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math., 94, 125-146 (1996) · Zbl 0851.35067
[11] Galaktionov, V. A.; Levine, H. A., A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34, 1005-1027 (1998) · Zbl 1139.35317
[12] Galaktionov, V. A.; Vázquez, J. L., The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. A, 8, 399-433 (2002) · Zbl 1010.35057
[13] Giga, Y.; Kohn, R. V., Characterizing blow-up using similarity variables, Indiana Univ. Math. J., 36, 1-40 (1987) · Zbl 0601.35052
[14] Levine, H. A., The role of critical exponents in blow-up theorems, SIAM Rev., 32, 262-288 (1990) · Zbl 0706.35008
[15] Levine, H. A.; Sacks, P., Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differential Equations, 52, 135-161 (1984) · Zbl 0487.34003
[16] de Pablo, A.; Quirós, F.; Rossi, J. D., Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition, IMA J. Appl. Math., 67, 69-98 (2002) · Zbl 1001.35071
[17] Pinsky, R. G., Existence and nonexistence of global solutions for \(u_t = \Delta u + a(x) u^p\) in \(R^d\), J. Differential Equations, 133, 152-177 (1997) · Zbl 0876.35048
[18] Polubarinova-Kochina, P. Ya., On a nonlinear differential equation encountered in the theory of infiltration, Dokl. Akad. Nauk SSSR, 63, 623-627 (1948)
[19] Qi, Y. W., The critical exponents of parabolic equations and blow-up in \(R^n\), Proc. Roy. Soc. Edinburgh Sect. A, 128, 123-136 (1998) · Zbl 0892.35088
[20] Quirós, F.; Rossi, J. D., Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions, Indiana Univ. Math. J., 50, 629-654 (2001) · Zbl 0994.35027
[21] Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P., Blow-up in Problems for Quasilinear Parabolic Equations (1995), Nauka: Nauka Moscow: Walter de Gruyter: Nauka: Nauka Moscow: Walter de Gruyter Berlin, (in Russian); English translation: · Zbl 1020.35001
[22] Souplet, Ph., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153, 374-406 (1999) · Zbl 0923.35077
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.