Research article Special Issues

A nonlinear diffusion equation with reaction localized in the half-line

  • Received: 21 May 2021 Accepted: 19 July 2021 Published: 10 August 2021
  • We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line

    $ u_t = (u^m)_{xx}+a(x) u^p, $

    $ m, p > 0 $ and $ a(x) = 1 $ for $ x > 0 $, $ a(x) = 0 $ for $ x < 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p > 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p > m $ or $ p = 1\neq m $.

    Citation: Raúl Ferreira, Arturo de Pablo. A nonlinear diffusion equation with reaction localized in the half-line[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022024

    Related Papers:

  • We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line

    $ u_t = (u^m)_{xx}+a(x) u^p, $

    $ m, p > 0 $ and $ a(x) = 1 $ for $ x > 0 $, $ a(x) = 0 $ for $ x < 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p > 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p > m $ or $ p = 1\neq m $.



    加载中


    [1] J. Aguirre, M. Escobedo, A Cauchy problem for $u_t-\Delta u = u^p$ with $0<p<1$. Asymptotic behaviour of solutions, Ann. Fac. Sci. Toulouse Math., 8 (1986), 175–203.
    [2] X. Bai, S. Zhou, S. Zheng, Cauchy problem for fast diffusion equation with localized reaction, Nonlinear Anal., 74 (2011), 2508–2514. doi: 10.1016/j.na.2010.12.006
    [3] R. Ferreira, A. de Pablo, Grow-up for a quasilinear heat equation with a localized reaction, J. Differ. Equations, 268 (2020), 6211–6229. doi: 10.1016/j.jde.2019.11.033
    [4] R. Ferreira, A. de Pablo, J. L. Vázquez, Blow-up for the porous medium equation with a localized reaction, J. Differ. Equations, 231 (2006), 195–211. doi: 10.1016/j.jde.2006.04.017
    [5] R. Ferreira, A. de Pablo, F. Quirós, J. D. Rossi, The blow-up profile for a fast diffusion equation with a nonlinear boundary condition, Rocky Mt. J. Math., 33 (2003), 123–146.
    [6] M. Fila, P. Souplet, The blow-up rate for semilinear parbolic problems on general domains, Nonlinear Differ. Equ. Appl., 8 (2001), 473–480. doi: 10.1007/PL00001459
    [7] V. A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita's exponents, P. Roy. Soc. Edinb. A, 124 (1994), 517–525. doi: 10.1017/S0308210500028766
    [8] B. H. Gilding, L. A. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl., 55 (1976), 351–364. doi: 10.1016/0022-247X(76)90166-9
    [9] M. A. Herrero, M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ when $0 < m < 1$, T. Am. Math. Soc., 291 (1985), 145–158.
    [10] H. A. Levine, P. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differ. Equations, 52 (1984), 135–161. doi: 10.1016/0022-0396(84)90174-8
    [11] A. de Pablo, J. L. Vázquez, The balance between strong reaction and slow diffusion, Commun. Part. Diff. Eq., 15 (1990), 159–183. doi: 10.1080/03605309908820682
    [12] A. de Pablo, J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differ. Equations, 93 (1991), 19–61. doi: 10.1016/0022-0396(91)90021-Z
    [13] R. G. Pinsky, Existence and nonexistence of global solutions for $u_t = \Delta u + a(x)u^p$ in $R^d$, J. Differ. Equations, 133 (1997), 152–177. doi: 10.1006/jdeq.1996.3196
    [14] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Blow-up in problems for quasilinear parabolic equations, Berlin: Walter de Gruyter, 1995.
    [15] J. L. Vázquez, The porous medium equation. Mathematical theory, Oxford: The Clarendon Press, Oxford University Press, 2007.
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(333) PDF downloads(53) Cited by(0)

Article outline

Figures and Tables

Figures(1)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog