Research article Special Issues

Blow-up for degenerate nonlinear parabolic problem

  • Received: 11 July 2019 Accepted: 29 August 2019 Published: 23 September 2019
  • MSC : 35K55, 35K57, 35K60, 35K65

  • In this paper, we deal with the existence, uniqueness, and finite time blow-up of the solution to the degenerate nonlinear parabolic problem: $u_{\tau}=\left( \xi^{r}u^{m}u_{\xi}\right) _{\xi}/\xi^{r}+u^{p}$ for$\;0 < \xi < a$,$\;0 < \tau < \Gamma$, $u\left( \xi,0\right) =u_{0}\left( \xi\right) $ for $0\leq\xi\leq a$, and $u\left( 0,\tau\right) =0=u\left( a,\tau\right) $ for $0 < \tau < \Gamma$, where $u_{0}\left( \xi\right) $ is a positive function and $u_{0}\left( 0\right) =0=u_{0}\left( a\right) $. In addition, we prove that $u$ exists globally if $a$ is small through constructing a global-exist upper solution, and $u_{\tau}$ blows up in a finite time.

    Citation: W. Y. Chan. Blow-up for degenerate nonlinear parabolic problem[J]. AIMS Mathematics, 2019, 4(5): 1488-1498. doi: 10.3934/math.2019.5.1488

    Related Papers:

  • In this paper, we deal with the existence, uniqueness, and finite time blow-up of the solution to the degenerate nonlinear parabolic problem: $u_{\tau}=\left( \xi^{r}u^{m}u_{\xi}\right) _{\xi}/\xi^{r}+u^{p}$ for$\;0 < \xi < a$,$\;0 < \tau < \Gamma$, $u\left( \xi,0\right) =u_{0}\left( \xi\right) $ for $0\leq\xi\leq a$, and $u\left( 0,\tau\right) =0=u\left( a,\tau\right) $ for $0 < \tau < \Gamma$, where $u_{0}\left( \xi\right) $ is a positive function and $u_{0}\left( 0\right) =0=u_{0}\left( a\right) $. In addition, we prove that $u$ exists globally if $a$ is small through constructing a global-exist upper solution, and $u_{\tau}$ blows up in a finite time.


    加载中


    [1] C. Y. Chan, W. Y. Chan, Existence of classical solutions of nonlinear degenerate parabolic problems, Proc. Dynam. Systems Appl., 5 (2008), 85-91.
    [2] C. Y. Chan, C. S. Chen, A numerical method for semilinear singular parabolic quenching problems, Q. Appl. Math., 47 (1989), 45-57. doi: 10.1090/qam/987894
    [3] W. Deng, Z. Duan, C. Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl., 264 (2001), 577-597. doi: 10.1006/jmaa.2001.7696
    [4] V. A. Galaktionov, Boundary-value problem for the nonlinear parabolic equation ut = △uσ+1+uβ, Differ. Uravn., 17 (1981), 836-842.
    [5] J. Gratton, F. Minotti, S. M. Mahajan, Theory of creeping gravity currents of a non-Newtonian liquid, Phy. Rev. E., 60 (1999), 6960-6967. doi: 10.1103/PhysRevE.60.6960
    [6] M. E. Gurtin, R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1
    [7] H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech., 121 (1982), 43-58. doi: 10.1017/S0022112082001797
    [8] H. A. Levine, P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate paraoblic equations, J. Differ. Equations, 52 (1984), 135-161. doi: 10.1016/0022-0396(84)90174-8
    [9] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York: Plenum Press, 1992.
    [10] M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, New York:Springer-Verlag, 1984.
    [11] M. I. Roux, Numerical solution of nonlinear reaction diffusion processes, SIAM J. Numer. Anal.,37 (2000), 1644-1656. doi: 10.1137/S0036142998335996
    [12] P. L. Sachdev, Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems, Florida:Chapman and Hall/CRC, 2000.
    [13] P. E. Sacks, Global behavior for a class of nonlinear evolution equations, SIAM J. Math. Anal.,16 (1985), 233-250. doi: 10.1137/0516018
    [14] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, et al. Blow-up in Quasilinear Parabolic Equations, New York: Walter de Gruyter, 1995.
    [15] A. D. Solomon, Melt time and heat flux for a simple PCM body, Sol. Energy, 22 (1979), 251-257. doi: 10.1016/0038-092X(79)90140-3
    [16] W. Walter, Differential and Integral Inequalities, New York: Springer-Verlag, 1970.
  • Reader Comments
  • © 2019 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(2845) PDF downloads(377) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog