Research Article
BibTex RIS Cite

Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes

Year 2018, Volume: 8, 65 - 69, 30.06.2018

Abstract

In this paper, we study a nonlinear diffusion equation (φ(u))_{t}=u_{xx}, 0<x<a, t>0 with singular boundary outfluxes u_{x}(0,t)=u^{-p}(0,t), u_{x}(a,t)=-u^{-q}(a,t). Firstly, we get the quenchnig occurs in a finite time at the boundary x=a under certain conditions. Finally, we show the time derivative blows up at the quenching time and we also establish results on quenching time and rate for certain nonlinearities.

References

  • Deng, K., Xu, M., Quenching for a nonlinear diffusion equation with a singular boundary condition, Z. Angew. Math. Phys., 50(1999), 574–584.
  • Fila, M., Levine, H.A., Quenching on the boundary, Nonlinear Anal., 21(1993), 795–802.
  • Pao, C.V., Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 66(2007), 639–662.
  • Selcuk, B., Ozalp, N., Quenching behavior of semilinear heat equations with singular boundary conditions, Electron. J. Diff. Equ., 2015(311)(2015), 1–13.
  • Selcuk, B., Ozalp, N., Quenching for a semilinear heat equation with a singular boundary outflux, Int. J. Appl. Math, 29(4)(2016), 451–464.
  • Yang, Y., Quenching phenomenon for a non-Newtonian filtration equation with singular boundary flux, Boundary Value Problems, (2015), 2015:233.
  • Yang, Y., Yin, J., Jin, C., Quenching phenomenon of positive radial solutions for p-Laplacian with singular boundary flux, J. Dyn.Control Syst., 22(2016), 653–660.
  • Yang, Y., Finite time quenching for a nonlinear diffusion equation with singular boundary flux, Journal of Physics: Conference Series, 814(1)(2017), 1–6.
  • Zhi, Y., Mu, C., The quenching behavior of a nonlinear parabolic equation with a nonlinear boundary outflux, Appl. Math.Comput., 184(2007), 624–630.
Year 2018, Volume: 8, 65 - 69, 30.06.2018

Abstract

References

  • Deng, K., Xu, M., Quenching for a nonlinear diffusion equation with a singular boundary condition, Z. Angew. Math. Phys., 50(1999), 574–584.
  • Fila, M., Levine, H.A., Quenching on the boundary, Nonlinear Anal., 21(1993), 795–802.
  • Pao, C.V., Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 66(2007), 639–662.
  • Selcuk, B., Ozalp, N., Quenching behavior of semilinear heat equations with singular boundary conditions, Electron. J. Diff. Equ., 2015(311)(2015), 1–13.
  • Selcuk, B., Ozalp, N., Quenching for a semilinear heat equation with a singular boundary outflux, Int. J. Appl. Math, 29(4)(2016), 451–464.
  • Yang, Y., Quenching phenomenon for a non-Newtonian filtration equation with singular boundary flux, Boundary Value Problems, (2015), 2015:233.
  • Yang, Y., Yin, J., Jin, C., Quenching phenomenon of positive radial solutions for p-Laplacian with singular boundary flux, J. Dyn.Control Syst., 22(2016), 653–660.
  • Yang, Y., Finite time quenching for a nonlinear diffusion equation with singular boundary flux, Journal of Physics: Conference Series, 814(1)(2017), 1–6.
  • Zhi, Y., Mu, C., The quenching behavior of a nonlinear parabolic equation with a nonlinear boundary outflux, Appl. Math.Comput., 184(2007), 624–630.
There are 9 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Burhan Selçuk

Nuri Özalp This is me

Publication Date June 30, 2018
Published in Issue Year 2018 Volume: 8

Cite

APA Selçuk, B., & Özalp, N. (2018). Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes. Turkish Journal of Mathematics and Computer Science, 8, 65-69.
AMA Selçuk B, Özalp N. Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes. TJMCS. June 2018;8:65-69.
Chicago Selçuk, Burhan, and Nuri Özalp. “Quenching for A Nonlinear Diffusion Equation With Singular Boundary Outfluxes”. Turkish Journal of Mathematics and Computer Science 8, June (June 2018): 65-69.
EndNote Selçuk B, Özalp N (June 1, 2018) Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes. Turkish Journal of Mathematics and Computer Science 8 65–69.
IEEE B. Selçuk and N. Özalp, “Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes”, TJMCS, vol. 8, pp. 65–69, 2018.
ISNAD Selçuk, Burhan - Özalp, Nuri. “Quenching for A Nonlinear Diffusion Equation With Singular Boundary Outfluxes”. Turkish Journal of Mathematics and Computer Science 8 (June 2018), 65-69.
JAMA Selçuk B, Özalp N. Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes. TJMCS. 2018;8:65–69.
MLA Selçuk, Burhan and Nuri Özalp. “Quenching for A Nonlinear Diffusion Equation With Singular Boundary Outfluxes”. Turkish Journal of Mathematics and Computer Science, vol. 8, 2018, pp. 65-69.
Vancouver Selçuk B, Özalp N. Quenching for A Nonlinear Diffusion Equation with Singular Boundary Outfluxes. TJMCS. 2018;8:65-9.