# Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions

## Main Article Content

## Abstract

This study concerns with the quenching features of solutions of the non-Newtonian filtration equation. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined are certain cases. Numerical experiments are provided to illustrate and provide additional validation of the theoretical predictions to the quenching rates and times.

## Article Details

*Tamkang Journal of Mathematics*. https://doi.org/10.5556/j.tkjm.55.2024.5009

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## References

Beauregard, M. A., Numerical solutions to singular reaction diffusion equation over elliptical domains, Appl. Math. Comput., Vol. 254, 75-91, 2015.

Beauregard, M. A., Sheng, Q., An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids, J. Comput. Appl. Math., Vol. 241, 30-44, 2013.

Beauregard, M.A., Sheng, Q., Explorations and expectations of equidistribution adaptations for nonlinear quenching problems, Adv. Appl. Math. Mech., Vol. 5, 407-422, 2013.

Beauregard, M. A., Padgett, J., Parshad, R. D., A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion, J. Comput. Appl. Math., Vol. 31, 8-25, 2017.

Chan, C.Y., Yuen, S.I., Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput., 121, 203-209, 2001.

C. Y. Chan and L. Ke, Parabolic quenching for nonsmooth convex domains, J. Math. Anal. Appl. 186 (1994), pp.~52-65.

C..Y. Chan, A quenching criterion for a multi-dimensional parabolic problem due to a concentrated nonlinear source, J. Comput. Appl. Math., 235 (2011), pp.~3724-3727.

H. Cheng, P. Lin, Q. Sheng and R.C.E. Tan, Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting, SIAM J. Sci. Comput. 25 (2003), pp.~1273-1292.

Hundsdorfer, W., Unconditional convergence of some Crank-Nicolson LOD method for initial-boundary value problems, Math. Comput., Vol. 58, 35-53, 1992.

H. Kawarada, On solutions of initial-boundary problem for u_{t}=u_{xx}+1/(1-u), Publ. Res. Inst. Math. Sci., 10

(1975), pp.~729-736.

Z. Li, C. Mu, Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux, J. Math. Anal. Appl., Vol. 346, Iss. 1, pp. 55-64, 2008.

X. Li, C. Mu, Q. Zhang, and S. Zhou, Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition, Abstr. Appl. Anal., Vol. 2012, Article ID 539161, doi:10.1155/2012/539161

Y. Mi, X. Wang and C. Mu, Blow-up set for the non-Newtonian polytropic filtration equation subjected to nonlinear Neumann

boundary condition, Applicable Analysis (2013) Vol. 92, No. 6, 1332--1344.

Selcuk, B., Ozalp, N., The quenching behavior of a semilinear heat equation with a singular boundary outflux, Quarterly of

Applied Mathematics, Vol. 72, Iss. 4, 747-752, 2014.

Selcuk, B., Ozalp, N., Quenching behavior of semilinear heat equations with singular boundary conditions, Electron. J.

Diff. Equ., Vol. 2015, No. 311, 1-13, 2015.

Q. Sheng and A. Q. M. Khaliq, A revisit of the semi-adaptive method for singular degenerate reaction-diffusion equations, East Asia J. Appl. Math., 2 (2012), pp.~185-203.

Strikwerda, John C., Finite difference schemes and partial differential equations, Wadsworth Publ. Co., Belmont, CA, ISBN: 0-534-09984, 112-134, 1989.

Ozalp N, Selcuk B, The quenching behavior of a nonlinear parabolic equation with a singular parabolic with a singular

boundary condition, Hacettepe Journal of Mathematics and Statistics,44: 615--621, 2015.

Z. Wang, J. Yin, and C. Wang, Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary

condition, Applied Mathematics Letters, vol. 20, no. 2, pp. 142--147, 2007.

Y. Zhi, C. Mu, The quenching behavior of a nonlinear parabolic equation with a nonlinear boundary outflux}, Appl. Math.

Comput., 184, 624-630, 2007.