Skip to main content
×
Home
    • Aa
    • Aa

Evolution of interfaces for the non-linear parabolic p-Laplacian type reaction–diffusion equations

  • UGUR G. ABDULLA (a1) and ROQIA JELI (a1)
Abstract

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction–diffusion equation of non-Newtonian elastic filtration: $$\begin{equation*} u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0. \end{equation*}$$ The interface may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In all cases, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. The methods of the proof are based on non-linear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves.

Copyright
References
Hide All
[1] AbdullaU. G. & KingJ. R. (2000) Interface development and local solutions to reaction-diffusion equations. SIAM J. Math. Anal. 32 (4), 235260.
[2] AbdullaU. G. (2002) Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption. Nonlinear Anal.: Theory, Methods Appl. 50 (4), 541560.
[3] AbdullaU. G. (2000) Reaction-diffusion in irregular domains. J. Differ. Equ. 164, 321354.
[4] AbdullaU. G. (2000) Reaction-diffusion in a closed domain formed by irregular curves. J. Math. Anal. Appl. 246, 480492.
[5] AbdullaU. G. (2001) On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains. J. Math. Anal. Appl. 260 (2), 384403.
[6] AbdullaU. G. (2005) Well-posedness of the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains. Trans. Amer. Math. Soc. 357 (1), 247265.
[7] AbdullaU. G. (2007) Reaction-diffusion in nonsmooth and closed domains. Boundary Value Problems, 2007, 28 (Special issue: Harnack Estimates, Positivity and Local Behaviour of Degenerate and Singular Parabolic Equations).
[8] AntontsevS. N., DazJ. I. & ShmarevS. (2002) Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Applications, Vol. 48, Birkhauser Boston, Inc., Boston.
[9] BarenblattG. I. (1952) On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mech. 16, 6778.
[10] BarenblattG. I. (1996) Scaling, Self-similarity, and Intermediate Asymptotics. With a Foreword by Zeldovich Ya. B., Cambridge Texts in Applied Mathematics, Vol. 14, Cambridge University Press, Cambridge, pp. xxii+386.
[11] DegtyarevS. P. & TedeevA. F. (2012) On the solvability of the Cauchy problem with growing initial data for a class of anisotropic parabolic equations. J. Math. Sci. 181 (1), 2846.
[12] DiBenedettoE. (1993) Degenerate Parabolic Equations, Universitext, Springer Verlag, New York.
[13] DiBenedettoE. (1986) On the local behaviour of solutions of degenerate parabolic equtions with measurable coefficients. Ann. Sc. Norm. Sup. Pisa Cl. Sci. Ser IV XIII (3), 487535.
[14] DiBenedettoE. & HerreroM. A. (1989) On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. AMS 314, 187224.
[15] EstebanJ. R. & VazquezJ. L. (1986) On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal.: Theory, Methods Appl. 10 (11), 13031325.
[16] GrundyR. E. & PeletierL. A. (1987) Short time behaviour of a singular solution to the heat equation with absorption. Proc. R. Soc. Edinburgh: Section A Math. 107 (3–4), 271288.
[17] GrundyR. E. & PeletierL. A. (1990) The initial interface development for a reaction-diffusion equation with power-law initial data. Q. J. Mech. Appl. Math. 43 (4), 535559.
[18] HerreroM. A. & VazquezJ. L. (1982) On the propagation properties of a nonlinear degenerate parabolic equation. Commun. Partial Differ. Equ. 7 (12), 13811402.
[19] IshigeK. (1996) On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation. SIAM J. Math. Anal. 27 (5), 12351260.
[20] KalashnikovA. S. (1987) Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russ. Math. Surv. 42 (2), 169222.
[21] KalashnikovA. S. (1978) On a nonlinear equation appearing in the theory of non-stationary filtration. Trud. Semin. I.G.Pertovski 4, 137146 (in Russian).
[22] KalashnikovA. S. (1982) On the propagation of perturbations in the first boundary value problem of a doubly-nonlinear degenerate parabolic equation. Trud. Semin. I.G.Pertovski 8, 128134 (in Russian).
[23] KingJ. R. (1995) Development of singularities in some moving boundary problems. Euro. J. Appl. Math. 6, 491507.
[24] LiZ., DuW. & MuC. (2013) Travelling-wave solutions and interfaces for non-Newtonian diffusion equations with strong absorption. J. Math. Res. Appl. 33 (4), 451462.
[25] ShmarevS., VdovinV. & VlasovA. (2015) Interfaces in diffusion-absorption processes in nonhomogeneous media. Math. Comput. Simul. 118, 360378.
[26] TsutsumiM. (1988) On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J. Math. Anal. Appl. 132 (1), 187212.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 20 *
Loading metrics...

Abstract views

Total abstract views: 214 *
Loading metrics...

* Views captured on Cambridge Core between 13th December 2016 - 22nd October 2017. This data will be updated every 24 hours.