Geodesic
The decay rate of the solution to the Cauchy problem for doubly nonlinear parabolic equation with absorption
Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 12-32 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This work deals with the Cauchy problem for a wide class of quasilinear second-order degenerate parabolic equations with inhomogeneous density and absorption terms. It is well known that for the problem under consideration but without absorption term and when the density tends to zero at infinity not very fast the mass conservation law holds true. However that fact is not always valid with an absorption term. In this paper, the precise conditions on both the structure of nonlinearity and inhomogeneous density which guarantee the decay to zero of the total mass of solution as time goes to infinity is established. In other words the criteria of stabilization to zero of the total mass for a large time is established in terms of critical exponents. As a consequence of obtained results and local Nash-Mozer estimates the sharp sup bound of a solution is done as well. 
@article{VMJ_2020_22_1_a1,
     author = {Z. V. Besaeva and A. F. Tedeev},
     title = {The decay rate of the solution to the {Cauchy} problem for doubly nonlinear parabolic equation with absorption},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {12--32},
     year = {2020},
     volume = {22},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2020_22_1_a1/}
}
TY  - JOUR
AU  - Z. V. Besaeva
AU  - A. F. Tedeev
TI  - The decay rate of the solution to the Cauchy problem for doubly nonlinear parabolic equation with absorption
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2020
SP  - 12
EP  - 32
VL  - 22
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMJ_2020_22_1_a1/
LA  - ru
ID  - VMJ_2020_22_1_a1
ER  - 
%0 Journal Article
%A Z. V. Besaeva
%A A. F. Tedeev
%T The decay rate of the solution to the Cauchy problem for doubly nonlinear parabolic equation with absorption
%J Vladikavkazskij matematičeskij žurnal
%D 2020
%P 12-32
%V 22
%N 1
%U http://geodesic.mathdoc.fr/item/VMJ_2020_22_1_a1/
%G ru
%F VMJ_2020_22_1_a1
Z. V. Besaeva; A. F. Tedeev. The decay rate of the solution to the Cauchy problem for doubly nonlinear parabolic equation with absorption. Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 12-32. http://geodesic.mathdoc.fr/item/VMJ_2020_22_1_a1/

[1] Tedeev A. F., “The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations”, Appl. Anal., 86:6 (2007), 755–782 | DOI | MR | Zbl

[2] Andreucci D., Tedeev A. F., Ughi M., “The Cauchy problem for degenerate parabolic equations with source and damping”, Ukr. Math. Bull., 1:1 (2004), 1–23 | MR

[3] Ben-Artzi B., Koch H., “Decay of mass for a semilinear parabolic equation”, Commun. Partial Differ. Equ., 24:5–6 (1999), 869–881 | DOI | MR | Zbl

[4] Skrypnik I., Tedeev A. F., “Decay of the mass of the solution to Cauchy problem of the degenerate parabolic equation with nonlinear potential”, Complex Var. Elliptic Equ., 63:1 (2018), 90–115 | DOI | MR | Zbl

[5] Kamin S., Rosenau P., “Propagation of thermal waves in an inhomogeneous medium”, Commun. Pure Appl. Math., 34:6 (1981), 831–852 | DOI | MR | Zbl

[6] Kamin S., Rosenau P., “Nonlinear diffusion in finite mass medium”, Commun. Pure Appl. Math., 35:1 (1982), 113–127 | DOI | MR | Zbl

[7] Kamin S., Kersner R., “Disappearance of interfaces in finite time”, Mechanica, 28:2 (1993), 117–120 | DOI | MR | Zbl

[8] Eidus D., Kamin S., “The filtration equation in class of functions decreasing at infinity”, Proc. Amer. Math. Soc., 120:3 (1994), 825–830 | DOI | MR | Zbl

[9] Galaktionov V. A., Kamin S., Kersner R., Vazquez J. L., “Intermediate asymptotics for inhomogeneous nonlinear heat conduction”, J. Math. Sci., 120:3 (2004), 1277–1294 | DOI | MR

[10] Guedda M., Hihorst D., Peletier M. A., “Disappearing interfaces in nonlinear diffussion”, Adv. Math. Sci. Appl., 7:2 (1997), 695–710 | MR | Zbl

[11] Martynenko A. V., Tedeev A. F., “Cauchy Problem for Quasilinear Parabolic Equation with a Source Term and an Inhomogeneous Density”, Computational Mathematics and Mathematical Physics, 47:2 (2007), 238–248 | DOI | MR | Zbl

[12] Martynenko A. V., Tedeev A. F., “On the Behavior of Solutions to the Cauchy Problem for a Degenerate Parabolic Equation with Inhomogeneous Density and a Source”, Computational Mathematics and Mathematical Physics, 48:7 (2008), 1145–1160 | DOI | MR

[13] Reyes G., Vazquez J. L., “The inhomogeneous PME in several space dimensions. existence and uniqueness of finite energy solutions”, Commun. Pure Appl. Anal., 7:6 (2008), 1275–1294 | DOI | MR

[14] Reyes G., Vazquez J. L., “Long time behavior for the inhomogeneous PMI in a medium with slowly decaying density”, Commun. Pure Appl. Anal., 8:2 (2009), 493–508 | DOI | MR | Zbl

[15] Kamin S., Reyes G., Vazquez J. L., “Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density”, Discrete Contin. Dyn. Syst. A, 26:2 (2010), 521–549 | DOI | MR | Zbl

[16] Benachour S., Laurentcot Ph., “Global Solutions to viscous Hamilton–Jacobi equations with irregular initial data”, Commun. Partial Differ. Equ., 24:11–12 (1999), 1999–2021 | DOI | MR | Zbl

[17] Di Benedetto E., Degenerate parabolic equations, Springer–Verlag, New York, 1993, 387 pp. | MR

[18] Andreucci D., Tedeev A. F., “Universal bounds at the blow-up time for nonlinear parabolic equations”, Adv. Differ. Equ., 10:1 (2005), 89–120 | MR | Zbl