Abstract: 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.
For citation: Besaeva, Z. V. and Tedeev, A. F. The Decay Rate of the Solution to the Cauchy Problem for Doubly Nonlinear Parabolic Equation with Absortion, Vladikavkaz Math. J., 2020, vol. 22, no. 1, pp. 13-32 (in Russian).
DOI 10.23671/VNC.2020.1.57535
1. Tedeev, A. F. The Interface Blow-up Phenomenon and Local Estimates
for Doubly Degenerate Parabolic Equations, Applicable Analysis, 2007, vol. 86, no. 6, pp. 755-782. DOI: 10.1080/00036810701435711.
2. Andreucci, D., Tedeev, A. F. and Ughi, M. The Cauchy Problem for Degenerate Parabolic Equations with Source and Damping, Ukrainian Mathematical Bulletin, 2004, vol. 1, no. 1, pp. 1-23.
3. Ben-Artzi, B. and Koch, H. Decay of Mass for a Semilinear Parabolic Equation, Communications in Partial Differential Equations, 1999, vol. 24, no. 5-6, pp. 869-881. DOI: 10.1080/03605309908821450.
4. Skrypnik, I. and Tedeev A. F. Decay of the Mass of the Solution to Cauchy Problem of the Degenerate Parabolic Equation with Nonlinear Potential, Complex Variables and Elliptic Equations,
2018, vol. 63, no. 1, pp. 90-115. DOI: 10.1080/17476933.2017.1286331.
5. Kamin, S. and Rosenau, P. Propagation of Thermal Waves in an Inhomogeneous Medium, Communications on Pure and Applied Mathematics, 1981, vol. 34, no. 6, pp. 831-852.
DOI: 10.1002/cpa.3160340605.
6. Kamin, S. and Rosenau, P. Nonlinear Diffusion in Finite Mass Medium,
Communications on Pure and Applied Mathematics,
1982, vol. 35, no. 1, pp. 113-127. DOI: 10.1002/cpa.3160350106.
7. Kamin, S. and Kersner, R. Disappearance of Interfaces in FiniteTime,
Mechanica, 1993, vol. 28, no. 2, pp. 117-120. DOI: 10.1007/BF01020323.
8. Eidus, D. and Kamin, S. The Filtration Equation in Class of Functions Decreasing at Infinity, Proceedings of the American Mathematical Society,
1994, vol. 120, no. 3, pp. 825-830. DOI: 10.1090/S0002-9939-1994-1169025-2.
9. Galaktionov, V. A., Kamin, S., Kersner, R. and Vazquez, J. L.
Intermediate Asymptotics for Inhomogeneous Nonlinear Heat Conduction,
Journal of Mathematical Sciences, 2004, vol. 120, no. 3, pp. 1277-1294.
DOI: 10.1023/B:JOTH.0000016049.94192.aa.
10. Guedda, M., Hihorst, D. and Peletier, M. A. Disappearing Interfaces in Nonlinear Diffussion, Advances in Mathematical Sciences and Applications, 1997, vol. 7, no. 2, pp. 695-710.
11. Martynenko, A. V. and Tedeev, A. F. Cauchy Problem for Quasilinear
Parabolic Equation with a Source Term and an Inhomogeneous Density,
Computational Mathematics and Mathematical Physics,
2007, vol. 47, no. 2, pp. 238-248. DOI: 10.1134/s096554250702008x.
12. Martynenko, A. V. and 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, 2008, vol. 48, no. 7, pp. 1145-1160.
DOI: 10.1134/s0965542508070087.
13. Reyes, G. and Vazquez, J. L. The Inhomogeneous PME in Several Space Dimensions. Existence and Uniqueness of Finite Energy Solutions,
Communications on Pure and Applied Analysis,
2008, vol. 7, no. 6, pp. 1275-1294. DOI: 10.3934/cpaa.2008.7.1275.
14. Reyes, G. and Vazquez, J. L. Long Time Behavior for the Inhomogeneous PMI in a Medium with Slowly Decaying Density,
Communications on Pure and Applied Analysis,
2009, vol. 8, no. 2, pp. 493-508. DOI: 10.3934/cpaa.2009.8.493.
15. Kamin, S., Reyes, G. and Vazquez, J. L. Long Time Behavior for the Inhomogeneous PME in a Medium with Rapidly Decaying Density,
Discrete and Continuous Dynamical Systems-A, 2010, vol. 26, no. 2, pp. 521-549. DOI: 10.3934/dcdc.2010.26.521.
16. Benachour, S. and Laurentcot, Ph. Global Solutions to Viscous Hamilton-Jacobi Equations with Irregular Initial Data,
Communications in Partial Differential Equations,
1999, vol. 24, no. 11-12, pp. 1999-2021. DOI: 10.1080/03605309908821492.
17. Di Benedetto, E. Degenerate Parabolic Equations,
New York, Springer-Verlag, 1993, 387 p.
18. Andreucci, D. and Tedeev, A. F. Universal Bounds at the Blow-Up Time for Nonlinear Parabolic Equations, Advances in Differential Equations, 2005, vol. 10, no. 1, pp. 89-120.
