Abstract: In this paper we study the large time behaviour for solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density. Under the suitable assumptions on the data of the problem and on the behaviour of the density at infinity we establish new sharp bound of solutions for a large time. One of the main tool of the proof is new weighted embedding result which is of independent interest. In addition, the proof of uniform estimates of the solution is carried out by modified version of the classical method of De-Giorgi-Ladyzhenskaya-Uraltseva-DiBenedetto. Similar results in the case of power-like density was obtained by one of the author . The approach of this work can be applied for example when studying the qualitative properties of solutions to the Neumann problem for a doubly nonlinear parabolic equation with inhomogeneous density in domains with non-compact boundaries.
Keywords: degenerate parabolic equation, inhomogeneous density, weighted embedding, large time behavior
For citation: Dzagoeva, L. F. and Tedeev, A. F. Asymptotic Behavior of the Solution of Doubly Degenerate Parabolic
Equations with Inhomogeneous Density, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 78-86. DOI 10.46698/p6936-3163-2954-s
1. Kamin, S. and Rosenau, P. Non-Linear Diffusion in Finite Mass Medium,
Communications on Pure Applied Mathematics, 1982, vol. 35, no. 1, pp. 113-127.
2. 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.
3. 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.
4. Guedda, M., Hihorst, D. and Peletier, M. A. Disappearing Interfaces
in Nonlinear Diffusion, Advances in Mathematical Sciences and Applications,
1997, vol. 7, pp. 695-710.
5. Reyes, G. and Vazquez, J. L. The Cauchy Problem for the Inhomogeneous
Porous Medium Equation, Networks and Heterogeneous Media,
2006, vol. 1,† no. 2, pp. 337-351. DOI: 10.3934/nhm.2006.1.337.
6. Reyes, G. and Vazquez, J. L. Long Time Behavior for the Inhomogeneous PME 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.
7. 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,
2010, vol. 26,† no. 2, pp. 521-549. DOI: 10.3934/dcds.2010.26.521.
8. Kamin, S. and Kersner, R. Disappearance of Interfaces in Finite Time,
Meccanica, 1993, vol. 28, no. 2, pp. 117-120. DOI: 10.1007/BF01020323.
9. Tedeev, A. F. Conditions for the Time Global Existence and Nonexistence
of a Compact Support of Solutions to the Cauchy Problem for Quasilinear Parabolic Equations,
Siberian Mathematical Journal, 2004, vol. 45,† no. 1, pp. 155-164.
10. 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.
11. Martynenko, A. V. and Tedeev, A. F. On the Behaviour of Solutions
to the Cauchy Problem for a Degenerate Parabolic Equation with Inhomogeneous
Density And a Sources, Computational Mathematics and Mathematical Physics,
2008, vol. 48, no. 7, pp. 1145-1160. DOI: 10.1134/S0965542508070087.
12. Andreucci, D., Cirmi, G. R., Leonardi, S. and Tedeev, A. F.
Large Time Behavior of Solutions to the Neumann Problem for
a Quasilinear Second Order Degenerate Parabolic Equation
in Domains with Noncompact Boundary, Journal of Differential Equations,
2001, vol. 174, no. 2, pp. 253-288.
13. Kalashnikov, A. S. Some Problems of the Qualitative Theory of Non-Linear
Degenerate Second-Order Parabolic Equations, Russian Mathematical Surveys,
1987, vol. 42, no. 2, pp. 169-222. DOI: 10.1070/RM1987v042n02ABEH001309.
14. Caffarelli, L., Kohn, R. and Nirenberg, L. First Order Interpolation
Inequalities with Weights, Composito Mathematica, 1984, vol. 53, no. 3, pp. 259-275.
15. Di Benedetto, E. and Herrero, M. A. On the Cauchy Problem and Initial
Traces for a Degenerate Parabolic Equation, Transactions of the American Mathematical Society,
1989, vol. 314, no. 1, pp. 187-224. DOI: 10.2307/2001442.
16. 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.
17. Andreucci, D. and Tedeev, A. F. Optimal Decay Rate for Degenerate
Parabolic Equations on Noncompact Manifolds, Methods and Applications of Analysis,
2015, vol. 22, no. 4, pp. 359-376. DOI: 10.4310/MAA.2015.v22.n4.a2
18. Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural'ceva, N. N.
†Linear and Quasi-Linear Equations of Parabolic Type,
Translations of Mathematical Monographs, vol. 23,† Providence,
R.I., American Mathematical Society, 1968.