Abstract: In this paper we study the large time behaviour of the solution and compactification of support to the Cauchy problem for doubly degenerate parabolic equations with strong gradient damping. Under the suitable assumptions on the structure of the equation and data of the problem we establish new sharp bound of solutions for a large time. Moreover, when the support of initial datum is compact we prove that the support of the solution contains in the ball with radius which is independent in time variable. In the critical case of the behaviour of the damping term the support of the solution depends on time variable logarithmically for a sufficiently large time. The main tool of the proof is based on nontrivial use of cylindrical Gagliardo-Nirenberg type embeddings and recursive inequalities. The sup-norm estimates of the solution is carried out by modified version of the classical method of De-Giorgi-Ladyzhenskaya-Uraltseva-DiBenedetto. The approach of the paper is flexible enough and can be used when studying the Cauchy-Dirichlet or Cauchy-Neumann problems in domains with non compact boundaries.
Keywords: doubly degenerate parabolic equations, strong gradient damping, finite speed of propagation, large time behavior
For citation: Tedeev, Al. F. and Tedeev, An. F. Large Time Decay Estimates of the Solution to the Cauchy Problem of Doubly Degenerate Parabolic Equations with Damping, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 93-104. DOI 10.46698/t4621-4848-0414-e
1. Antontsev, S. N., Diaz, J. I. and Shmarev, S. I. Energy Methods for Free
Boundary Problems: Applications to Non-Linear PDEs and Fluid Mechanics,
Progress in Nonlinear Differential Equations and
Their Applications, vol. 48, Boston, Bikhauser, 2002.
2. Laurencot, Ph. and Vazquez, J. L. Localized non-Diffusive Asymptotic Patterns
for Nonlinear Parabolic Equations with Gradient Absorptionm, Journal of Dynamics
and Differential Equations, 2007, vol. 19, no. 4, pp. 985-1005. DOI: 10.1007/s10884-007-9093-y.
3. Andreucci, D. Degenerate Parabolic Equations with Initial Data Measure,
Transactions of the American Mathematical Society, 1997, vol. 349, no. 10, pp. 3911-3923.
4. Deng, L. and Shang, X. Doubly Degenerate Parabolic Equation with Time Gradient
Source and Initial Data Measure, Hindawi Journal of Function Spaces, 2020, pp. 1-11.
DOI: 10.1155/2020/1864087.
5. Benachour, S., Roynette, B. and Vallois, P. Asymptotic Estimates of
Solutions of \(u_{t}+\Delta u=-\vert \nabla u\vert\), Journal of Functional Analysis,
1997, vol. 144, no. 2, pp. 301-324. DOI: 10.1006/jfan.1996.2984.
6. Benachour, S. and Laurencot, 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.
7. Benachour, S., Laurencot, Ph. and Schmitt, D. Extinction and Decay
Estimates for Viscous Hamilton-Jacobi Equations in \(\mathbb{R^N}\), Proceedings
of the American Mathematical Society, 2001, vol. 130, no. 4, pp. 1103-1111.
DOI: 10.1090/S0002-9939-01-06140-8.
8. Ben-Artzi, M., Souplet, Ph. and Weissler, F. B. The Local Theory for Viscous
Hamilton-Jacobi Equations in Lebesgue Spaces, Journal de Mathematiques Pures et Appliques,
2002, vol. 81, no. 4, pp. 343-378. DOI: 10.1016/S0021-7824(01)01243-0.
9. Benachour, S., Laurencot, Ph., Schmitt, D. and Souplet, Ph. Extinction and
Nonextinction for Viscous Hamilton-Jacobi Equations in \(\mathbb{R^N}\),
Asymptotic Analysis, 2002, vol. 31, no. 3-4, pp. 229-246.
10. Gilding, B. H., Guedda, M. and Kersner, R. The Cauchy Problem for \(u_{t}=\Delta u+\vert \nabla u\vert ^{q}\), Journal of Mathematical Analysis and Applications, 2003,
vol. 284, no. 2, pp. 733-755. DOI: 10.1016/S0022-247X(03)00395-0.
11. Andreucci, D., Tedeev, A. F. and Ughi, M. The Cauchy Problem for
Degenerate Parabolic Equations with Source and Damping, Ukrainian Mathematical
Bulletin, 2004, no. 1, pp. 1-23.
12. Benachour, S., Karch, G. and Laurencot, Ph. Asymptotic Profiles of
Solutions to Viscous Hamilton-Jacobi Equations, Journal de Mathematiques Pures
et Appliques, 2004, vol. 83, no. 10, pp. 1275-1308. DOI: 10.1016/j.matpur.2004.03.002.
13. Biler, P., Guedda, M. and Karch, G. Asymptotic Properties of Solutions of
the Viscous Hamilton-Jacobi Equation, Journal of Evolution Equations, 2004,
vol. 4, pp. 75-97. DOI: 10.1007/s00028-003-0079-x.
14. Gilding, B. H. The Cauchy Problem for \(u_{t}=\Delta u+\left\vert
\nabla u\right\vert ^{q}\), Large-Time Behaviour, Journal de Mathematiques Pures et Appliques,
2005, vol. 84, no. 6, pp. 753-785. DOI: 10.1016/j.matpur.2004.11.003.
15. Gallay, Th. and Laurencot, Ph. Asymptotic Behavior for a Viscous
Hamilton-Jacobi Equation with Critical Exponent, Indiana University Mathematics Journal,
2007, vol. 56, pp. 459-479.
16. Iagar, R. and Laurencot, Ph. Positivity, Decay and Extinction for a
Singular Diffusion Equation with Gradient Absorption, Journal of Functional Analysis,
2012, vol. 262, no. 7, pp. 3186-3239. DOI: 10.1016/j.jfa.2012.01.013.
17. Bidaut-Veron, M.-F. and Dao, N. A. \(L_{\infty}\) Estimates and Uniqueness
Results for Nonlinear Parabolic Equations with Gradient Absorption Terms,
Nonlinear Analysis, 2013, vol. 91, pp. 121-152. DOI: 10.48550/arXiv.1202.2674.
18. Attouchi, A. Gradient Estimate and a Liouville Theorem for a \(P\)-Laplacian
Evolution Equation with a Gradient Nonlinearity, Differential and Integral Equations,
2016, vol. 29, no. 1-2, pp. 137-150. DOI: 10.48550/arXiv.1405.5896.
19. Iagar, R. G., Laurencot, Ph. and Stinner, Ch. Instantaneous Shrinking and
Single Point Extinction for Viscous Hamilton-Jacobi Equations with Fast
Diffusion, Mathematische Annalen, Springer-Verlag, 2017, vol. 368, pp. 65-109. DOI: 10.48550/arXiv.1510.00500.
20. Andreucci, D. and Tedeev, A. F. A Fujita Type Result for a Degenerate
Neumann Problem in Domains with Noncompact Boundary, Journal of Mathematical
Analysis and Applications, 1999, vol. 231, no. 2, pp. 543-567.
21. Andreucci, D. and Tedeev, A. F. Finite Speed of Propagation for the
Thin-Film Equation and other Higher-Order Parabolic Equations with General
Nonlinearity, Interfaces Free Bound, 2001, vol. 3, no. 3, pp. 233-264. DOI: 10.4171/IFB/40.
22. Andreucci, D. and Tedeev, A. F. Universal Bounds at the Blow-up Time for
Nonlinear Parabolic Equations, Advances in Difference Equations, 2005,
vol. 10, no. 1, pp. 89-120. DOI: 10.57262/ade/1355867897.
23. Ladyzhenskaja, O., Solonnikov, V. A. and Uralceva, N. V. Linear and
Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
