Large time behavior of ODE type solutions to nonlinear diffusion equations

Junyong Eom; Kazuhiro Ishige

doi:10.3934/dcds.2019229

Article Contents

2020, Volume 40, Issue 6: 3395-3409. Doi: 10.3934/dcds.2019229 open access

This issue Previous Article Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity Next Article Asymptotic population abundance of a two-patch system with asymmetric diffusion

Large time behavior of ODE type solutions to nonlinear diffusion equations

Junyong Eom^1, and
Kazuhiro Ishige^2, ,

1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
2.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

^* Corresponding author: Kazuhiro Ishige
^* Corresponding author: Kazuhiro Ishige

Received: August 31, 2018

Revised: January 31, 2019

Published: March 2020

The second author of this paper was supported in part by the Grant-in-Aid for Scientific Research (A)(No. 15H02058) from Japan Society for the Promotion of Science

Abstract Full Text(HTML) Related Papers Cited by

Abstract

Consider the Cauchy problem for a nonlinear diffusion equation

$ \begin{equation} \left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} $

where $ m>0 $, $ \alpha\in(-\infty,1) $, $ \lambda>0 $ and $ \varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N) $ with $ 1\le r<\infty $ and $ \inf_{x\in{\bf R}^N}\varphi(x)>-\lambda $. Then the positive solution to problem (P) behaves like a positive solution to ODE $ \zeta' = \zeta^\alpha $ in $ (0,\infty) $ and it tends to $ +\infty $ as $ t\to\infty $. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.

Keywords:

Mathematics Subject Classification: Primary: 35B40, 35K55.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	J. Aguirre and M. A. Escobedo, Cauchy problem for $u_t - \Delta u = u^p$ with $0 < p < 1$, Asymptotic behaviour of solutions, Ann. Fac, Sci. Toulouse Math., 8 (1986/87), 175-203.
[2]	D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.
[3]	J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142. doi: 10.1512/iumj.2000.49.1756.
[4]	A. Friedman and S. Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. doi: 10.2307/1999846.
[5]	S. Kamin and L. A. Peletier, Source-type solutions of degenerate diffusion equations with absorption, Israel J. Math., 50 (1985), 219-230. doi: 10.1007/BF02761403.
[6]	S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146. doi: 10.1007/BF02801989.
[7]	K. Ishige, M. Ishiwata and and T. Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J., 58 (2009), 2673-2708. doi: 10.1512/iumj.2009.58.3771.
[8]	K. Ishige and T. Kawakami, Refined asymptotic profiles for a semilinear heat equation, Math. Ann., 353 (2012), 161-192. doi: 10.1007/s00208-011-0677-9.
[9]	K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math., 121 (2013), 317-351. doi: 10.1007/s11854-013-0038-6.
[10]	K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel, J. Evol. Eqn., 14 (2014), 749-777. doi: 10.1007/s00028-014-0237-3.
[11]	K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differential Equations, 254 (2013), 1247-1268. doi: 10.1016/j.jde.2012.10.014.
[12]	K. Ishige, T. Kawakami and H. Michihisa, Asymptotic expansions of solutions of fractional diffusion equations, SIAM J. Math. Anal., 49 (2017), 2167-2190. doi: 10.1137/16M1101428.
[13]	R. Kajikiya, Stability and instability of stationary solutions for sublinear parabolic equations, J. Differential Equations, 264 (2018), 786-834. doi: 10.1016/j.jde.2017.09.023.
[14]	T. Kawanago, Existence and behaviour of solutions for $u_t = \Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.
[15]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968.
[16]	L. A. Peletier and J. Zhao, Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 17 (1991), 991-1009. doi: 10.1016/0362-546X(91)90059-A.
[17]	R. Suzuki, Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity, J. Differential Equations, 190 (2003), 150-181. doi: 10.1016/S0022-0396(02)00086-4.
[18]	N. Umeda, Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372. doi: 10.3836/tjm/1244208595.
[19]	J. L. Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space, J. Evol. Eqn., 3 (2003), 67-118. doi: 10.1007/s000280300004.
[20]	J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
[21]	L. Wang and J. Yin, Grow-up rate of solutions for the heat equation with a sublinear source, Bound. Value Probl., 96 (2012), 14 pp. doi: 10.1186/1687-2770-2012-96.