# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2019229

## Large time behavior of ODE type solutions to nonlinear diffusion equations

 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

Received  September 2018 Revised  February 2019 Published  June 2019

Fund Project: 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

Consider the Cauchy problem for a nonlinear diffusion equation
 $$$\left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right.$$$
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.
Citation: Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019229
##### 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.

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##### 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.
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