# American Institute of Mathematical Sciences

June  2020, 40(6): 3395-3409. 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, 2020, 40 (6) : 3395-3409. 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.  Google Scholar [2] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] T. Kawanago, Existence and behaviour of solutions for $u_t = \Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [2] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] T. Kawanago, Existence and behaviour of solutions for $u_t = \Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar [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.  Google Scholar
 [1] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [2] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [3] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [4] Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $-\Delta u = \lambda f(u)$ as $\lambda\to-\infty$. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293 [5] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [6] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [7] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [8] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [9] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [10] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [11] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [12] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354 [13] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [14] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [15] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [16] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [17] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [18] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [19] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [20] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

2019 Impact Factor: 1.338