# 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] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [2] Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009 [3] Mamoru Okamoto. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. Evolution Equations & Control Theory, 2019, 8 (3) : 567-601. doi: 10.3934/eect.2019027 [4] Takanobu Okazaki. Large time behaviour of solutions of nonlinear ode describing hysteresis. Conference Publications, 2007, 2007 (Special) : 804-813. doi: 10.3934/proc.2007.2007.804 [5] Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41 [6] Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 [7] Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383 [8] Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks & Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749 [9] Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027 [10] Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 [11] Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 [12] Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176 [13] Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767 [14] Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020247 [15] Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47 [16] Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012 [17] Naoki Fujino, Mitsuru Yamazaki. Burgers' type equation with vanishing higher order. Communications on Pure & Applied Analysis, 2007, 6 (2) : 505-520. doi: 10.3934/cpaa.2007.6.505 [18] Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087 [19] Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211 [20] Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590

2018 Impact Factor: 1.143

Article outline