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Monotonicity with respect to $p$ of the First Nontrivial Eigenvalue of the $p$-Laplacian with Homogeneous Neumann Boundary Conditions
September  2020, 19(9): 4373-4386. doi: 10.3934/cpaa.2020199

## Large time behavior of ODE type solutions to parabolic $p$-Laplacian type equations

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

* Corresponding author

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: The second author was partially supported by Grant-in-Aid for Early-Career Scientists JSPS KAKENHI Grant Number 18K13435

Let
 $u$
be a solution to the Cauchy problem for a nonlinear diffusion equation
 $\begin{equation*} \begin{cases} \partial_t u = \mathrm{div}\, (|\nabla u|^{p-2} \nabla u) + u^\alpha & \quad\mathrm{in}\quad{\bf R}^N\times(0, \infty), \\ u(x, 0) = \lambda+\varphi(x) & \quad\mathrm{in}\quad{\bf R}^N, \end{cases} \end{equation*}$
where
 $N \ge 1$
,
 $2N/(N+1) , $ \alpha \in (-\infty, 1) $, $ \lambda>0 $and $ \varphi\in BC({\bf R}^N)\, \cap\, L^1({\bf R}^N) $with $ \varphi\geq0 $in $ {\bf R}^{N} $. Then the solution $ u $behaves like a positive solution to ODE $ \zeta' = \zeta^\alpha $in $ (0, \infty) $. In this paper we show that the large time behavior of the solution $ u $is described by a rescaled Barenblatt solution. Citation: Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic$ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199 ##### References:  [1] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. Google Scholar [2] E. DiBenedetto and A. Friedman, Addendum to Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 363 (1985), 217-220. doi: 10.1515/crll.1985.363.217. Google Scholar [3] E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.2307/2001442. Google Scholar [4] E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution$p$-Laplacian equation. Initial traces and Cauchy problem when$1 < p < 2$, Arch. Ration. Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111. Google Scholar [5] J. Eom and K. Ishige, Large time behavior of ODE type solutions to a nonlinear parabolic system, Nonlinear Anal., 191 (2020), 19 pp. doi: 10.1016/j.na.2019.111631. Google Scholar [6] J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. Google Scholar [7] 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 [8] A. Gmira and L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in$\mathbf{R}^{N}$, J. Differ. Equ., 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1. Google Scholar [9] M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math., 3 (1981), 113-127. Google Scholar [10] K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differ. Equ., 254 (2013), 1247-1268. doi: 10.1016/j.jde.2012.10.014. Google Scholar [11] S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87. doi: 10.1007/BF02761536. Google Scholar [12] S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12 (1985), 393-408. Google Scholar [13] 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 [14] S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the$p$- Laplacian equation, Rev. Mat. Iberoam., 4 (1988), 339-354. doi: 10.4171/RMI/77. Google Scholar [15] L. A. Peletier and J. N. Zhao, Source-type solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 14 (1990), 107-121. doi: 10.1016/0362-546X(90)90018-C. Google Scholar [16] J. N. Zhao, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differ. Equ., 102 (1993), 33-52. doi: 10.1006/jdeq.1993.1020. Google Scholar show all references ##### References:  [1] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. Google Scholar [2] E. DiBenedetto and A. Friedman, Addendum to Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 363 (1985), 217-220. doi: 10.1515/crll.1985.363.217. Google Scholar [3] E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.2307/2001442. Google Scholar [4] E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution$p$-Laplacian equation. Initial traces and Cauchy problem when$1 < p < 2$, Arch. Ration. Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111. Google Scholar [5] J. Eom and K. Ishige, Large time behavior of ODE type solutions to a nonlinear parabolic system, Nonlinear Anal., 191 (2020), 19 pp. doi: 10.1016/j.na.2019.111631. Google Scholar [6] J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. Google Scholar [7] 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 [8] A. Gmira and L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in$\mathbf{R}^{N}$, J. Differ. Equ., 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1. Google Scholar [9] M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math., 3 (1981), 113-127. Google Scholar [10] K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differ. Equ., 254 (2013), 1247-1268. doi: 10.1016/j.jde.2012.10.014. Google Scholar [11] S. Kamin, The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87. doi: 10.1007/BF02761536. Google Scholar [12] S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12 (1985), 393-408. Google Scholar [13] 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 [14] S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the$p$- Laplacian equation, Rev. Mat. Iberoam., 4 (1988), 339-354. doi: 10.4171/RMI/77. Google Scholar [15] L. A. Peletier and J. N. Zhao, Source-type solutions of the porous media equation with absorption: the fast diffusion case, Nonlinear Anal., 14 (1990), 107-121. doi: 10.1016/0362-546X(90)90018-C. Google Scholar [16] J. N. Zhao, The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation, J. Differ. Equ., 102 (1993), 33-52. doi: 10.1006/jdeq.1993.1020. Google Scholar  [1] Junyong Eom, Kazuhiro Ishige. 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