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Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions
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 |
| $ u $ |
| $ \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*} $ |
| $ N \ge 1 $ |
| $ 2N/(N+1)<p\neq2 $ |
| $ \alpha \in (-\infty, 1) $ |
| $ \lambda>0 $ |
| $ \varphi\in BC({\bf R}^N)\, \cap\, L^1({\bf R}^N) $ |
| $ \varphi\geq0 $ |
| $ {\bf R}^{N} $ |
| $ u $ |
| $ \zeta' = \zeta^\alpha $ |
| $ (0, \infty) $ |
| $ u $ |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [11] |
S. Kamin,
The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.
doi: 10.1007/BF02761536. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
J. Eom and K. Ishige, Large time behavior of ODE type solutions to nonlinear diffusion equations, Discrete Contin. Dyn. Syst., to appear. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [11] |
S. Kamin,
The asymptotic behavior of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.
doi: 10.1007/BF02761536. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
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