-
Previous Article
Long-time behaviour of doubly nonlinear parabolic equations
- CPAA Home
- This Issue
-
Next Article
Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model
Asymptotic behavior of solutions for some semilinear heat equations in $R^N$
1. | Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan, Japan |
$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$
$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$
where $N \geq 1$, $f \in C^1([0,\infty))$, and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$. We study the asymptotic behavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$, by using the relative entropy methods.
[1] |
Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643 |
[2] |
Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic and Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003 |
[3] |
Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483 |
[4] |
Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027 |
[5] |
Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 |
[6] |
Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations and Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423 |
[7] |
Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160 |
[8] |
Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59 |
[9] |
Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure and Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197 |
[10] |
Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic and Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701 |
[11] |
Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems and Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008 |
[12] |
V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 |
[13] |
Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012 |
[14] |
Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60 |
[15] |
Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 |
[16] |
Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063 |
[17] |
Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149 |
[18] |
Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 |
[19] |
Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307 |
[20] |
Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]