American Institute of Mathematical Sciences

Advanced
November  2009, 8(6): 1917-1924. doi: 10.3934/cpaa.2009.8.1917

Stabilization towards the steady state for a viscous Hamilton-Jacobi equation

Yuxiang Li 1,

1. 

Department of Mathematics, Southeast University, Nanjing 210096, China

Received  November 2008 Revised  April 2009 Published  August 2009

In this short paper, we obtain the asymptotic behavior of the global solutions of a viscous Hamilton-Jacobi equation $u_t=\Delta u+|\nabla u|^p$ in $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M$ on $\partial B_R$. It is proved that there exists a constant $M_c>0$ such that the problem admits a unique steady state if and only if $M\leq M_c$. When $M < M_c$, the global solution converges in $C^1(\overline{B_{r,R}})$ to the unique regular steady state. When $M=M_c$, the global solution converges in $C(\overline{B_{r,R}})$ to the unique singular steady state, and the grow-up rate of $||u_\nu(t)||_{L^\infty(\partial B_r)}$ in infinite time is obtained.
Keywords: gradient blow-up., Viscous Hamilton-Jacobi equations, asymptotic behavior.
Mathematics Subject Classification: Primary: 35K60, 35K65, 35B4.
Citation: Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917
[1]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[2]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[3]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[4]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007
[5]

Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223
[6]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i
[7]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[8]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[9]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[10]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
[11]

Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647
[12]

Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[13]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447
[14]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[15]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167
[16]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
[17]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[18]

Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225
[19]

Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007
[20]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences