Rate of approach to the steady state for a diffusion-convection equation on annular domains
In this paper, we study the asymptotic behavior of global solutions of the equation $u_t=\Delta u+e^{|\nabla u|}$ in the annulus $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M\geq 0$ on $\partial B_R$. It is proved that there exists a constant $M_c>0$ such that the problem admits a unique...
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| Format: | Article |
| Language: | English |
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University of Szeged
2012-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1470 |
| _version_ | 1797830628209590272 |
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| author | Liping Zhu Zhengce Zhang |
| author_facet | Liping Zhu Zhengce Zhang |
| author_sort | Liping Zhu |
| collection | DOAJ |
| description | In this paper, we study the asymptotic behavior of global solutions of the equation $u_t=\Delta u+e^{|\nabla u|}$ in the annulus $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M\geq 0$ 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 blowup rate in infinite time is obtained. |
| first_indexed | 2024-04-09T13:40:12Z |
| format | Article |
| id | doaj.art-23c2f09fc08d4758af335a995470e670 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:12Z |
| publishDate | 2012-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-23c2f09fc08d4758af335a995470e6702023-05-09T07:53:02ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752012-04-0120123911010.14232/ejqtde.2012.1.391470Rate of approach to the steady state for a diffusion-convection equation on annular domainsLiping Zhu0Zhengce Zhang1Xi'an University of Architecture & Technology, Xi'an, P. R. ChinaXi'an Jiaotong University, Xi'an, P. R. ChinaIn this paper, we study the asymptotic behavior of global solutions of the equation $u_t=\Delta u+e^{|\nabla u|}$ in the annulus $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M\geq 0$ 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 blowup rate in infinite time is obtained.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1470convergencesteady stategradient blowup |
| spellingShingle | Liping Zhu Zhengce Zhang Rate of approach to the steady state for a diffusion-convection equation on annular domains Electronic Journal of Qualitative Theory of Differential Equations convergence steady state gradient blowup |
| title | Rate of approach to the steady state for a diffusion-convection equation on annular domains |
| title_full | Rate of approach to the steady state for a diffusion-convection equation on annular domains |
| title_fullStr | Rate of approach to the steady state for a diffusion-convection equation on annular domains |
| title_full_unstemmed | Rate of approach to the steady state for a diffusion-convection equation on annular domains |
| title_short | Rate of approach to the steady state for a diffusion-convection equation on annular domains |
| title_sort | rate of approach to the steady state for a diffusion convection equation on annular domains |
| topic | convergence steady state gradient blowup |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1470 |
| work_keys_str_mv | AT lipingzhu rateofapproachtothesteadystateforadiffusionconvectionequationonannulardomains AT zhengcezhang rateofapproachtothesteadystateforadiffusionconvectionequationonannulardomains |