On stability of two degenerate reaction–diffusion systems
In this paper, we construct a partially degenerate reaction–diffusion equation subject to the Neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable. In this way, we solve an open problem proposed by Casten and Holland (1977) [...
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| Published in | Journal of mathematical analysis and applications Vol. 390; no. 1; pp. 126 - 135 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.06.2012
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| ISSN | 0022-247X 1096-0813 |
| DOI | 10.1016/j.jmaa.2012.01.032 |
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| Abstract | In this paper, we construct a partially degenerate reaction–diffusion equation subject to the Neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable. In this way, we solve an open problem proposed by Casten and Holland (1977) [4]. Moreover, we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross-diffusion effects, which cannot be determined by a simple spectral analysis based on the well developed semigroup theory. |
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| AbstractList | In this paper, we construct a partially degenerate reaction–diffusion equation subject to the Neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable. In this way, we solve an open problem proposed by Casten and Holland (1977) [4]. Moreover, we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross-diffusion effects, which cannot be determined by a simple spectral analysis based on the well developed semigroup theory. |
| Author | Xu, Chuang Wei, Junjie |
| Author_xml | – sequence: 1 givenname: Chuang surname: Xu fullname: Xu, Chuang organization: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China – sequence: 2 givenname: Junjie surname: Wei fullname: Wei, Junjie email: weijj@hit.edu.cn organization: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China |
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| Cites_doi | 10.1016/S0304-3800(01)00255-1 10.1016/0022-0396(78)90033-5 10.1103/PhysRevE.78.026116 10.1007/BF00163879 10.1016/j.jmaa.2008.01.089 10.3934/dcds.1998.4.193 10.1016/j.cam.2008.10.064 10.1016/j.cnsns.2010.08.035 10.1007/s00285-006-0013-2 10.1006/jdeq.1996.0157 10.1016/j.jtbi.2009.01.001 10.1016/j.na.2010.09.035 10.57262/die/1371586185 10.1080/17513750802491849 10.1016/j.physd.2010.07.014 10.1016/j.physd.2004.10.001 10.1006/jdeq.1998.3559 10.1007/s11425-010-0141-4 10.1016/j.nonrwa.2007.06.001 10.1017/S0308210506001259 10.1016/j.nonrwa.2009.01.043 10.1098/rstb.1952.0012 10.1103/PhysRevE.77.036219 10.1016/j.mbs.2005.10.004 10.1016/0022-5193(79)90258-3 10.1016/j.nonrwa.2008.01.011 10.1016/j.mbs.2007.05.004 10.1137/0133023 10.1016/j.physleta.2007.04.114 10.1007/s10483-009-0607-x 10.1016/j.ecolmodel.2010.08.020 10.4134/CKMS.2008.23.2.211 10.32917/hmj/1206392779 10.1002/mana.200510624 10.1006/jmaa.2000.7220 10.1016/S0167-2789(02)00494-3 10.1007/s10884-009-9152-7 |
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| Keywords | Exponential asymptotic stability Partially degeneracy Asymptotic stability Reaction–diffusion equation Cross-diffusion effect Lugiato–Lefever equation |
| Language | English |
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| SubjectTerms | Asymptotic stability Cross-diffusion effect Exponential asymptotic stability Lugiato–Lefever equation Partially degeneracy Reaction–diffusion equation |
| Title | On stability of two degenerate reaction–diffusion systems |
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