Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
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DOI：
10.1051/m2an:2006025
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发表时间：
2006-05
期刊：
影响因子：
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通讯作者：
S. Luckhaus;Y. Sugiyama
中科院分区：
文献类型：
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作者：
S. Luckhaus;Y. Sugiyama
我们考虑下面的反应扩散方程：其中。在[杉山，非线性分析。63（2005）1051-1062; Submitted; J. Differential Equations（in press）]的情况下，上述问题（KS）对于“小数据”在时间上是全局可解的。此外，还证明了解（u，v）在中的衰减性.本文考虑了具有任意固定的“和小数据”的情形，证明了（i）（KS）存在时间整体解（u，v），并且当t趋于∞时，解（u，v）衰减到0;（ii）（KS）中第一个方程的解u，当t趋于∞时，渐近地表现为Barenblatt解，其中Barenblatt解是m>1的多孔介质方程的精确解（具有自相似性）。
We consider the following reaction-diffusion equation: where . In [Sugiyama, Nonlinear Anal. 63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of , the above problem (KS) is solvable globally in time for “small data”. Moreover, the decay of the solution (u,v) in was proved. In this paper, we consider the case of “ and small data” with any fixed and show that (i) there exists a time global solution (u,v ) of (KS) and it decays to 0 as t tends to ∞ and (ii) a solution u of the first equation in (KS) behaves like the Barenblatt solution asymptotically as t tends to ∞ , where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation with m>1 .