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Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems. (English) Zbl 1113.35028

Global existence and large time asymptotics of non-negative and integrable solutions are studied for the parabolic-elliptic system
\[ \begin{align*}{ \partial_t u & = \nabla\cdot \left( \nabla u^m - u^{q-1} \nabla v \right) \;\;\text{ in }\;\; (0,\infty)\times {\Bbb R}^N\,, \cr 0 & = \Delta v - v + u \;\;\text{ in }\;\; (0,\infty)\times{\Bbb R}^N\,. }\end{align*} \]
This system is an extended version of the simplified Keller-Segel model for chemotaxis which corresponds to the parameters \(m=1\) and \(q=2\). Assuming that \(m>1\), \(q\geq 2\), and \(q>m+(2/N)\), the global existence of a solution \((u,v)\) is established provided the \(L^{N(q-m)/2}\)-norm of the initial condition is sufficiently small. The solution \((u,v)\) thus obtained decays in \(L^p({\mathbb R}^N)\) at the same rate as the solution \(w\) to the porous medium equation \(\partial_t w - \Delta w^m=0\) in \((0,\infty)\times{\mathbb R}^N\) for \(p\in [1,\infty)\). In addition, the drift term becomes negligible for large times and \(u\) behaves as the solution \(w\) to \(\partial_t w - \Delta w^m=0\) in \((0,\infty)\times {\mathbb R}^N\). However, the estimates of the distance between \(u\) and \(w\) are not optimal and have been subsequently improved by the same authors [S. Luckhaus and Y. Sugiyama, “Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases”, to appear in Indiana Univ. J. Math.].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K45 Initial value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
35K65 Degenerate parabolic equations
92C17 Cell movement (chemotaxis, etc.)
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References:

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