Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Stephan Luckhaus; Yoshie Sugiyama

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 3, page 597-621
  • ISSN: 0764-583X

Abstract

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We consider the following reaction-diffusion equation: ( KS ) u t = ∇ · ∇ u m - u q - 1 ∇ v , x ∈ ℝ N , 0 < t < ∞ , 0 = Δ v - v + u , x ∈ ℝ N , 0 < t < ∞ , u ( x , 0 ) = u 0 ( x ) , x ∈ ℝ N , where N ≥ 1 , m > 1 , q ≥ max { m + 2 N , 2 } .
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of q ≥ max { m + 2 N , 2 } , the above problem (KS) is solvable globally in time for “small L N ( q - m ) 2 data”. Moreover, the decay of the solution (u,v) in L p ( ℝ N ) was proved. In this paper, we consider the case of “ q ≥ max { m + 2 N , 2 } and small L ℓ data” with any fixed ℓ ≥ N ( q - m ) 2 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 u t = Δ u m with m>1.

How to cite

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Luckhaus, Stephan, and Sugiyama, Yoshie. "Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems." ESAIM: Mathematical Modelling and Numerical Analysis 40.3 (2006): 597-621. <http://eudml.org/doc/249685>.

@article{Luckhaus2006,
abstract = { We consider the following reaction-diffusion equation:$$ \{\rm (KS)\} \left\\{ \begin\{array\}\{llll\} u\_t = \nabla \cdot \Big( \nabla u^m - u^\{q-1\} \nabla v \Big), & x \in \mathbb\{R\}^N, \ 0<t<\infty, \nonumber 0 = \Delta v - v + u, & x \in \mathbb\{R\}^N, \ 0<t<\infty, \nonumber u(x,0) = u\_0(x), & x \in \mathbb\{R\}^N, \end\{array\} \right. $$ where $N \ge 1, \ m > 1, \ q \ge \max\\{m+\frac\{2\}\{N\},2\\}$.
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of $q \ge \max\\{m+\frac\{2\}\{N\},2\\}$, the above problem (KS) is solvable globally in time for “small $L^\{\frac\{N(q-m)\}\{2\}\}$ data”. Moreover, the decay of the solution (u,v) in $L^p(\mathbb\{R\}^N)$ was proved. In this paper, we consider the case of “$q \ge \max\\{m+\frac\{2\}\{N\},2\\}$ and small $L^\{\ell\}$ data” with any fixed $\ell \ge \frac\{N(q-m)\}\{2\}$ 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 $u_t = \Delta u^m$ with m>1. },
author = {Luckhaus, Stephan, Sugiyama, Yoshie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Degenerate parabolic system; chemotaxis; Keller-Segel model; drift term; decay property; asymptotic behavior; Fujita exponent; porous medium equation; Barenblatt solution.; global existence; asymptotic behaviour; degenerate parabolic equation; self-similarity; parabolic-elliptic system},
language = {eng},
month = {7},
number = {3},
pages = {597-621},
publisher = {EDP Sciences},
title = {Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems},
url = {http://eudml.org/doc/249685},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Luckhaus, Stephan
AU - Sugiyama, Yoshie
TI - Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 3
SP - 597
EP - 621
AB - We consider the following reaction-diffusion equation:$$ {\rm (KS)} \left\{ \begin{array}{llll} u_t = \nabla \cdot \Big( \nabla u^m - u^{q-1} \nabla v \Big), & x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber 0 = \Delta v - v + u, & x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber u(x,0) = u_0(x), & x \in \mathbb{R}^N, \end{array} \right. $$ where $N \ge 1, \ m > 1, \ q \ge \max\{m+\frac{2}{N},2\}$.
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)] it was shown that in the case of $q \ge \max\{m+\frac{2}{N},2\}$, the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”. Moreover, the decay of the solution (u,v) in $L^p(\mathbb{R}^N)$ was proved. In this paper, we consider the case of “$q \ge \max\{m+\frac{2}{N},2\}$ and small $L^{\ell}$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ 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 $u_t = \Delta u^m$ with m>1.
LA - eng
KW - Degenerate parabolic system; chemotaxis; Keller-Segel model; drift term; decay property; asymptotic behavior; Fujita exponent; porous medium equation; Barenblatt solution.; global existence; asymptotic behaviour; degenerate parabolic equation; self-similarity; parabolic-elliptic system
UR - http://eudml.org/doc/249685
ER -

References

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