Green’s function of a chemotaxis model in the half space and its applications

Shi, Renkun

doi:10.1016/j.na.2014.04.009

Cited by 1 publication

(11 citation statements)

References 14 publications

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“…Let us denote the Green's function of () by F b ( y , t ; x ). From , we can write F b ( y , t ; x ) as follows:

F_{b} (y, t; x) = \sum_{i = 1}^{4} F_{i} (y, t; x),

where

F_{1} (y, t; x) = F (x - y, t),

F_{2} (y, t; x) = e^{l y} F (x + y, t),

F_{3} (y, t; x) = l e^{l y} {falsefalse}_{\int}^{0} F (y, t - τ) F (x, τ) d τ,

and

F_{4} (y, t; x) = l e^{l y} {falsefalse}_{\int}^{0} \partial_{y} F (y, t - τ) {falsefalse}_{\int}^{0} \partial_{x} F (x, σ) d σ d τ .

Here,

F (x, t) = \frac{1}{\sqrt{4 π t}} e^{- \frac{{(x + l t)}^{2}}{4 t}} .

By the Duhamel's principle, the solution u ( x , t ) of the problem () can be expressed as

…”

Section: Preliminariesmentioning

confidence: 99%

“…Then applying the coordinate transform x n → x n + l t in (), we obtain

({, \begin{matrix} \partial_{t} u - l \partial_{x_{n}} u - Δ u = - β \nabla \cdot (u \nabla v), & x_{n} > 0, t > 0, \\ λ v - Δ v = u, & x_{n} > 0, t > 0, \\ (l + \partial_{x_{n}}) u |_{x_{n} = 0} = \partial_{x_{n}} v |_{x_{n} = 0} = 0, & t > 0, \\ u (x, 0) = u_{0} (x), & x_{n} > 0 . \end{matrix})

We can see that the system () is () when n = 1. The system () was first considered by Shi in , where the problem in high dimensions was studied, and the long time behavior of solutions was obtained. The results in infer that the optimal L ∞ decay rate of solutions is (1 + t ) − n /2 when l < 0 and (1 + t ) − ( n − 1)/2 when l > 0.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we will study the problem () with small initial data in two cases: l < 0 and l > 0. When l < 0, we can follow directly the process of for the case of l < 0 to prove that the L ∞ decay rate of the solution ( u , v ) is (1 + t ) − 1/2 . This implies that [, Theorem 1.1] is also valid for n = 1( n is the space dimension).…”

Section: Introductionmentioning

confidence: 99%

“…When l < 0, we can follow directly the process of for the case of l < 0 to prove that the L ∞ decay rate of the solution ( u , v ) is (1 + t ) − 1/2 . This implies that [, Theorem 1.1] is also valid for n = 1( n is the space dimension). When l > 0, Theorem in this paper infers that the previous conjecture is right, and furthermore, we can prove that the solution ( u , v ) of () with small initial data tends to a conserved stationary solution( u ∗ , v ∗ ) in an exponential decay rate.…”

Section: Introductionmentioning

confidence: 99%

“…We can follow the proof of [, Theorem 1.1] to get Theorem , so we omit the proof here. In the following, we only consider the case of l > 0.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

Shi

2016

Math Methods in App Sciences

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In this paper, we are interested in a model derived from the 1‐D Keller‐Segel model on the half line x > as follows: ut−lux−uxx=−β(uvx)x,x>0,t>0,λv−vxx=u,x>0,t>0,lu(0,t)+ux(0,t)=vx(0,t)=0,t>0,u(x,0)=u0(x),x>0, where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l < 0. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Let us denote the Green's function of () by F b ( y , t ; x ). From , we can write F b ( y , t ; x ) as follows:

F_{b} (y, t; x) = \sum_{i = 1}^{4} F_{i} (y, t; x),

where

F_{1} (y, t; x) = F (x - y, t),

F_{2} (y, t; x) = e^{l y} F (x + y, t),

F_{3} (y, t; x) = l e^{l y} {falsefalse}_{\int}^{0} F (y, t - τ) F (x, τ) d τ,

and

F_{4} (y, t; x) = l e^{l y} {falsefalse}_{\int}^{0} \partial_{y} F (y, t - τ) {falsefalse}_{\int}^{0} \partial_{x} F (x, σ) d σ d τ .

Here,

F (x, t) = \frac{1}{\sqrt{4 π t}} e^{- \frac{{(x + l t)}^{2}}{4 t}} .

By the Duhamel's principle, the solution u ( x , t ) of the problem () can be expressed as

…”

Section: Preliminariesmentioning

confidence: 99%

“…Then applying the coordinate transform x n → x n + l t in (), we obtain

({, \begin{matrix} \partial_{t} u - l \partial_{x_{n}} u - Δ u = - β \nabla \cdot (u \nabla v), & x_{n} > 0, t > 0, \\ λ v - Δ v = u, & x_{n} > 0, t > 0, \\ (l + \partial_{x_{n}}) u |_{x_{n} = 0} = \partial_{x_{n}} v |_{x_{n} = 0} = 0, & t > 0, \\ u (x, 0) = u_{0} (x), & x_{n} > 0 . \end{matrix})

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We can follow the proof of [, Theorem 1.1] to get Theorem , so we omit the proof here. In the following, we only consider the case of l > 0.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

Shi

2016

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

Green’s function of a chemotaxis model in the half space and its applications

Cited by 1 publication

References 14 publications

Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line

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