抛物-抛物趋化系统的行波解存在性
偏微分方程分析
2016-11-28 v2
摘要
本文致力于研究如下抛物-抛物趋化系统的行波解，{ u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + u ( a − b u ) , x ∈ R N τ v t = Δ v − v + u , x ∈ R N , \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N \tau v_t=\Delta v-v+u, \quad x\in\mathbb{R}^N, \end{cases} { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + u ( a − b u ) , x ∈ R N τ v t = Δ v − v + u , x ∈ R N , u ( x , t ) u(x,t) u ( x , t ) v ( x , t ) v(x,t) v ( x , t ) χ \chi χ τ > 0 \tau >0 τ > 0 0 < χ τ ∗ < b 2 0<\chi_{\tau}^*<\frac{b}{2} 0 < χ τ ∗ < 2 b 0 < χ < χ τ ∗ 0<\chi<\chi_{\tau}^* 0 < χ < χ τ ∗ 2 a ≤ c ∗ ( χ , τ ) < c ∗ ∗ ( χ , τ ) 2\sqrt a \le c^{*}(\chi,\tau)<c^{**}(\chi,\tau) 2 a ≤ c ∗ ( χ , τ ) < c ∗∗ ( χ , τ ) c ∈ [ c ∗ ( χ , τ ) c ∗ ∗ ( χ , τ ) ) c\in [ c^{*}(\chi,\tau)\,\ c^{**}(\chi,\tau)) c ∈ [ c ∗ ( χ , τ ) c ∗∗ ( χ , τ )) ξ ∈ S N − 1 \xi\in S^{N-1} ξ ∈ S N − 1 c c c ( a b , a b ) (\frac{a}{b},\frac{a}{b}) ( b a , b a ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ( u ( x , t ) , v ( x , t ) ) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ ) ) (u(x,t),v(x,t))=(U(x\cdot\xi-ct;\tau),V(x\cdot\xi-ct;\tau)) ( u ( x , t ) , v ( x , t )) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ )) 2 a 2\sqrt a 2 a lim χ → 0 + c ∗ ∗ ( χ , τ ) = ∞ , \lim_{\chi\to 0^+}c^{**}(\chi,\tau)=\infty, χ → 0 + lim c ∗∗ ( χ , τ ) = ∞ , lim χ → 0 + c ∗ ( χ , τ ) = { 2 a if 0 < a ≤ 1 + τ a ( 1 − τ ) + 1 + τ a ( 1 − τ ) + + a ( 1 − τ ) + 1 + τ a if a ≥ 1 + τ a ( 1 − τ ) + , \lim_{\chi\to 0^+}c^{*}(\chi,\tau)=\begin{cases} 2\sqrt{a}\qquad \qquad \qquad\ \text{if} \quad 0<a\leq \frac{1+\tau a}{(1-\tau)_+}\cr \frac{1+\tau a}{(1-\tau)_{+}}+\frac{a(1-\tau)_{+}}{1+\tau a}\quad \text{if} \quad a\geq \frac{1+\tau a}{(1-\tau)_+}, \end{cases} χ → 0 + lim c ∗ ( χ , τ ) = { 2 a if 0 < a ≤ ( 1 − τ ) + 1 + τ a ( 1 − τ ) + 1 + τ a + 1 + τ a a ( 1 − τ ) + if a ≥ ( 1 − τ ) + 1 + τ a , lim x → − ∞ U ( x ; τ ) e − μ x = 1 , \lim_{x\to -\infty}\frac{U(x;\tau)}{e^{-\mu x}}=1, x → − ∞ lim e − μx U ( x ; τ ) = 1 , μ \mu μ μ + a μ = c \mu+\frac{a}{\mu}=c μ + μ a = c ( 0 , min { a , 1 + τ a ( 1 − τ ) + } ) (0, \min\{\sqrt a, \sqrt{\frac{1+\tau a}{(1-\tau)_+}}\}) ( 0 , min { a , ( 1 − τ ) + 1 + τ a }) lim τ → 0 + χ τ ∗ = b 2 \lim_{\tau\to 0^+}\chi_{\tau}^*=\frac{b}{2} lim τ → 0 + χ τ ∗ = 2 b
引用
@article{arxiv.1610.05215,
title = {Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems},
author = {Rachidi B. Salako and Wenxian Shen},
journal= {arXiv preprint arXiv:1610.05215},
year = {2016}
}
备注
arXiv admin note: substantial text overlap with arXiv:1609.05387