Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system

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• 作者：Hao Yu ^{yuhaomuu@126.com" class="auth_mail" title="E-mail the corresponding author} ; Qian Guo ^{424096826@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Sining Zheng ; ^{snzheng@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Attraction&ndash ; repulsion ; Blow-up ; Nonradial solutions ; Keller&ndash ; Segel system
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：34
• 期：Complete
• 页码：335-342
• 全文大小：600 K

文摘

This paper considers the attraction–repulsion chemotaxis system: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si1.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=e9cdd5bb6211d51fec1b2435f86ff550" title="Click to view the MathML source">u_t=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)class="mathContainer hidden">class="mathCode">$u_t=\Delta u-\chi \nabla \cdot \left(u\nabla v\right)+\xi \nabla \cdot \left(u\nabla w\right)$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si2.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=ffb8554605c30e8741b05c3d9628a419" title="Click to view the MathML source">0=Δv+αu−βvclass="mathContainer hidden">class="mathCode">$0=\Delta v+\alpha u-\beta v$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si3.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=9135ec374a39a4c273d5903e486e207a" title="Click to view the MathML source">0=Δw+γu−δwclass="mathContainer hidden">class="mathCode">$0=\Delta w+\gamma u-\delta w$, subject to the non-flux boundary condition in a smooth bounded domain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si4.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=ca4751a3cb085d0e04a3806f24b0cc44" title="Click to view the MathML source">Ω⊂R²class="mathContainer hidden">class="mathCode">$\Omega \subset {\mathbb{R}}^2$, with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si5.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=6fecee6be2b996f69f311a61a2983b82" title="Click to view the MathML source">χ,ξ≥0class="mathContainer hidden">class="mathCode">$\chi ,\xi \ge 0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si6.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=abec769d1806b0eb80617a1cdee8a6f9" title="Click to view the MathML source">α,β,γ,δ>0class="mathContainer hidden">class="mathCode">$\alpha ,\beta ,\gamma ,\delta >0$. We establish the finite time blow-up conditions for nonradial solutions that the finite time blow-up occurs at class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si7.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=7591bf74e3ea631e0c1d6bbdde889ca1" title="Click to view the MathML source">x₀∈Ωclass="mathContainer hidden">class="mathCode">$x_0\in \Omega$ whenever class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si8.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=fe7815fd945f857db081e0141b4d3ed4" title="Click to view the MathML source">∫_Ωu₀(x)dx>8π/(χα−ξγ)class="mathContainer hidden">class="mathCode">${\int }_{\Omega }{u}_0\left(x\right)dx>8\pi /(\chi \alpha -\xi \gamma )$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si9.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=2fdb454b8a99440d3d46a1e7a22a95b5" title="Click to view the MathML source">χα−ξγ>0class="mathContainer hidden">class="mathCode">$\chi \alpha -\xi \gamma >0$, under class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si10.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=70d6e088bf9ef47b4cdd9f021edbe7e6" title="Click to view the MathML source">∫_Ωu₀(x)|x−x₀|²dxclass="mathContainer hidden">class="mathCode">${\int }_{\Omega }{u}_0\left(x\right){|x-x_0|}^{2}dx$ sufficiently small. This does agree with the known blow-up conditions for radial solutions of the same model. The previous blow-up conditions for nonradial solutions are more complicated involving a classification to the sign of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816301109&_mathId=si11.gif&_user=111111111&_pii=S1468121816301109&_rdoc=1&_issn=14681218&md5=c7e7994532203f7ea4f42f9ef5863450" title="Click to view the MathML source">δ−βclass="mathContainer hidden">class="mathCode">$\delta -\beta$.