具次线性机制的趋化系统古典解的整体有界性

Global Boundedness in A Parabolic-Parabolic Chemotaxis System with Singular Sensitivity and Sublinear Production

摘要: 本文研究光滑有界区域Ω⊂ ℝⁿ（n≥2）中带有奇异敏感性与次线性产生项的抛物-抛物趋化系统 $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot \left({\displaystyle \frac{u}{v^{\alpha }}}\mathrm{\nabla }v\right),\\ v_t=\mathrm{\Delta }v-v+u^{\beta },\end{array}$$ 其中α∈(0,1)，,β∈(0,1),x > 0.本文证明：当α∈(0,1),β ∈(0,2/n) 且χ > 0，则系统存在唯一的整 体有界古典解. 这表明次线性产生项有助于保证具有奇异趋化机制的抛物-抛物趋化系统整体有界古 典解的存在性。

Abstract: We consider a parabolic-parabolic chemotaxis system with singular sensitivity and sublinear production in a smooth bounded domain Ω⊂ ℝⁿ（n≥2）$$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot \left({\displaystyle \frac{u}{v^{\alpha }}}\mathrm{\nabla }v\right),\\ v_t=\mathrm{\Delta }v-v+u^{\beta },\end{array}$$ where α∈(0,1)，,β∈(0,1),x > 0. It is proven that the system has a globally bounded classical solution under the conditions α∈(0,1),β ∈(0,2/n) , and χ > 0. This shows that the sublinear production effect is indeed beneficial in ensuring the existence of a globally bounded classical solution for the parabolic-parabolic chemotaxis system with a singular chemotactic mechanism.