In this paper, we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source {ut=△uχ·(u/v■v)+ru-μu^(k),x∈Ω,t>0,O=△u-v+u,x∈Ω,t&...In this paper, we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source {ut=△uχ·(u/v■v)+ru-μu^(k),x∈Ω,t>0,O=△u-v+u,x∈Ω,t>0 in a smooth bounded domain ? ? R~n(n > 2) with the non-flux boundary, where χ, r, μ > 0,k ≥ 2. It is proved that the global bounded classical solution will exponentially converge to((r/μ)^(1/k-1),(r/μ)^(1/k-1)) as t →∞ if r is suitably large.展开更多
In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="&q...In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="" />(0.1)<br /> is considered in a bounded domain with smooth boundary, Ω ⊂R<sup>n</sup> (n ≥ 1), where d<sub>1</sub> > 0, d<sub>2</sub> > 0 with parameter χ ∈ R. When d<sub>1</sub> = d<sub>2</sub> + χ, satisfying for all initial data 0 ≤ n<sub>0</sub> ∈ C<sup>0</sup><img src="Edit_4898c7a9-f047-4856-b9ad-8d42ecf262a2.bmp" alt="" /> and 0 < v<sub>0</sub>∈ W<sup>1,∞</sup> (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞).展开更多
This paper deals with the singular chemotaxis-Navier-Stokes system with indirect signal consumption n_(t)+u·▽v=△n-Х▽·(n/v▽u);v_(t)+u·▽v=△v-uw;w_(t)+u·▽w=△w-w+n;u_(t)+(u·▽)u=△u-▽P+...This paper deals with the singular chemotaxis-Navier-Stokes system with indirect signal consumption n_(t)+u·▽v=△n-Х▽·(n/v▽u);v_(t)+u·▽v=△v-uw;w_(t)+u·▽w=△w-w+n;u_(t)+(u·▽)u=△u-▽P+n▽Ф;▽·u=0,x∈Ω,t>0 in a bounded and smooth domainΩ⊂ℝ2 with no-flux/no-flux/no-flux/no-slip boundary conditions,whereΦ∈W2,∞(Ω).A recent literature[Dai F,Liu B.J Differential Equations,2023,369:115–155]has proved that for all reasonably regular initial data,the associated initial-boundary value problem possesses a global classical solution,but qualitative information on the behavior of solution has never been touched so far.In stark contrast to the positive effect of indirect signal consumption mechanism on the global solvability of system,the analysis of asymptotic behavior of solution to the system with indirect signal consumption is essentially complicated than that with direct signal consumption because the favorable coupled structure between cells and signal is broken down by the indirect signal consumption mechanism.The present study shows that the global classical solution exponentially stabilizes toward the corresponding spatially homogeneous equilibria under a smallness condition on the initial cell mass.In comparison to the previously known result concerning the uniform convergence of solution to the system with direct signal consumption,our result inter alia provides a more in-depth understanding on the asymptotic behavior of solution.展开更多
基金Supported by Innovation Team of China West Normal University (Grant No. CXTD2020-5)the Meritocracy Research Funds of China West Normal University (Grant No. 17YC372)。
文摘In this paper, we study the asymptotic behavior of solutions to the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source {ut=△uχ·(u/v■v)+ru-μu^(k),x∈Ω,t>0,O=△u-v+u,x∈Ω,t>0 in a smooth bounded domain ? ? R~n(n > 2) with the non-flux boundary, where χ, r, μ > 0,k ≥ 2. It is proved that the global bounded classical solution will exponentially converge to((r/μ)^(1/k-1),(r/μ)^(1/k-1)) as t →∞ if r is suitably large.
文摘In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity <img src="Edit_4b941130-fc1e-4c9b-9626-4fd5a1f03836.bmp" alt="" />(0.1)<br /> is considered in a bounded domain with smooth boundary, Ω ⊂R<sup>n</sup> (n ≥ 1), where d<sub>1</sub> > 0, d<sub>2</sub> > 0 with parameter χ ∈ R. When d<sub>1</sub> = d<sub>2</sub> + χ, satisfying for all initial data 0 ≤ n<sub>0</sub> ∈ C<sup>0</sup><img src="Edit_4898c7a9-f047-4856-b9ad-8d42ecf262a2.bmp" alt="" /> and 0 < v<sub>0</sub>∈ W<sup>1,∞</sup> (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞).
文摘This paper deals with the singular chemotaxis-Navier-Stokes system with indirect signal consumption n_(t)+u·▽v=△n-Х▽·(n/v▽u);v_(t)+u·▽v=△v-uw;w_(t)+u·▽w=△w-w+n;u_(t)+(u·▽)u=△u-▽P+n▽Ф;▽·u=0,x∈Ω,t>0 in a bounded and smooth domainΩ⊂ℝ2 with no-flux/no-flux/no-flux/no-slip boundary conditions,whereΦ∈W2,∞(Ω).A recent literature[Dai F,Liu B.J Differential Equations,2023,369:115–155]has proved that for all reasonably regular initial data,the associated initial-boundary value problem possesses a global classical solution,but qualitative information on the behavior of solution has never been touched so far.In stark contrast to the positive effect of indirect signal consumption mechanism on the global solvability of system,the analysis of asymptotic behavior of solution to the system with indirect signal consumption is essentially complicated than that with direct signal consumption because the favorable coupled structure between cells and signal is broken down by the indirect signal consumption mechanism.The present study shows that the global classical solution exponentially stabilizes toward the corresponding spatially homogeneous equilibria under a smallness condition on the initial cell mass.In comparison to the previously known result concerning the uniform convergence of solution to the system with direct signal consumption,our result inter alia provides a more in-depth understanding on the asymptotic behavior of solution.
