Abstract
This paper is devoted to the following quasilinear chemotaxis system: \(\bigl\{\scriptsize{ \begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v)+uf(u),\quad x\in\Omega,t>0, \\ v_{t}=\Delta v-ug(v),\quad x\in\Omega,t>0, \end{array} }\bigr. \) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^{N}\). The given functions \(D(s)\), \(\chi(s)\), \(g(s)\), and \(f(s)\) are assumed to be sufficiently smooth for all \(s\ge0\) and such that \(f(s)\le\kappa-\mu s^{\tau}\). It is proved that the corresponding initial boundary value problem possesses a unique global classical solution for any \(\mu>0\) and \(\tau \ge1\), which is uniformly bounded in \(\Omega\times(0,+\infty)\). Moreover, when \(\kappa=0\), the decay property of the solution is also discussed in this paper.
Highlights
In this paper, we consider the fully parabolic chemotaxis system: ⎧ ⎪⎪⎪⎨ ut vt = =∇· v (D(u)∇u) – ug(v), – x ∇ ∈ ·(uχ (v)∇, t >, v) +
The following chemotaxis-(Navier)-Stokes model which is a generalized version of the model proposed in[ ], describes the motion of oxygen-driven swimming cells in an incompressible fluid, which is closely related to
The aim of this paper is to study the global existence and boundedness of the solutions for the parabolic-parabolic chemotaxis system with linear or nonlinear diffusion and logistic source ( . )
Summary
We consider the fully parabolic chemotaxis system:. ∂ ∂ν denotes the derivative with respect to the outer normal of ∂. The following chemotaxis-(Navier)-Stokes model which is a generalized version of the model proposed in[ ], describes the motion of oxygen-driven swimming cells in an incompressible fluid, which is closely related to. Χ > , Tao [ ] proved that if v L∞( ) is sufficiently small, the corresponding initial boundary value problem possesses a unique global solution that is uniformly bounded. Showed that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded provided that ⊂ RN is a bounded convex domain and some other technical conditions are fulfilled. The aim of this paper is to study the global existence and boundedness of the solutions for the parabolic-parabolic chemotaxis system with linear or nonlinear diffusion and logistic source + κu ln( + u) – μu +τ ln( + u) dx for all t ∈ , T∗
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.