Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion

This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u χ ( v ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v - u f ( v ) , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded...

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Published in	Zeitschrift für angewandte Mathematik und Physik Vol. 65; no. 6; pp. 1137 - 1152
Main Authors	Wang, Liangchen, Mu, Chunlai, Zhou, Shouming
Format	Journal Article
Language	English
Published	Basel Springer Basel 01.12.2014
Subjects	Bacteria Boundary conditions Boundary value problems Concentration gradient Diffusion Engineering Mathematical analysis Mathematical Methods in Physics Mathematical models Nonlinearity Theoretical and Applied Mechanics Global existence 92C17 35B35 Chemotaxis 35K55 Boundedness
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ISSN	0044-2275 1420-9039
DOI	10.1007/s00033-013-0375-4

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Abstract	This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u χ ( v ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v - u f ( v ) , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R n , with nonnegative initial data u 0 ∈ W 1 , θ ( Ω ) (for some θ > n ) and v 0 ∈ W 1 , ∞ ( Ω ) . The given functions D ( s ) , χ ( s ) and f ( s ) are supposed to be sufficiently smooth for all s ≥ 0 and such that f (0) = 0. This model describes the motion of the cells (e.g., bacteria) under the effect of gradients of the concentration of the oxygen that is consumed by the cells. It is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in Ω × ( 0 , + ∞ ) provided that some technical conditions are fulfilled.
AbstractList	This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u χ ( v ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v - u f ( v ) , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R n , with nonnegative initial data u 0 ∈ W 1 , θ ( Ω ) (for some θ > n ) and v 0 ∈ W 1 , ∞ ( Ω ) . The given functions D ( s ) , χ ( s ) and f ( s ) are supposed to be sufficiently smooth for all s ≥ 0 and such that f (0) = 0. This model describes the motion of the cells (e.g., bacteria) under the effect of gradients of the concentration of the oxygen that is consumed by the cells. It is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in Ω × ( 0 , + ∞ ) provided that some technical conditions are fulfilled. This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system $$\left\{\begin{array}{ll}u_t=\nabla \cdot (D(u) \nabla u)- \nabla \cdot (u \chi(v) \nabla v), \quad \quad x \in \Omega, \quad t > 0,\\v_t = \Delta v-u f(v), \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \Omega, \quad t > 0,\end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain ${\Omega\subset \mathbb{R} Delta }$ , with nonnegative initial data ${u_0 \in W1,\theta}{(\Omega)}}$ (for some theta > n) and ${v_0 \in W1,\infty}{(\Omega)}}$ . The given functions ${D(s), \chi(s)}$ and f(s) are supposed to be sufficiently smooth for all s greater than or equal to 0 and such that f(0) = 0. This model describes the motion of the cells (e.g., bacteria) under the effect of gradients of the concentration of the oxygen that is consumed by the cells. It is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in ${\Omega \times (0, +\infty)}$ provided that some technical conditions are fulfilled.
Author	Zhou, Shouming Wang, Liangchen Mu, Chunlai
Author_xml	– sequence: 1 givenname: Liangchen surname: Wang fullname: Wang, Liangchen email: liangchenwang0110@126.com organization: College of Mathematics and Statistics, Chongqing University – sequence: 2 givenname: Chunlai surname: Mu fullname: Mu, Chunlai organization: College of Mathematics and Statistics, Chongqing University – sequence: 3 givenname: Shouming surname: Zhou fullname: Zhou, Shouming organization: College of Mathematics Science, Chongqing Normal University
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Snippet	This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u χ ( v ) ∇... This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system $$\left\{\begin{array}{ll}u_t=\nabla \cdot...
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SubjectTerms	Bacteria Boundary conditions Boundary value problems Concentration gradient Diffusion Engineering Mathematical analysis Mathematical Methods in Physics Mathematical models Nonlinearity Theoretical and Applied Mechanics
Title	Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion
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