Uniform boundedness and convergence of solutions to cross-diffusion systems
type: Item
instance of: scholarly article
title: Uniform boundedness and convergence of solutions to cross-diffusion systems
zbMATH Open document ID: 1032.35090
author: Seong-A Shim
published in: Journal of Differential Equations
publication date: 2003-03-18T00:00:00Z
review text: Consider the system of quasilinear parabolic partial differential equations \[ \begin{aligned} u_t &= (d_1u + \alpha_{11} u^2 + \alpha_{12} uv)_{xx} + u(a_1- b_1u- c_1v) \\ v_t &= (d_2v + \alpha_{22} v^2 + \alpha_{21} uv)_{xx} + v(a_2- b_2u- c_2v) \end{aligned} \] for \(0 < x < 1, 0 < t\), which has been proposed as a model for spatial segregation of competing and diffusing species. The author studies this problem together with zero Neumann boundary conditions and non-negative initial conditions for the unknown functions \(u\) and \(v\) under the assumption \(d_1 = d_2 = d\), in the two cases \(c_1/c_2 < a_1/a_2 < b_1/b_2\) (weak competition) and when these inequalities are reversed (strong competition). In the weak competition case, there is a stable equilibrium at \(\bar u > 0, \bar v > 0\) for the corresponding system of ordinary differential equations; in the case of strong competition there are two stable equilibria of non-negative solutions, one on the \(u\)-axis and one on the \(v\)-axis. The author proves a global existence result and global bounds for the Sobolev norms \(\|u(\cdot,t)\|_{1,2}\) and \(\|v(\cdot,t)\|_{1,2}\) without additional assumptions relating \(d, \alpha_{12}, \alpha_{21}\), using combinations of a priori estimates and calculus inequalities for the system of equations satisfied by \(u\), \(v\), and \(\zeta = u-v\). The bound is independent of \(d\) if this diffusion coefficient is large. In the weak competition case, it is shown that \((\bar u, \bar v)\) is globally asymptotically stable if \(d\) is suitable large. In the strong competition case, if \(d\) is sufficiently large and some additional technical conditions hold for the other coefficients, it is shown that a cone-like set such as \(\{(u,v) |u \geq 0, v \geq 0, \alpha_{12}v \leq \alpha_{21}u \}\) is invariant and that the equilibrium \((\bar u, 0), \bar u > 0\) is stable for all initial data with values in the cone. The proofs use suitable Lyapunov functionals.
reviewed by: Hans Engler
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1884204
zbMATH Keywords: one space dimension
zbMATH Keywords: spatial segregation of competing and diffusing species
zbMATH Keywords: a priori bounds
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: zero Neumann boundary conditions
zbMATH Keywords: Lyapunov functionals
MaRDI profile type: Publication
full work available at URL: jdeq.2002.4169
OpenAlex ID: W2035184624
cites work: Equivalent Norms for Sobolev Spaces
cites work: Dynamic theory of quasilinear parabolic systems. III: Global existence
cites work: Dynamic theory of quasilinear parabolic equations. II: Reaction-diffusion systems
cites work: Q4284440
cites work: An initial-boundary-value problem for a certain density-dependent diffusion system
cites work: Q5630031
cites work: Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
cites work: On the global existence of a cross-diffusion system
cites work: Q3260749
cites work: Q3856484
cites work: Q5533735
cites work: Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics
cites work: Global solution to some quasilinear parabolic system in population dynamics
DOI: 10.1006/JDEQ.2002.4169
description: scientific article; zbMATH DE number 1884204