Uniform boundedness and convergence of solutions to cross-diffusion systems
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.
spatial segregation of competing and diffusing species
scientific article; zbMATH DE number 1884204