Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source
The authors study two-species chemotaxis system with Lotka-Volterra competitive kinetics \begin{align*} u_t&=d_1\Delta u-\nabla\cdot (u\chi_1(w)\nabla w)+\mu_1u(1-u-a_1v) \\ v_t&=d_2\Delta v-\nabla\cdot (v\chi_2(w)\nabla w)+\mu_2v(1-v-a_2u) \\ w_t&=d_3\Delta w-(\alpha u+\beta v)w \quad \text{in }\Omega\times (0,\infty) \end{align*} with \[ \left. \frac{\partial}{\partial n}(u,v,w)\right\vert_{\partial\Omega}=0, \quad 0\leq \left. (u,v,w)\right\vert_{t=0}=(u_0, v_0, w_0) \in W^{1,\infty}(\Omega)^3, \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\) and \(d_j\) for \(j=1,2,3\), \(\mu_i\), \(a_i\) for \(i=1,2\), \(\alpha\), \(\beta\) are positive constants, and \(0<\chi_i\in C^{1,\theta}([0, \infty))\), \(i=1,2\), Then there is \(A=A(d_1, d_3)>0\) explicitly given such that if \[ \Vert w_0\Vert_\infty <\overline{w}\equiv \frac{A(d_1, d_3)}{\Vert \chi_1\Vert_{L^\infty[0,\Vert w_0\Vert_\infty]}}, \] there is a weak solution global-in-time which belogs to \begin{align*} u&\in L^\infty(0,\infty; L^1(\Omega))\cap L^2_{\mathrm{loc}}([0, \infty; W^{1,2}(\Omega)) \\ v&\in L^\infty(0,\infty; L^1(\Omega)) \\ w&\in L^\infty(\Omega\times (0,\infty))\cap L^2_{\mathrm{loc}}([0,\infty); W^{1,2}(\Omega)), \end{align*} where this weak solution is defined in the standard weak form with respect to \(u\) and \(w\), while a combination of an inequality derived from the multiplication of \(\frac{\psi}{v+1}\) for the test function \(\psi\) to the super-solution to the second equation and the other inequality valid for the sub-solution to it, \[ \int_\Omega v\leq \mu_1\int_\Omega v(1-v-a_2u). \] The proof relies on the approximate solution classical global-in-time for the truncated nonlinearity, and then the compactness argument for the passage to the limit.
scientific article; zbMATH DE number 7191484
scientific article