Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis (Q1989647)

Let \((u,v)\) be a global classical solution to the one-dimensional Keller-Segel chemotaxis system \[ \begin{aligned} \partial_t u & = \partial_x \left( \partial_ x u - \chi u \partial_x v \right) \;\text{ in }\; (0,\infty)\times I\;, \cr \partial_t v & = \partial_x^2 v - \gamma v + \alpha u \;\text{ in }\; (0,\infty)\times I\;, \\ \partial_x u & = \partial_x v = 0 \;\text{ on }\; (0,\infty)\times \partial I\;, \\ (u,v)(0) & = (\bar{u}, \bar{v}) \;\text{ in }\; I\;, \end{aligned} \] where \(I=(a,b)\) is a bounded interval of \(\mathbb{R}\), \(\chi\), \(\gamma\), and \(\alpha\) are positive real numbers, and \((\bar{u}, \bar{v})\) are positive initial conditions in \(W^{1,2}(I)\) with \(M(b-a)=\|\bar{u}\|_{L^1}\). The author proves the following conditional theorem: if there is \(t_*>0\) such that \[ \chi C \|\partial_x v(t)\|_{L^\infty} < 1\;, \qquad t\geq t_*\;, \] the constant \(C\) being that in Poincaré's inequality, and if \[ \lim_{t\to\infty} \|\partial_x^2 u(t)\|_{L^2} = 0\;, \] then \((u(t),v(t))\) converges to \((M,\alpha M/\gamma)\) in \(L^2(I;\mathbb{R}^2)\) as \(t\to\infty\). No examples of initial data to which the previous result may apply are given. Moreover, the intermediate results are either well-known and standard properties of linear parabolic equations or classical differential inequalities and their proofs could have been omitted.