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تسجيل مستخدم جديدBoundary spike-layer solutions of the singular Keller-Segel system: existence and stability
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We exploit the existence and nonlinear stability of boundary spike/layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero. Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation, to a nonlinear nonlocality which is resolved by the technique of taking antiderivatives, i.e. working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardys inequality that fully captures the dissipative structure of the system.
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Perhaps the most classical diffusion model for chemotaxis is the Patlak-Keller-Segel system begin{equation} label{ks0} left{ begin{aligned} u_t =& Delta u - abla cdot(u abla v) quad inn R^2times(0,infty), v =& (-Delta_{R^2})^{-1} u := frac 1{2pi}
int_{R^2} , log frac 1{|x-z|} u(z,t), dz, & qquad u(cdot ,0) = u_0 ge 0quadhbox{in } R^2. end{aligned} right. end{equation} We consider the {em critical mass case} $int_{R^2} u_0(x), dx = 8pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of equ{ks0} is globally defined and blows-up in infinite time. As $tto+infty $ it has the approximate profile $$ u(x,t) approx frac 1{la^2} U_0left (frac {x-xi(t)}{la(t)} right ), quad U_0(y)= frac{8}{(1+|y|^2)^2}, $$ where $la(t) approx frac c{sqrt{log t}}, xi(t)to q $ for some $c>0$ and $qin R^2$
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