Emergence of large densities in a chemotaxis system with signal-dependent motility

In this paper, we focus on the emergence of large densities of a cell population in finite time in the following parabolic-elliptic Keller-Segel system with signal-dependent motility and a source term of logistic type {equation} {cases} uε t = Δ(e-v_εu_ε) + ε(κu_ε - μu_ε^2) ,\\ 0 =Δvε-v_ε+ uε. {cases} {equation} in a smooth bounded domain Ω² with homogenuous Neumann boundary conditions and parameters $κ,μ>0$ and ε∈(0,1). We will show that the solution u_ε can exceed any certain value $M>0$ before a time $T(M)>0$ under the condition that ε>0 is sufficiently small. To prove the result, we show convergence of classical solutions of the first system to classical solutions of the corresponding limit system as ε 0 {equation} {cases} u_t=Δ(e⁻ᵛu),\\ -Δv+v=u. {cases} {equation} The main challenge is to find suitable bounds for u_ε and v_ε, that are ε-independent, in Lebesgue and Hölder spaces, whereas for fixed ε>0 global existence and boundedness is known.