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Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening
Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) +Expand
Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems
We study the finite-time blow-up in two variants of the parabolic--elliptic Keller--Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} Expand
On the optimality of upper estimates near blow-up in quasilinear Keller--Segel systems
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ \tau v_t = \Delta v - v + u \end{cases}Expand
Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source
Abstract The Neumann initial–boundary problem for the chemotaxis system ( ⋆ ) u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ ( | x | ) u − μ ( | x | ) u p , 0 = Δ v − m ( t ) | Ω | + u , m ( t ) ≔ ∫ Ω u ( ⋅ , t ) isExpand
Analysis of a chemotaxis model with indirect signal absorption
Abstract We consider the chemotaxis model { u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − v w , w t = − δ w + u in smooth, bounded domains Ω ⊂ R n , n ∈ N , where δ > 0 is a given parameter. If either n ≤Expand
Boundedness enforced by mildly saturated conversion in a chemotaxis-May–Nowak model for virus infection
Abstract We study the system ( ⋆ ) { u t = Δ u − ∇ ⋅ ( u ∇ v ) − u − f ( u ) w + κ , v t = Δ v − v + f ( u ) w , w t = Δ w − w + v , which models the virus dynamics in an early stage of an HIVExpand
Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model
Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= \Delta n - \nabla \cdot (n \nabla c), \\ 0 &= \Delta c - nc, \end{align*} underExpand
Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity
Systems of the type { ut = ∇ · (D1(u)∇u − S1(u)∇v) + f1(u, v), vt = ∇ · (D2(v)∇v + S2(v)∇u) + f2(u, v) (⋆) can be used to model pursuit–evasion relationships between predators and prey. Apart fromExpand
Global Solutions near Homogeneous Steady States in a Multidimensional Population Model with Both Predator- and Prey-Taxis
  • Mario Fuest
  • Computer Science, Physics
  • SIAM J. Math. Anal.
  • 9 April 2020
TLDR
Migration-influenced predator–prey interaction can mathematically be described by the system { ut = D1∆u+ ∇ · (ρ1(u, v)∇v) + f(U, v), vt = D2∆v. Expand
Blow-up profiles in quasilinear fully parabolic Keller--Segel systems
We examine finite-time blow-up solutions $(u, v)$ to \begin{align} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \nabla \cdot (D(u, v) \nabla u - S(u, v) \nabla v), v_t = \Delta v - v + u Expand
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