Global stability of prey-taxis systems

We consider the chemotaxis system ut = Δu −∇⋅u v∇v, vt = Δv − uv, as originally introduced in 1971 by Keller and Segel in the second of their seminal works. This system constitutes a prototypical model for taxis-driven pattern formation and front propagation in various biological contexts such as tumor angiogenesis, but in the higher-dimensional context any global existence theory for large-data solutions is yet lacking. In this work it is shown that in bounded planar domains Ω with smooth boundary, for all reasonably regular initial data u0 ≥ 0 and v0 > 0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Thus particularly addressing arbitrarily large initial data, this goes beyond previously gained results asserting global existence of solutions only in spatial one-dimensional problems, or under certain smallness conditions on the initial data. The derivation of this result is based on a priori estimates for the quantities ∇ln(u + 1) and ∇v in spatio-temporal L2 spaces, where further boundedness and compactness properties are derived from the former by relying on the planar spatial setting in using an associated Moser–Trudinger inequality. Furthermore, some further boundedness and relaxation properties are derived, inter alia indicating that for any such solution we have v(⋅,t) → 0 in Lp(Ω) as t →∞ for all finite p > 1, and that in an appropriate generalized sense the quantities u and ∇ln v eventually enter bounded sets in Lp(Ω) and L2(Ω), respectively, with diameters only determined by the total population size ∫Ωu0. Finally, some numerical experiments illustrate the analytically obtained results.

The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties

VARIATIONAL CONVERGENCE FOR FUNCTIONS AND OPERATORS (Applicable Mathematics Series)

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

Application of an Adams type inequality to a two-chemical substances chemotaxis system

This paper deals with the chemotaxis system
\[
	\begin{cases}
	u_t=\Delta u - \nabla \cdot (u\nabla v),
	\qquad x\in \Omega, \ t>0, \\
	v_t=\Delta v + wz,
	\qquad x\in \Omega, \ t>0, \\
	w_t=-wz,
	\qquad x\in \Omega, \ t>0, \\
	z_t=\Delta z - z + u,
 \qquad x\in \Omega, \ t>0,
	\end{cases}
\]
in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$,
that has recently been proposed as a model for tumor invasion
in which the role of an active extracellular matrix is accounted for.
 
It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding
initial-boundary value problem of Neumann type possesses a global solution which is bounded.
Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain
spatially homogeneous equilibrium in the sense that as $t\to\infty$,
 
	$u(x,t)\to \overline{u_0}$ ,  &nbsp
	$v(x,t) \to \overline{v_0} + \overline{w_0}$,  &nbsp
	$w(x,t) \to 0$  &nbsp and &nbsp $z(x,t) \to \overline{u_0}$, &nbsp
uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$,
$\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ &nbspand  &nbsp $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

Stabilization in a chemotaxis model for tumor invasion

We study the systems of nonlinear parabolic PDEs for Penrose–Fife models of non-isothermal phase transitions. Both models with conserved and non-conserved order parameters are studied with Neumann boundary conditions imposed on the systems in multi-dimensional spatial domain.

Non-isothermal phase transition models with Neumann boundary conditions

Large time behavior in a chemotaxis model with nonlinear general diffusion for tumor invasion

We study the coupled Cahn-Hilliard/Allen-Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence-uniqueness result and to construct the global attractor for the related dynamical system.

Global attractor for the Cahn--Hilliard/Allen--Cahn system

Phase Separation Model of Penrose-Fife Type With Signorini Boundary Condition

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