Nonlinear stability of phase transition steady states to a hyperbolic-parabolic system modelling vascular networks

Experiments of in vitro blood vessel formation demonstrate that endothelial cells randomly dispersing on a gel substrate (matrix) can spontaneously organize into a coherent vascular network (see [1, 13] and figures therein), which is called angiogenesis - a major factor driving the tumor growth. How endothelial cells self-organize geometrically into capillary networks and how separate individual cells cooperate in the formation of coherent patterns remain poorly understood biologically up to date. These networking patterns can not be explained by the macroscopic aggregation models such as Keller-Segel type chemotaxis models that lead to point-wise blowup or rounded aggregates, nor by the microscopic kinetic models that describe individual cell behaviors, as commented in [6]. Strikingly they can be numerically reproduced by a hydrodynamic (hyperbolic-parabolic) models of chemotaxis proposed in [1, 13] as follows

where $\rho$ denotes the endothelial cell density, $u$ the cell velocity and $\phi$ the concentration of chemoattractant secreted by cells; $p$ is a pressure function accounting for the fact that closely packed cells resist to compression due to the impenetrability of cellular matter, the parameter $\mu>0$ measures the intensity of cell response to the chemoattractant and $\alpha\rho u$ corresponds to a damping (friction) force due to the interaction between cells and underlying substratum with a drag coefficient $\alpha>0$; $a$ and $b$ are positive constants accounting for the growth and death rates of the chemoattractant. The convective term $\nabla\cdot(\rho u\otimes u)$ models the cell persistence (inertia effect). We refer more detailed biological interpretations on the model (1.4) to [1, 13]. The hydrodynamic system (1.4) has been (formally) derived from the following kinetic equation in [7] via the mean-field approximation

where $f(t,x,v)$ denotes cell density distribution at time $t$, position $x\in{\mathbb{R}}^{d}$ moving with a velocity $v$ from a compact set $V$ in $\mathbb{R}^{d}$, and tumbling kernel $T[\phi](v^{\prime},v)$ describes the frequency of changing trajectories from velocity $v^{\prime}$ (anterior) to $v$ (posterior) in response to a chemical concentration $\phi$.

At first glance from mathematical point of view, the above hydrodynamic system (1.4) is analogous to the well-known damped Euler-Poisson system where the $\phi$-equation is elliptic (i.e. $-\Delta\phi=\alpha\rho$) which appears in various important applications depending on the sign of $\mu$, such as the propagation of electrons in semiconductor devices (cf. [22]) and the transport of ions in plasma physics (cf. [8]) and the collapse of gaseous stars due to self-gravitation [5]. However the parabolic $\phi$-equation in (1.4) will bring substantial differences in mathematical analysis and many existing mathematical frameworks developed for the Euler-Poisson system are inapplicable directly to (1.4). Due to the competing interactions between parabolic dissipation and hyperbolic anti-dissipation effect plus nonlinear convection, the global well-posedness and regularity of solutions to (1.4) is very complicated as can be glimpsed from the Euler-Poisson equations for which the understanding of solution behaviors is rather incomplete despite of numerous studies attempted. Up to date, there are only few results obtained for (1.4) in the literature. First when the initial value $(\rho_{0},u_{0},\phi_{0})$ is a small perturbation of a constant ground state $(\bar{\rho},0,\bar{\phi})$ in $H^{s}(\mathbb{R}^{d})(s>d/2+1)$ with $\bar{\rho}>0$ sufficiently small, the global existence and stability of solutions with non-vacuum (i.e. $\inf\limits_{x\in\mathbb{R}^{d}}\rho>0$) to (1.4) was established in [11, 9]. The linear stability of constant ground state $(\bar{\rho},0,\bar{\phi})$ was obtained under the condition $p^{\prime}(\bar{\rho})>\frac{a\mu}{b}\bar{\rho}$ in [16] where an additional viscous term $\nabla^{2}u$ is supplied to the second equation of (1.4). The stationary solutions of (1.4) with vacuum (bump solutions) in a bounded interval with zero-flux boundary condition and in $\mathbb{R}$ were constructed in [2] and further elaborated in [3]. The model (1.4) with $p(\rho)=\rho$ and periodic boundary conditions in one dimension was numerically explored in [12]. These appear to be the only results available to the system (1.4) in the literature and further studies are in demand. For results on some other types of hyperbolic-parabolic chemotaxis models, we refer to [10, 17, 18, 24, 27] and references therein.

Note that the above-mentioned mathematical works on (1.4) prescribe initial data as a small perturbation of constant equilibria and the large-time profile of solutions is also constant, which can not explain the experimental observations of [1, 13] showing prominent phase transition patterns connecting regions clear (or low density) of cells. This motivates us to explore the possible non-constant phase transition profiles of solutions. The aim of this paper is to study the existence and stability of phase transition steady states without vacuum to the system (1.4) in one dimensional half space $\mathbb{R}_{+}=(0,\infty)$. For the convenience of presentation in the sequel, we set $m=\rho u$, namely $m$ denotes the momentum of cells, and recast the one-dimensional system (1.4) in $\mathbb{R}_{+}$ as

As in [16], we assume the pressure function $p$ depends only on the density and satisfies

A typical form of $p$ is $p(\rho)=\frac{K}{2}\rho^{2}$ with $K>\frac{a\mu}{b}$. We supplement the system (1.5a)–(1.5c) with the following boundary conditions

and the initial value

where $\phi_{-}$, $\rho_{+}>0$, $\phi_{+}>0$ are constants and $\phi_{-}\neq\phi_{+}$.

In this paper, we shall first use delicate analysis to show that the system (1.5a)–(1.5c) has a unique non-constant stationary solution $(\bar{\rho},0,\bar{\phi})$ without vacuum satisfying

where $\bar{\rho}$ and $\bar{\phi}$ are monotone. Then we show that the stationary solution $(\bar{\rho},0,\bar{\phi})$ is asymptotically stable, namely the solution of (1.5a)–(1.5c) converges to $(\bar{\rho},0,\bar{\phi})$ point-wisely as $t\to\infty$ if the initial value $(\rho_{0},m_{0},\phi_{0})$ is an appropriate small perturbation of $(\bar{\rho},0,\bar{\phi})$. The monotonicity of $\bar{\rho}$ and $\bar{\phi}$ indicates that the steady states have phase transition profiles. To the best of our knowledge, there are not such results available in the literature for (1.5a)–(1.5c) and even for the Euler-Poisson equations (cf. [26, 21, 20, 19, 14]. To prove our results, we fully capture the structure of (1.5a)–(1.5c) where the stationary solution has exponential decay at far field, which is not enjoyed by the Euler-Poisson (or Euler) equations. Though part of the proof of our results is inspired by some ideas of [14, 25, 26, 15] on Euler-Poisson or Euler equations, we have added lof of extra efforts to deal with complicated couplings and boundary effects. The coupling term $\rho\phi_{x}$ leads to two linear terms in the linearized system around stationary solutions and how to make these linear terms under control is key to the asymptotic stability against small perturbations. We resolve this issue by the structural assumption (1.6) to take up the dissipation and use the exponentially weighted Hardy inequality in half space to compensate for the lack of dissipation in the hyperbolic equations. Due to the couplings and boundary effects, the energy estimates are very sophisticated, where the lower-order estimates involve higher-order estimates and vice versa. We use the delicate energy estimates along with the technique of a priori assumptions to unravel these tangles and gradually achieve our results.

The rest of this paper is organized as follows. In section 2, we state our main results on the existence of non-constant stationary solutions (Theorem 2.1) and stability of stationary solutions (Theorem 2.2). In section 3, we study the stationary problem and prove Theorem 2.1. The proof of Theorem 2.2 is given in Section 4.

In this section, we shall state the main results of this paper. To be precise, we first introduce some notations used. Throughout the paper, we use $\|\cdot\|_{L^{\infty}}$, $\|\cdot\|$ and $\|\cdot\|_{k}$ to denote the norms of usual $L^{\infty}(\mathbb{R}_{+})$, $L^{2}(\mathbb{R}_{+})$ and the standard Sobolev space $H^{k}(\mathbb{R}_{+})$, respectively. We also use $\|(f_{1},\cdots,f_{n})\|~{}(\mathrm{resp}.~{}\|(f_{1},\cdots,f_{n})\|_{k})$ to denote $\|f_{1}\|+\cdots+\|f_{n}\|~{}(\mathrm{resp}.~{}\|f_{1}\|_{k}+\cdots+\|f_{n}\|_{k})$ for some $n\in\mathbb{Z}_{+}$. We denote by $C$ a generic constant that may vary in the context, and by $C_{\eta}$ a constant depending on $\eta$. Occasionally, we simply write $f\sim g$ if $C^{-1}\leq f\leq Cg$ for some constant $C>0$.

It can be verified that the system (1.5a)–(1.5c) possesses the following energy functional (cf. [2, 7])

which, subject to the boundary condition (1.7), satisfies that

where $\rho G^{\prime\prime}(\rho)=p^{\prime}(\rho)$. Thus the stationary solution satisfying $\frac{\mathrm{d}}{\mathrm{d}t}F[\rho,u,\phi]=0$ gives rise to $\rho u=0$ and $\phi_{t}=0$ in $\mathbb{R}_{+}$. Since we are interested in non-constant profile for $\rho$, $u\equiv 0$ is the only (physical) stationary profile for the velocity $u$. Therefore stationary solutions of (1.5a)–(1.5c) without vacuum must possess the form $(\bar{\rho},0,\bar{\phi})$, where $(\bar{\rho},\bar{\phi})$ satisfies

Here the pressure $p$ satisfies (1.6) and the constants $\rho_{+}$ and $\phi_{\pm}$ are the same as in (1.7) and (1.8).

Then our first result concerning the existence and uniqueness of solutions to the stationary problem (2.1a)–(2.1d) is given below.

Our second result is the asymptotic stability of the stationary solutions obtained in Theorem 2.1, which is stated in the following theorem.

In this section, we shall study the stationary problem (2.1a)–(2.1d) and complete the proof of Theorem 2.1. To this end, we first reformulate our problem (2.1a)–(2.1d), and then prove the existence and uniqueness of solutions. Finally, we derive the monotone and decay properties of solutions.