How hysteresis produces discontinuous patterns in degenerate reaction

Abstract

In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction–diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.

Keywords

Reaction-diffusion degenerate discontinuous pattern asymptotic behavior hysteresis

1. Introduction

In this paper, we study the long time behaviour of the solutions to the following degenerate reaction–diffusion system: $\begin{matrix} (1) & \{\begin{matrix} \frac{\partial u}{\partial t} = α w - q (u) & in (0, + \infty) \times Ω, \\ \frac{\partial w}{\partial t} = δ Δ w - β w + α u & in (0, + \infty) \times Ω, \\ \frac{\partial w}{\partial ν} = 0 & on (0, + \infty) \times \partial Ω, \\ u (0, x) = u_{0} (x), w (0, x) = w_{0} (x) & in Ω . \end{matrix} \end{matrix}$ Here, Ω is a one-dimensional bounded domain. The unknown functions $u = u (t, x)$ , $w = w (t, x)$ are defined in $(0, + \infty) \times Ω$ , and the initial conditions $u_{0}$ , $w_{0}$ are defined in Ω. Δ denotes the Laplace diffusion operator defined as $Δ w = \frac{\partial^{2} w}{\partial x^{2}}$ . The outward normal vector to $\partial Ω$ at point x is denoted by $ν (x)$ . The parameters α, β, δ are positive coefficients, and the function $q (u)$ is a non-invertible cubic function defined by $\begin{matrix} (2) & q (u) = u (u^{2} - 1) . \end{matrix}$

System (1) belongs to the class of systems that can be written under the form $\begin{matrix} (3) & \frac{\partial u}{\partial t} = f (u, w), \frac{\partial w}{\partial t} = δ Δ w + g (u, w), \end{matrix}$ which are commonly called degenerate reaction–diffusion systems or sometimes partly dissipative reaction–diffusion systems, since the diffusion operator acts only on one of the two equations. These systems have been widely studied. They admit several common properties with non-degenerate reaction–diffusion systems, for which the literature is abundant (see for instance [1,7,16,19–21] and the references cited therein). In particular, the existence of the global attractor has been established in [18] for the degenerate reaction–diffusion systems, under the assumption that $f (u, w)$ is linear with respect to u; the existence of exponential attractors for degenerate reaction–diffusion systems satisfying similar assumptions has been proved in [6]. Furthermore, the spatial diffusion effect has been proved to act indirectly on the non-diffusive species in [5]. However, the degeneracy of the diffusion process in one equation can provoke singular dynamics, which profoundly differ from the non-degenerate case, almost when the non-diffusive equation admits a strong nonlinearity. One of the most interesting singular properties of the degenerate reaction–diffusion systems is the existence of discontinuous stationary solutions, which has been already analyzed in several papers (notably in [15], and in [13] where the local stability of the discontinuous stationary solutions has been investigated). These discontinuous stationary solutions are seen to be the result of a hysteresis process and are incompatible with the existence of the global attractor, as proved recently in [2]. However, they can still attract the orbits of the system in a slightly weakened topology, as shown in [10], where a Lyapunov function is used for studying a degenerate system quite similar to (1). Nonetheless, the asymptotic behaviour of these degenerate systems is not fully understood. Indeed, the elegant result presented in [10] proves that the orbits converge to a discontinuous stationary solution, but, among the infinite number of such discontinuous solutions, it is not known to which one. It is precisely the aim of the present paper to answer this question, and to characterize a particular discontinuous pattern as the convergence point of a non trivial set of trajectories. To establish this novel convergence result, we use a principle of preservation of an initial distribution within locally invariant regions, under the flow induced by the degenerate reaction–diffusion system (1). Afterwards, we look for a discontinuous stationary solution which is compatible with this principle of preservation. Finally, we prove the convergence of the trajectories towards this particular discontinuous pattern in the $L^{2}$ norm and explain why it cannot occur in the $L^{\infty}$ norm; to control the hysteresis effect of the non-invertible cubic function (2), we rely on a macroscopic mass effect, which roughly states that fast particles help slow particles to converge.

Our paper is organized as follows. In Section 2, we show that the dynamics of the degenerate reaction–diffusion system (1) can be described by means of a potential; we present the functional context which guarantees the existence of global solutions (Theorem 1) and state the principle of preservation of an initial distribution within locally invariant regions (Theorem 2). In Section 3, we show that the degenerate reaction–diffusion system (1) admits a continuum of discontinuous stationary solutions (Theorem 3) and identify in this continuum the only discontinuous stationary solution which is compatible with the locally invariant regions (Proposition 1). Finally, two novel convergence results are presented and proved in Section 4 (Theorems 4 and 5).

2. Potential, global solutions and positively invariant regions

In this section, we prove that the degenerate reaction–diffusion system (1) admits a potential that partially governs the dynamics of its trajectories. Then we establish the existence of global solutions and of locally invariant regions.

2.1. Homogeneous stationary solutions and potential

First, we observe that system (1) admits three homogeneous stationary solutions, which are denoted $O (0, 0)$ , $U^{+} (u^{*}, w^{*})$ , $U^{-} (- u^{*}, - w^{*})$ , with $u^{*} = \sqrt{1 + \frac{α^{2}}{β}}$ and $w^{*} = \frac{α}{β} u^{*}$ . Next, we remark that system (1) admits a potential $H (u, w)$ given by $\begin{matrix} (4) & H (u, w) = β \frac{w^{2}}{2} - α u w + Q (u), \end{matrix}$ where Q is a primitive of q on $R$ , so that the two first equations of (1) can be rewritten $\begin{matrix} \frac{\partial u}{\partial t} = - \frac{\partial H}{\partial u}, \frac{\partial w}{\partial t} = δ Δ w - \frac{\partial H}{\partial w} . \end{matrix}$ The energy levels of the potential $H (u, w)$ are shown in Fig. 1(b). The stationary homogeneous solutions O, $U^{+}$ and $U^{-}$ of the degenerate reaction–diffusion system (1) correspond to the critical points of the potential $H (u, w)$ : O is a saddle of $H (u, w)$ , whereas $U^{+}$ , $U^{-}$ are sinks of equal levels.

Fig. 1.

(a) Curve of the non-invertible cubic function $q (u)$ defined by (2). The intersection of the cubic with the line $\frac{α^{2}}{β} u$ determines three homogeneous stationary solutions denoted $O (0, 0)$ , $U^{+} (u^{*}, w^{*})$ , $U^{-} (- u^{*}, - w^{*})$ . (b) Energy levels of the potential $H (u, w)$ defined by (4). O is a saddle of $H (u, w)$ , whereas $U^{+}$ , $U^{-}$ are sinks of equal levels.

In absence of diffusion, that is, if $δ = 0$ , the degenerate reaction–diffusion system (1) reduces to a system of ordinary differential equations which can be written in a gradient form: $\begin{matrix} (5) & \dot{u} = - \frac{\partial H}{\partial u}, \dot{w} = - \frac{\partial H}{\partial w} . \end{matrix}$ Depending on their initial condition, the orbits of the latter gradient system are attracted to one of the sinks $U^{-}$ or $U^{+}$ of the potential $H (u, w)$ , provided this initial condition does not coincide with the saddle point O. Furthermore, it can easily be proved that the rate of convergence of the orbits towards one of the sinks tends to 0 if the initial condition tends to the saddle point 0. However, in presence of diffusion, that is, if $δ > 0$ , the dynamics of system (1) are completely modified and do not resemble the dynamics of system (5). In particular, it is observed that discontinuous patterns connecting the sinks $U^{+}$ and $U^{-}$ emerge. Roughly speaking, these discontinuous patterns succeed in crossing the homoclinic loops encircling $U^{-}$ and $U^{+}$ in a jump discontinuity, and seem to attract some particular trajectories. It is precisely the purpose of this paper to identify a non trivial class of trajectories which converge towards such a discontinuous pattern, and to explain the role of hysteresis, provoked by the non-invertible cubic function $q (u)$ , in that process.

Remark (Applications in life sciences).

The degenerate reaction–diffusion system (1) is a generic model for a great number of phenomena arising in biology or ecology, which are characterized by the cohabitation of diffusing species or particles with static ones. One important example of application of degenerate reaction–diffusion systems similar to system (1) arises in the study of cellular or intracellular dynamics, as found in [9,11–13] or [17] for instance. Another original application of system (1) is given by the forest dynamics model presented in [14], and further studied in numerous papers (see [2,3,10,15] and the references cited therein): in this forest model, the static species correspond to trees, whereas diffusing particles correspond to seeds.

2.2. Global solutions

Now we intend to show the existence of global solutions to the degenerate reaction–diffusion system (1). To that aim, we handle system (1) in the Banach space X given by $\begin{matrix} X = L^{\infty} (Ω) \times L^{2} (Ω), \end{matrix}$ equipped with the product norm, and consider the diagonal operator A defined by $\begin{matrix} (6) & A = (\begin{matrix} 1 & 0 \\ 0 & - δ Λ + β \end{matrix}), \end{matrix}$ where Λ denotes the Laplacian operator in $L^{2} (Ω)$ , with the Neumann boundary condition on $\partial Ω$ . It is known that A is a sectorial operator of angle strictly less than $\frac{π}{2}$ , definite-positive, self-adjoint, with domain $D (A) = L^{\infty} (Ω) \times H_{N}^{2} (Ω)$ (see for instance [22]). Here $H_{N}^{2} (Ω) = {w \in W^{2, 2} (Ω) ∣ \frac{\partial w}{\partial ν} = 0 on \partial Ω}$ , where $W^{2, 2} (Ω)$ denotes the usual Sobolev space of $L^{2} (Ω)$ functions admitting derivatives in $L^{2} (Ω)$ up to order 2. The following theorem establishes the existence of global solutions and of a positively invariant region. The proof can be found in [2].

Theorem 1 (Global solutions).

For each $U_{0} \in X$ , the degenerate reaction–diffusion system ( 1 ) admits a unique global solution $U = (u, w)$ defined on $[0, + \infty)$ and satisfying $\begin{array}{l} u \in C ([0, + \infty), L^{\infty} (Ω)) \cap C^{1} ((0, + \infty), L^{\infty} (Ω)), \\ w \in C ((0, + \infty), H_{N}^{2} (Ω)) \cap C ([0, + \infty), L^{2} (Ω)) \cap C^{1} ((0, + \infty), L^{2} (Ω)) . \end{array}$ Furthermore, the region $R^{+}$ defined by $\begin{matrix} (7) & R^{+} = {(u, w) \in X; (u (x), w (x)) \in [- u^{*}, u^{*}] \times [- w^{*}, w^{*}] a.e. in Ω}, \end{matrix}$ is positively invariant under the flow induced by the degenerate reaction–diffusion ( 1 ).

The latter theorem proves that the trajectories of the degenerate reaction–diffusion system (1) starting in the region $R^{+}$ remain in $R^{+}$ for all future time. We will now establish a stronger property, stating that some particular initial distributions within local sub-regions of $R^{+}$ are preserved.

2.3. Locally invariant regions

Since Ω is a one-dimensional domain, we can without loss of generality write $Ω = (- ℓ, ℓ)$ with $ℓ > 0$ . For the rest of the paper, we set $\begin{matrix} (8) & Ω = Ω_{1} \cup Γ \cup Ω_{2} with Ω_{1} = (- ℓ, 0), Γ = {0}, Ω_{2} = (0, ℓ) . \end{matrix}$ Next, we introduce the rectangles $R_{1}$ and $R_{2}$ defined by $\begin{matrix} (9) & R_{1} = [- u^{*}, 0] \times [- w^{*}, 0], R_{2} = [0, u^{*}] \times [0, w^{*}] . \end{matrix}$ The rectangles $R_{1}$ and $R_{2}$ are symmetric with respect to O in the $(u, w)$ plane and satisfy $\begin{matrix} {U \in X; U (x) \in R_{1} \forall x \in Ω} \subset R^{+}, {U \in X; U (x) \in R_{2} \forall x \in Ω} \subset R^{+} . \end{matrix}$ Finally, we consider the set $X_{0} \subset X$ of initial conditions $U_{0}$ defined in Ω and satisfying $\begin{matrix} (10) & \begin{array}{l} U_{0} (x) \in R_{1}, \forall x \in Ω_{1}, \\ U_{0} (x) \in R_{2}, \forall x \in Ω_{2}, \\ U_{0} (x) = - U_{0} (- x), \forall x \in Ω . \end{array} \end{matrix}$ In this way, we have $X_{0} \subset R^{+}$ . Note that the third property in (10) implies that $U_{0} (0) = (0, 0)$ . Roughly speaking, $X_{0}$ is the set of initial conditions which are symmetrically distributed within the regions $R_{1}$ and $R_{2}$ . We emphasize that $X_{0}$ is a non-empty set, since it contains any initial condition $(u_{0}, w_{0})$ of the form $\begin{matrix} u_{0} (x) = u^{*} \frac{k}{ℓ} x, w_{0} (x) = w^{*} \frac{k}{ℓ} x, x \in Ω, \end{matrix}$ where k is a real number such that $0 ⩽ k ⩽ 1$ .

The following theorem proves that the global solution $U (t, x)$ of system (1) stemming from $U_{0} \in X_{0}$ is itself symmetrically distributed in $R_{1}$ and $R_{2}$ for all future time. We skip the proof which can be adapted from that of Theorem 5 in [2].

Theorem 2.
Let $U_{0} \in X_{0}$ and denote the global solution of the degenerate reaction–diffusion system ( 1 ) stemming from $U_{0}$ by $U (t, x)$ . Then we have: $\begin{array}{l} (11) & U (t, x) = - U (t, - x), \\ (12) & \nabla U (t, x) = \nabla U (t, - x), \end{array}$ for each $t > 0$ and all $x \in Ω$ .

Furthermore, the regions $R_{1}$ and $R_{2}$ are preserved under the flow induced by the degenerate reaction–diffusion ( 1 ), that is $\begin{matrix} (13) & U (t, x) \in R_{1}, \forall (t, x) \in (0, \infty) \times Ω_{1}, U (t, x) \in R_{2}, \forall (t, x) \in (0, \infty) \times Ω_{2} . \end{matrix}$

Note that properties (11) and (12) imply that both the fluxes $\frac{\partial w}{\partial x} (t, 0) \times w (t, 0)$ and $\frac{\partial u}{\partial x} (t, 0) \times u (t, 0)$ at the saddle point O are null. In other words, the diffusing species w never crosses the saddle point O and remains in one of the rectangles $R_{1}$ or $R_{2}$ . In parallel, the same holds for the static species u, although it is subject to an indirect diffusion effect induced by the coupling with the diffusing species w.
3. A continuum of discontinuous stationary solutions

One of the very interesting properties of the degenerate reaction–diffusion system (1) is that it admits an infinite number of non-homogeneous stationary solutions, some of them being even discontinuous in space. This property is a consequence of the non-invertibility of the cubic function $q (u)$ defined by (2). It could happen that these discontinuous stationary solutions are isolated. However, it can be proved that the set of non-homogeneous stationary solutions of system (1) contains a continuum of discontinuous solutions.

We can construct this continuum as follows. First, basic computations show that the cubic function $q (u)$ defined by (2) admits a local maximum m such that $\begin{matrix} (14) & m = q (- \frac{1}{\sqrt{3}}) = \frac{2}{3 \sqrt{3}} . \end{matrix}$ Next, let $σ \in (- m, m)$ . For each $z \in R$ such that $| z | ⩽ m$ , the equation $q (u) = z$ admits three solutions $u_{1} ⩽ u_{2} ⩽ u_{3}$ . We then consider the function $h_{σ}$ defined by $\begin{matrix} h_{σ} (z) = \{\begin{matrix} u_{1} & if - m ⩽ z < σ, \\ u_{2} & if z = σ, \\ u_{3} & if σ < z ⩽ m . \end{matrix} \end{matrix}$ Finally, if $| z | > m$ , we define $h_{σ} (z)$ as the unique $u \in R$ such that $q (u) = z$ . In this way, $h_{σ}$ is a piecewise continuous and increasing function defined on $R$ , admitting a unique jump discontinuity at $z = σ$ , and satisfying $q (h_{σ} (z)) = z$ for all $z \in R$ and $h_{σ} (q (u)) = u$ for all $u \in R$ . In other words, the one-parameter family ${h_{σ}}_{σ \in (- m, m)}$ is made of inverses of the cubic function $q (u)$ . Note that $q (u)$ admits an infinite number of other inverses, which are not necessarily monotone nor piecewise continuous.

Afterwards, we search for non-homogeneous stationary solutions $(u (x), w (x))$ of the degenerate reaction–diffusion system (1) by solving the elliptic system $\begin{matrix} (15) & \{\begin{matrix} 0 = α w (x) - q (u (x)), & x \in Ω, \\ 0 = δ \frac{d^{2} w}{d x^{2}} (x) - β w (x) + α u (x), & x \in Ω, \\ \frac{\partial w}{\partial ν} (- ℓ) = \frac{\partial w}{\partial ν} (ℓ) = 0 . \end{matrix} \end{matrix}$ Hence, we are led to consider the following ordinary differential equation of order 2: $\begin{matrix} (16) & δ \ddot{w} (x) - β w (x) + α h_{σ} (α w (x)) = 0, x \in R . \end{matrix}$ Obviously, each solution $w (x)$ of the latter differential equation, which moreover satisfies the boundary condition $\dot{w} (- ℓ) = \dot{w} (ℓ) = 0$ , determines a solution of system (15), by setting $u (x) = h_{σ} (α w (x))$ for all $x \in Ω$ . Equation (16) can be solved by introducing the variable $v = \dot{w}$ , which leads to the following system: $\begin{matrix} (17) & \{\begin{matrix} \dot{v} = \frac{β}{δ} w - \frac{α}{δ} h_{σ} (α w), \\ \dot{w} = v . \end{matrix} \end{matrix}$ Although its right-hand side contains the discontinuous function $h_{σ}$ , system (17) can be solved by using the method of Filippov for differential equations of Carathéodory type [8]. As an example, the phase portrait of system (17) for $σ = 0$ is illustrated in Fig. 2(a). In this phase portrait, we observe three critical points: $(0, 0)$ is a center, whereas $(0, - w^{*})$ and $(0, w^{*})$ are saddle points. Next, following the method presented in [13], it can be proved that a unique orbit of system (17) determines a monotone solution $w_{σ} (x)$ of the second order equation (16), with $x \in \bar{Ω}$ , and satisfying the boundary condition ${\dot{w}}_{σ} (- ℓ) = {\dot{w}}_{σ} (ℓ) = 0$ . In turn, this solution $w_{σ} (x)$ determines a discontinuous stationary solution of system (1), by setting $u_{σ} (x) = h_{σ} (α w_{σ} (x))$ for all $x \in Ω$ . We denote this discontinuous stationary solution $\begin{matrix} (18) & U_{σ} (x) = {(u_{σ} (x), w_{σ} (x))}^{⊤}, x \in Ω . \end{matrix}$ Since $α w_{σ} (x) = q (u_{σ} (x))$ , we emphasize that the discontinuous solution $U_{σ} (x)$ determines a curve parametrized by $x \in Ω$ which is contained in the curve of the cubic function $q (u)$ , as illustrated in Fig. 2(b).

Finally, the principle of continuity of the solution of an ordinary differential equation under the variation of a small parameter guarantees that $U_{σ}$ varies smoothly with σ (note that the principle of continuity is also valid for differential equations admitting a discontinuous right-hand side, as proved in [8], Chapter 2, Section 8). Overall, we obtain the following theorem.

Fig. 2.

(a) Phase portrait of the system of ordinary differential equations (17) for $σ = 0$ , admitting a discontinuous right-hand side: $(0, 0)$ is a center, whereas $(0, - w^{*})$ and $(0, w^{*})$ are saddle points. A unique orbit determines a monotone solution $w_{σ} (x)$ of the second order equation (16), with $x \in \bar{Ω}$ , and satisfying the boundary condition ${\dot{w}}_{σ} (- ℓ) = {\dot{w}}_{σ} (ℓ) = 0$ . (b) For all $σ \in (- m, m)$ , the degenerate reaction–diffusion system (1) admits a discontinuous stationary solution $U_{σ} (x) = {(u_{σ} (x), w_{σ} (x))}^{⊤}$ , contained in the curve of the cubic function $q (u)$ . The family ${U_{σ}}_{σ \in (- m, m)}$ forms a continuum of discontinuous stationary solutions.

Theorem 3 (Continuum of discontinuous stationary solutions).

Let m be the local maximum of the cubic function $q (u)$ , given by ( 14 ). Then, for all $σ \in (- m, m)$ , the degenerate reaction–diffusion system ( 1 ) admits a unique non-homogeneous stationary solution $U_{σ} (x) = {(u_{σ} (x), w_{σ} (x))}^{⊤}$ satisfying the following properties:

$u_{σ}$ and $w_{σ}$ are increasing functions on Ω;

$u_{σ}$ admits a unique discontinuity at $x_{σ} \in Ω$ such that $u_{σ} (x) > \frac{1}{\sqrt{3}}$ if $x > x_{σ}$ and $u_{σ} (x) < \frac{- 1}{\sqrt{3}}$ if $x < x_{σ}$ .

Furthermore, the one parameter family ${U_{σ}}_{σ \in (- m, m)}$ forms a continuum in $L^{\infty} {(Ω)}^{2}$ , that is, for each $ε > 0$ , there exist $σ_{1}$ , $σ_{2}$ in $(- m, m)$ such that $σ_{1} \neq σ_{2}$ and $\begin{matrix} ‖ U_{σ_{1}} - U_{σ_{2}} ‖_{L^{\infty} {(Ω)}^{2}} < ε . \end{matrix}$

Although Theorem 3 establishes the existence of discontinuous stationary solutions whose components are monotone, we emphasize that non monotone discontinuous stationary solutions can also be proved to exist. For these non monotone discontinuous stationary solutions, multiple discontinuities are observed. Next, in the family ${U_{σ}}_{σ \in (- m, m)}$ , the discontinuous stationary solution obtained for $σ = 0$ , which we denote $\bar{U} (x) = (\bar{u} (x), \bar{w} (x))$ in order to avoid any confusion with an initial condition $U_{0}$ , plays an important role. The following proposition summarizes its characteristic properties, which are depicted in Fig. 3.

Proposition 1.
The discontinuous stationary solution $\bar{U} (x) = (\bar{u} (x), \bar{w} (x))$ is the only one in the family ${U_{σ}}_{σ \in (- m, m)}$ which satisfies the following properties:
$\bar{U} (x) = - \bar{U} (- x)$ for all $x \in Ω$ ;

$\bar{U} (x) \in R_{1}$ for all $x \in Ω_{1}$ and $\bar{U} (x) \in R_{2}$ for all $x \in Ω_{2}$ .

Fig. 3.
Particular discontinuous stationary $\bar{U} (x) = (\bar{u} (x), \bar{w} (x))$ of the degenerate reaction–diffusion system (1), satisfying the properties of theorem 3 and proposition 1. (A) view in the $(u, α w)$ plane. (B) view with respect to $x \in Ω$ .
4. Convergence towards discontinuous stationary solutions

We are now interested in studying the asymptotic behaviour of the orbits of the degenerate reaction–diffusion system (1). The following novel theorem states that the particular discontinuous stationary solution $\bar{U} (x) = (\bar{u} (x), \bar{w} (x))$ given in Proposition 1 attracts in the $L^{2}$ norm the orbits of the degenerate reaction–diffusion system (1) starting from any initial condition in the set $X_{0}$ given by (10).

Theorem 4.
Let $U_{0}$ be an initial condition in the set $X_{0}$ defined by ( 10 ), and let $U (t, x)$ denote the global solution of the degenerate reaction–diffusion system ( 1 ) stemming from $U_{0}$ . Assume that $\begin{matrix} (19) & q^{'} (1) > \frac{α^{2}}{β} . \end{matrix}$ Then there exist $t^{} > 0$ and* $ρ^{} > 0$ such that* $\begin{matrix} (20) & {‖ U (t, x) - \bar{U} (x) ‖}_{L^{2} {(Ω)}^{2}} ⩽ {‖ U_{0} (x) - \bar{U} (x) ‖}_{L^{2} {(Ω)}^{2}} e^{- ρ^{} t}, \forall t > t^{}, \end{matrix}$ where $\bar{U} (x)$ is the particular discontinuous stationary solution given in Proposition 1 .

Before proving Theorem 4, some remarks are useful to provide intuition on the dynamics of the degenerate reaction–diffusion system (1).

First, we observe that the behaviour of the orbits of system (1) starting in $X_{0}$ can be divided into two periods, separated by the critical time $t^{}$ established in Theorem 4. Before $t^{}$ , the hysteresis effect induced by the non-invertible cubic function $q (u)$ acts as a separator of the orbits, which are equally pushed to one of the two rectangles $R_{1}$ and $R_{2}$ defined by (9), according to Theorem 2. After $t^{}$ , the asymptotic phase begins and the particular discontinuous solution $\bar{U} (x)$ attracts the orbits at an exponential rate, which is uniform in space.

Next, for studying the asymptotic phase of system (1) and determining the critical time $t^{}$ , we consider the energy functions $E_{i}$ and $F_{i}$ defined for $i \in {1, 2}$ and $t ⩾ 0$ by $\begin{matrix} (21) & E_{i} (t) = \frac{1}{2} \int_{Ω_{i}} {| u (t, x) - \bar{u} (x) |}^{2} d x, F_{i} (t) = \frac{1}{2} \int_{Ω_{i}} {| w (t, x) - \bar{w} (x) |}^{2} d x, \end{matrix}$ where $Ω_{1}$ and $Ω_{2}$ are given by (8), and the total energy $E (t)$ defined for $t ⩾ 0$ by $\begin{matrix} (22) & E (t) = E_{1} (t) + E_{2} (t) + F_{1} (t) + F_{2} (t) . \end{matrix}$ Using the first equation in system (1), we compute, for $i \in {1, 2}$ and $t ⩾ 0$ : $\begin{matrix} \frac{d E_{i}}{d t} (t) = α \int_{Ω_{i}} (w (t) - \bar{w}) (u (t) - \bar{u}) d x - \int_{Ω_{i}} [q (u (t)) - q (\bar{u})] (u (t) - \bar{u}) d x, \end{matrix}$ where we omit the space variable x in order to lighten the notations.

Here, we highlight that the difficult point for proving Theorem 4 is to find a positive coefficient $κ_{0}$ in order to control the quantity $- [q (u (t)) - q (\bar{u})] (u (t) - \bar{u})$ , under the integral symbol, by an expression of the form $- κ_{0} | u (t) - \bar{u} |^{2}$ . Indeed, it is easy to prove that the quantity $J (t, x)$ defined by $\begin{matrix} J (t, x) = [q (u (t, x)) - q (\bar{u} (x))] (u (t, x) - \bar{u} (x)) \end{matrix}$ is non negative for all $t ⩾ 0$ and all $x \in Ω$ . However, it could happen that $J (t, x)$ tends to 0 as x tends to 0, since in that case, $U (t, x)$ would approach (by virtue of Equation (11) in Theorem 2) the saddle point $(0, 0)$ and loose its mobility, in the sense that the gradient $\nabla U (t, x)$ would vanish. Hence, it is to crucial to take into account a macroscopic mass effect, which roughly means that the particles $U (t, x)$ at some place $x \in Ω$ located far from the saddle point $(0, 0)$ help other particles, which are located close to the saddle, to move away from that saddle. Note that the mass effect is direct on $w (t, x)$ , due to the action of the diffusion operator Δ, whereas it acts indirectly on $u (t, x)$ , through the couplings $\pm α u w$ in system (1). This mass effect is visible below in the proof of Lemma 1, namely in Equation (28). In absence of diffusion (that is, if $δ = 0$ ), the mass effect does not hold, and the convergence result stated in Theorem 4 is not valid any longer.

Therefore, we introduce, for $i \in {1, 2}$ and $t ⩾ 0$ : $\begin{matrix} (23) & Ψ_{i} (t) = \int_{Ω_{i}} [q (u (t, x)) - q (\bar{u} (x))] (u (t, x) - \bar{u} (x)) d x . \end{matrix}$ It is easily observed that $E_{i} (t) > 0$ for all $t ⩾ 0$ and $i \in {1, 2}$ , provided $U_{0} \neq \bar{U}$ . Thus we can consider, for all $t ⩾ 0$ and $i \in {1, 2}$ : $\begin{matrix} (24) & ψ_{i} (t) = \frac{Ψ_{i} (t)}{E_{i} (t)} . \end{matrix}$ The following lemma determines the asymptotic behaviour of the functions $ψ_{1}$ , $ψ_{2}$ .
Lemma 1 (Mass effect).

Assume that $U_{0} \neq \bar{U}$ . Then the functions $ψ_{1}$ , $ψ_{2}$ defined by ( 24 ) satisfy $\begin{matrix} \underset{t \to + \infty}{lim inf} ψ_{i} (t) ⩾ 2 q^{'} (1), i \in {1, 2} . \end{matrix}$

Proof of Lemma 1.
Let $x \in Ω_{2}$ . We have, by virtue of the first equation in system (1): $\begin{matrix} \frac{\partial u}{\partial t} (t, x) = α w (t, x) - q (u (t, x)), \forall t > 0 . \end{matrix}$ Next, since $U_{0} \in X_{0}$ , Theorem 2 applies and guarantees that $w (t, x) ⩾ 0$ for all $t ⩾ 0$ . Hence we have $\begin{matrix} \frac{\partial u}{\partial t} (t, x) ⩾ - q (u (t, x)), \forall t > 0 . \end{matrix}$ Now, we compare $u (t, x)$ with the solution of the Cauchy problem for the ordinary differential equation given by $\begin{matrix} (25) & \{\begin{matrix} \dot{z} (t) = - q (z (t)), t > 0, \\ z (0) = u_{0} (x) . \end{matrix} \end{matrix}$ We denote by $z (t, x)$ the solution of the latter Cauchy problem. Standard comparison principles [4] guarantee that $\begin{matrix} u (t, x) ⩾ z (t, x), \end{matrix}$ for all $t ⩾ 0$ .

Afterwards, since $q (z) = z^{3} - z$ , it is easy to describe the asymptotic behaviour of the solutions to the Cauchy problem (25). Indeed, since $u_{0} (x) > 0$ , it can be deduced that $z (t, x)$ tends to 1 as t tends to $+ \infty$ . Furthermore, the function $t \mapsto z (t, x)$ is an increasing function of t if $u_{0} (x) \in (0, 1)$ , whereas it is a decreasing function of t if $u_{0} (x) > 1$ . In addition, we have $- q (a) ⩾ 0$ for all real number $a \in [0, 1]$ . Consequently, the function $t \mapsto u (t, x)$ is increasing while $0 ⩽ u (t, x) ⩽ 1$ (if $u (t, x)$ becomes greater than 1, it may become decreasing).

Now we fix $η > 0$ . For each $t ⩾ 0$ , we write $Ω_{2}$ as the disjoint union $\begin{matrix} (26) & Ω_{2} = Ω_{2}^{⩾ 1 - η} (t) \cup Ω_{2}^{< 1 - η} (t), \end{matrix}$ with $\begin{matrix} Ω_{2}^{⩾ 1 - η} (t) = {x \in Ω_{2}; u (t, x) ⩾ 1 - η}, Ω_{2}^{< 1 - η} (t) = {x \in Ω_{2}; u (t, x) < 1 - η} . \end{matrix}$ It is easily seen that the following properties are fulfilled: $\begin{matrix} (27) & \begin{matrix} Ω_{2}^{⩾ 1 - η} (t) \subset Ω_{2}^{⩾ 1 - η} (t^{'}), Ω_{2}^{< 1 - η} (t) \supset Ω_{2}^{< 1 - η} (t^{'}), if t < t^{'}, \\ ⋃_{t > 0} Ω_{2}^{⩾ 1 - η} (t) = Ω_{2}, ⋂_{t > 0} Ω_{2}^{< 1 - η} (t) = \emptyset . \end{matrix} \end{matrix}$ Afterwards, we define for all $t ⩾ 0$ and all $x \in Ω_{2}$ : $\begin{matrix} S (t, x) = \frac{q (u (t, x)) - q (\bar{u} (x))}{u (t, x) - \bar{u} (x)}, \end{matrix}$ which allows to rewrite the function $Ψ_{2} (t)$ defined in (23) as $\begin{matrix} Ψ_{2} (t) = \int_{Ω_{2}} S (t, x) {| u (t, x) - \bar{u} (x) |}^{2} d x . \end{matrix}$ Note that $S (t, x)$ gives the rate of increase of the line that joins the points of coordinates $(\bar{u} (x), q (\bar{u} (x)))$ and $(u (t, x), q (u (t, x)))$ in the $(u, α w)$ plane, as depicted in Fig. 3(a). Using the splitting of $Ω_{2}$ given by (26), we write, for all $t ⩾ 0$ : $\begin{matrix} Ψ_{2} (t) = \int_{Ω_{2}^{< 1 - η} (t)} S (t, x) {| u (t, x) - \bar{u} (x) |}^{2} d x + \int_{Ω_{2}^{⩾ 1 - η} (t)} S (t, x) {| u (t, x) - \bar{u} (x) |}^{2} d x . \end{matrix}$ Since $S (t, x) ⩾ q^{'} (1 - η)$ for all $x \in Ω_{2}^{⩾ 1 - η} (t)$ , we obtain $\begin{matrix} Ψ_{2} (t) ⩾ \int_{Ω_{2}^{< 1 - η} (t)} S (t, x) {| u (t, x) - \bar{u} (x) |}^{2} d x + q^{'} (1 - η) \int_{Ω_{2}^{⩾ 1 - η} (t)} {| u (t, x) - \bar{u} (x) |}^{2} d x . \end{matrix}$ We divide the latter inequality by $E_{2} (t)$ and deduce $\begin{matrix} (28) & \begin{matrix} ψ_{2} (t) & ⩾ \frac{\int_{Ω_{2}^{< 1 - η} (t)} S (t, x) | u (t, x) - \bar{u} (x) |^{2} d x}{E_{2} (t)} + q^{'} (1 - η) \frac{\int_{Ω_{2}^{⩾ 1 - η} (t)} | u (t, x) - \bar{u} (x) |^{2} d x}{E_{2} (t)}, \end{matrix} \end{matrix}$ which implies $\begin{matrix} (29) & ψ_{2} (t) ⩾ 2 q^{'} (1 - η) \frac{\int_{Ω_{2}^{⩾ 1 - η} (t)} | u (t, x) - \bar{u} (x) |^{2} d x}{\int_{Ω_{2}} | u (t, x) - \bar{u} (x) |^{2} d x} . \end{matrix}$ By virtue of (27), passing to the limit in (29) as t tends to $+ \infty$ leads to $\begin{matrix} \underset{t \to + \infty}{lim inf} ψ_{2} (t) ⩾ 2 q^{'} (1 - η) . \end{matrix}$ Since the latter inequality is valid for any η, and using the continuity of the cubic function q, we finally obtain $\begin{matrix} \underset{t \to + \infty}{lim inf} ψ_{2} (t) ⩾ 2 q^{'} (1) . \end{matrix}$ The same reasoning holds in $Ω_{1}$ , thus the same estimate can be deduced for $ψ_{1} (t)$ and Lemma 1 is proved. □

Afterwards, it can easily be proved that the functions $ψ_{1}$ , $ψ_{2}$ are continuous on $[0, + \infty)$ . Since $ψ_{1} (0) > 0$ and $ψ_{2} (0) > 0$ , provided $U_{0} \neq \bar{U}$ , we obtain the following lemma.
Lemma 2.
Assume that $U_{0} \neq \bar{U}$ . Then the functions $ψ_{1}$ , $ψ_{2}$ defined by ( 24 ) satisfy the following properties.
If $q^{'} (1) > \frac{α^{2}}{β}$ , then for all $k^{}$ such that* $q^{'} (1) > k^{} > \frac{α^{2}}{β}$ , a positive time* $t^{}$ can be found such that* $ψ_{i} (t) ⩾ 2 k^{}$ , for all* $t ⩾ t^{}$ and* $i \in {1, 2}$ .

There exists $k_{0} > 0$ such that $ψ_{i} (t) ⩾ 2 k_{0}$ for all $t ⩾ 0$ and $i \in {1, 2}$ .

We are now ready to prove Theorem 4.
Proof of Theorem 4.
If $U_{0} = \bar{U}$ , the conclusion is immediate, since $\bar{U}$ is a stationary solution. Hence, we suppose that $U_{0} \neq \bar{U}$ . We first estimate the derivative of $E_{2}$ , omitting again the space variable x for short: $\begin{matrix} \frac{d E_{2}}{d t} (t) = α \int_{Ω_{2}} (w (t) - \bar{w}) (u (t) - \bar{u}) d x - \int_{Ω_{2}} [q (u (t)) - q (\bar{u})] (u (t) - \bar{u}) d x . \end{matrix}$ Applying the Young inequality $a b ⩽ \frac{ε a^{2}}{2} + \frac{b^{2}}{2 ε}$ , which is valid for all a, b in $R$ and all $ε > 0$ , we have $\begin{matrix} α \int_{Ω_{2}} (w (t) - \bar{w}) (u (t) - \bar{u}) d x ⩽ α ε F_{2} (t) + \frac{α}{ε} E_{2} (t) . \end{matrix}$ In parallel, by virtue of the first assertion in Lemma 2, we have $\begin{matrix} \int_{Ω_{2}} [q (u (t)) - q (\bar{u})] (u (t) - \bar{u}) d x = Ψ_{2} (t) ⩾ 2 k^{} E_{2} (t), \end{matrix}$ for all $t ⩾ t^{}$ , with $t^{} > 0$ and $k^{} > \frac{α^{2}}{β}$ . It follows that $\begin{matrix} \frac{d E_{2}}{d t} (t) ⩽ α ε F_{2} (t) + \frac{α}{ε} E_{2} (t) - 2 k^{} E_{2} (t), \end{matrix}$ for all $t ⩾ t^{}$ .

Next, we estimate the derivative of $F_{2}$ : $\begin{matrix} \frac{d F_{2}}{d t} (t) & = δ \int_{Ω_{2}} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x \\ + α \int_{Ω_{2}} (w (t) - \bar{w}) (u (t) - \bar{u}) - β \int_{Ω_{2}} {| w (t, x) - \bar{w} |}^{2} d x \\ ⩽ δ \int_{Ω_{2}} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x + α ε F_{2} (t) + \frac{α}{ε} E_{2} (t) - 2 β F_{2} (t) . \end{matrix}$ The same reasoning applies to the energy functions $E_{1}$ and $F_{1}$ defined over $Ω_{1}$ , which can be written: $\begin{matrix} \frac{d E_{1}}{d t} (t) ⩽ α ε F_{1} (t) + \frac{α}{ε} E_{1} (t) - 2 k^{} E_{1} (t), \\ \frac{d F_{1}}{d t} (t) ⩽ δ \int_{Ω_{1}} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x + α ε F_{1} (t) + \frac{α}{ε} E_{1} (t) - 2 β F_{1} (t) . \end{matrix}$ A simple sum of the latter inequalities leads to $\begin{matrix} \frac{d E}{d t} (t) & ⩽ δ \int_{Ω} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x \\ - 2 (k^{} - \frac{α}{ε}) (E_{1} (t) + E_{2} (t)) - 2 (β - α ε) (F_{1} (t) + F_{2} (t)) . \end{matrix}$ Now we use the Green–Riemann formula and the Neumann boundary condition to write $\begin{matrix} \int_{Ω} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x = - \int_{Ω} {| \nabla (w (t) - \bar{w}) |}^{2} d x . \end{matrix}$ We obtain $\begin{matrix} \frac{d E}{d t} (t) ⩽ - 2 (k^{} - \frac{α}{ε}) (E_{1} (t) + E_{2} (t)) - 2 (β - α ε) (F_{1} (t) + F_{2} (t)) . \end{matrix}$ Since $k^{} > \frac{α^{2}}{β}$ , we can choose $ε^{} > 0$ such that $k^{} > \frac{α}{ε^{}} > \frac{α^{2}}{β}$ , which guarantees that $k^{} - \frac{α}{ε^{}} > 0$ and $β - α ε^{} > 0$ . We then introduce $ρ^{} = min {k^{} - \frac{α}{ε^{}}, β - α ε^{}}$ and finally obtain $\begin{matrix} \frac{d E}{d t} (t) ⩽ - 2 ρ^{} E (t), \end{matrix}$ for all $t ⩾ t^{}$ . Applying Gronwall lemma leads to $\begin{matrix} E (t) ⩽ E (0) e^{- 2 ρ^{} t}, \end{matrix}$ for all $t ⩾ t^{}$ , from which (20) is deduced. The proof is complete. □

We emphasize that the exponential decay of the energy of system (1) in the $L^{2}$ norm cannot be proved to occur in the $L^{\infty}$ norm, since the uniform limit of continuous functions is necessarily continuous. This very simple remark is the main argument for proving the non-existence of the global attractor for the flow induced by the degenerate reaction–diffusion system (1), as presented in [2].

Next, Theorem 4 has been established under assumption (19), which is fulfilled if $q^{'} (1)$ is sufficiently large. If assumption (19) is not satisfied, we can however prove a second convergence result, which is valid if in counterpart the diffusion coefficient δ is sufficiently large. Theorem 5.
Let $U_{0}$ be an initial condition in the set $X_{0}$ defined by ( 10 ), and let $U (t, x)$ denote the global solution of the degenerate reaction–diffusion system ( 1 ) stemming from $U_{0}$ . Then there exist $δ_{0} > 0$ and $ρ_{0} > 0$ such that, if $δ ⩾ δ_{0}$ , the following estimate holds: $\begin{matrix} (30) & {‖ U (t, x) - \bar{U} (x) ‖}_{L^{2} {(Ω)}^{2}} ⩽ {‖ U_{0} (x) - \bar{U} (x) ‖}_{L^{2} {(Ω)}^{2}} e^{- ρ_{0} t}, \forall t > 0 . \end{matrix}$
Proof.
We use the same reasoning as in the proof of Theorem 4, but the second item of Lemma 2 instead of the first one, which leads to: $\begin{matrix} \frac{d E}{d t} (t) & ⩽ δ \int_{Ω} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x \\ - 2 (k_{0} - \frac{α}{ε}) (E_{1} (t) + E_{2} (t)) - 2 (β - α ε) (F_{1} (t) + F_{2} (t)) . \end{matrix}$ Now, we observe, by virtue of Theorem 2, that the mean value of $w (t) - \bar{w}$ over Ω is null. Consequently, the Poincaré–Wirtinger inequality guarantees that $\begin{matrix} δ \int_{Ω} Δ (w (t) - \bar{w}) (w (t) - \bar{w}) d x ⩽ - 2 δ μ (Ω) (F_{1} (t) + F_{2} (t)), \end{matrix}$ for all $t > 0$ , where $μ (Ω) > 0$ denotes the smallest positive eigenvalue of the Laplace operator on Ω with the Neumann boundary condition on $\partial Ω$ . We obtain $\begin{matrix} \frac{d E}{d t} (t) ⩽ - 2 (k_{0} - \frac{α}{ε}) (E_{1} (t) + E_{2} (t)) - 2 (δ μ (Ω) + β - α ε) (F_{1} (t) + F_{2} (t)) . \end{matrix}$ Hence, we choose $ε_{0} > 0$ such that $k_{0} - \frac{α}{ε_{0}} > 0$ . Then a positive $δ_{0}$ can be found such that $δ_{0} μ (Ω) + β - α ε_{0} > 0$ .

Finally, we introduce $ρ_{0} = min {k_{0} - \frac{α}{ε_{0}}, δ_{0} μ (Ω) + β - α ε_{0}}$ , and achieve the proof as for Theorem 4. □

5. Conclusion

The main result we have proved in this paper is the convergence of a non trivial set of trajectories of the degenerate reaction–diffusion system (1) in the $L^{2}$ norm. We have identified the limit as a discontinuous pattern which is the only one, among a continuum of discontinuous stationary solutions, to be compatible with the preservation of locally invariant regions by the flow induced by the degenerate system. Since the uniform limit of continuous functions is necessarily continuous, the convergence cannot occur in the $L^{\infty}$ norm.

This novel convergence result has however been established in the context of a symmetric cubic non-linearity. In the case of an asymmetric non-linearity, we can prove that the existence of a continuum of discontinuous patterns and the macroscopic mass effect still hold. Nevertheless, the principle of preservation of an initial distribution within locally invariant regions seems to fail in the asymmetric case, hence represents an exciting and promising research perspective.

References

J.M.

Arrieta,

Lopez-Fernandez and

Zuazua, Approximating travelling waves by equilibria of non-local equations, Asymptotic Analysis 78(3) (2012), 145–186. doi:10.3233/ASY-2011-1088.

Cantin, Non-existence of the global attractor for a partly dissipative reaction–diffusion system with hysteresis, Journal of Differential Equations 299 (2021), 333–361. doi:10.1016/j.jde.2021.07.023.

Cantin,

Ducrot and

B.M.

Funatsu, Mathematical modeling of forest ecosystems by a reaction–diffusion–advection system: Impacts of climate change and deforestation, Journal of Mathematical Biology 83(6) (2021), 1–45.

W.A.

Coppel , Stability and Asymptotic Behavior of Differential Equations, Heath, 1965.

Einav,

J.J.

Morgan and

B.Q.

Tang, Indirect diffusion effect in degenerate reaction–diffusion systems, SIAM Journal on Mathematical Analysis 52(5) (2020), 4314–4361. doi:10.1137/20M1319930.

Fabrie and

Galusinski, Exponential attractors for a partially dissipative reaction system, Asymptotic Analysis 12(4) (1996), 329–354. doi:10.3233/ASY-1996-12403.

P.C.

Fife, Asymptotic states for equations of reaction and diffusion, Bulletin of the American Mathematical Society 84(5) (1978), 693–726. doi:10.1090/S0002-9904-1978-14502-9.

A.F.

Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, Vol. 18, Springer Science & Business Media, 2013.

Hagberg and

Meron, Pattern formation in non-gradient reaction–diffusion systems: The effects of front bifurcations, Nonlinearity 7(3) (1994), 805. doi:10.1088/0951-7715/7/3/006.

10.

Iwasaki, Asymptotic convergence of solutions to the forest kinematic model, Nonlinear Analysis: Real World Applications 62 (2021), 103382. doi:10.1016/j.nonrwa.2021.103382.

11.

Kazmierczak and

Volpert, Travelling waves in partially degenerate reaction–diffusion systems, Mathematical Modelling of Natural Phenomena 2(2) (2007), 106–125. doi:10.1051/mmnp:2008021.

12.

Klika,

R.E.

Baker,

Headon and

E.A.

Gaffney, The influence of receptor-mediated interactions on reaction–diffusion mechanisms of cellular self-organisation, Bulletin of Mathematical Biology 74(4) (2012), 935–957. doi:10.1007/s11538-011-9699-4.

13.

Köthe,

Marciniak-Czochra and

Takagi, Hysteresis-driven pattern formation in reaction–diffusion-ODE systems, Discrete & Continuous Dynamical Systems-A 40(6) (2020), 3595. doi:10.3934/dcds.2020170.

14.

Y.A.

Kuznetsov,

M.Y.

Antonovsky,

V.N.

Biktashev and

E.A.

Aponina, A cross-diffusion model of forest boundary dynamics, Journal of Mathematical Biology 32(3) (1994), 219–232. doi:10.1007/BF00163879.

15.

Le Huy,

Tsujikawa and

Yagi, Stationary solutions to forest kinematic model, Glasgow Mathematical Journal 51(1) (2009), 1–17. doi:10.1017/S0017089508004485.

16.

Mallet-Paret and

G.R.

Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, Journal of the American Mathematical Society 1(4) (1988), 805–866. doi:10.1090/S0894-0347-1988-0943276-7.

17.

Marciniak-Czochra and

Kimmel, Reaction–diffusion model of early carcinogenesis: The effects of influx of mutated cells, Mathematical Modelling of Natural Phenomena 3(7) (2008), 90–114. doi:10.1051/mmnp:2008043.

18.

Martine, Finite-dimensional attractors associated with partly dissipative reaction–diffusion systems, SIAM Journal on Mathematical Analysis 20(4) (1989), 816–844. doi:10.1137/0520057.

19.

Smoller, Shock Waves and Reaction–Diffusion Equations, Vol. 258, Springer Science & Business Media, 2012.

20.

Strani, Slow dynamics in reaction–diffusion systems, Asymptotic Analysis 98(1–2) (2016), 131–154. doi:10.3233/ASY-161364.

21.

Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68, Springer Science & Business Media, 2012.

22.

Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Science & Business Media, 2009.

How hysteresis produces discontinuous patterns in degenerate reaction–diffusion systems