Global attractor and exponential attractor for a Parabolic system of Cahn-Hilliard with a proliferation term

1.   Introduction

The generalization of the Cahn-Hilliard equation

where $g$ is the proliferation term, has been proposed in ^[3,5] as a model for the growth of cancerous tumors and other biological entities.

Generally, $g$ can be the linear function $g(s) = \alpha s$, $\alpha > 0$ in which case $(1.1)$ is known as the Cahn-Hilliard-Oono equation and accounts for long-ranged interactions in the phase equations and in the phase separation process (see ^[8]). The other possibility is the quadratic function $g(s) = \alpha s(s-1)$, $\alpha > 0$ (nonlinear). In that case $(1.1)$ has applications in biology and, precisely, models wound healing and tumour growth (see ^[12]). To be more precise, such a model deals with cells which move, proliferate and interact via diffusion and cell-cell adhesion. Here, $u$ is the order parameter, $f$ is the local free energy and $\alpha$ is the proliferation rate.

These equation has been studied; we refer to, e.g., ^[3,5,9], in which authors have proved the existence and the uniqueness of the solution and the existence of the finite-dimensional attractors.

Precisely, we are going to study the model for the growth of cancerous tumors and other biological entities, model obtained by combinaison of (1.1) with the following temperature equation

The same kind of model without proliferation term are knwon as the conserved phase field model and has been studied (see for instance D. Brochet, X. Chen and D. Hilhost G ^[1], L. Cherfils and A. Miranville ^[2], Gilardi ^[6], C. Giorgi, M. Grasseli, and V. Pata A ^[7], Miranville ^[10], A. J. Ntsokongo and N. Batangouna ^[11]).

This work is structured as follows: firstly, we have the setting of the problem followed by a priori estimates which allow us to construct the dissipative semigroup associated to the problem, and finally we prove the existence of the exponential attractors and, thus, of finite-dimensional global attractors.

2.   Setting of the problem

We consider the following parabolic system of Cahn-Hilliard with a proliferation term

in a bounded and regular domain $\Omega \subset \mathbb{R}^{n}$($n = 1, 2$ or $3$) with boundary $\Gamma$.

$f$ and $g$ verifies the following properties:

where $F(s) = {\int^{s}_{0}}f(\xi)d\xi$ and for all $u\in L^{2}(\Omega)$, such that ${\int_{ \Omega} }F(u) dx < +\infty$, we have

Notation. We denote by $(., .)$ the usual $L^{2}$-scalar product, with associated norm $\Vert. \Vert$, and we set

$\Vert. \Vert_{-1} = \Vert (-\Delta)^{-\frac{1}{2}}.\Vert$, where $-\Delta$ denotes the minus Laplace operator with Dirichlet boundary conditions. More generally, $\Vert. \Vert_{X}$ denotes the norm in the Banach space $X$.

Remark 2.1. The properties (2.8) and (2.9) essentially mean that $g$ is subordinated to $f$. In particular, they hold for the usual choices $f(s) = s^{3}-s$ and $g(s) = \alpha s$ or $g(s) = \alpha s(s-1)$, $\alpha > 0$. More generally, (2.8) and (2.9) hold when $f$ and $g$ have polynomial growths of order $2p+1$, $p\geqslant 1$ and $q\geqslant 1$ such that $q\leqslant p+1$. Indeed, let us assume, for simplicity, that $f(s) = {\sum\limits_{i = 1}^{2p+1} a_{i}s^{i}}$, $a_{2p+1} > 0$ and $g(s) = {\sum\limits_{i = 1}^{q} b_{i}s^{i}}$, $b_{q} > 0$. Then,

since $2q\leqslant2p+2$ and $q+1 < 2p+2$. Furthermore,

In this article, we assume $f(s) = s^{3}-s$ and $g(s) = \alpha s(s-1)$, $\alpha > 0$.

Finally, the same letter $c$, $c^{'}$ and $c^{''}$ denotes constants which may vary from line to line, or even in a same line. Similary, the same letter $Q$ denotes monotone increasing functions which may vary from line to line, or even in a same line.

3.   A priori estimates

In what follows, the Poincaré, Hölder and Young inequalities are extensively used, without further referring to them.

We multiply (2.1) by $(-\Delta)^{-1}u$ and integrate over $\Omega$, we obtain

owing to (2.7) and (2.8), we obtain

hence

We now multiply (2.2) by $\theta$ and integrate over $\Omega$, we find

We then multiply (2.1) by $\dfrac{\partial u}{\partial t}$ and integrate over $\Omega$, we have

Here,

Furthermore,

which yields

We recal that $H^{2}(\Omega)$ is continuously embedded in $C(\overline{\Omega})$ and owing to (2.5),

and inserting (3.6) into (3.5), we obtain

Summing finally (3.3) and (3.7), we find

In particular, we deduce

We set

we deduce from (3.8) an inequation of the form

Let $z$ be the solution to the ordinary differential equation

It follows from the comparison principle, that there exists a time $T_{0} = T_{0}(\Vert u_{0} \Vert_{H^{2}(\Omega)}, \Vert \theta_{0} \Vert) > 0$

belonging to, say $(0, \dfrac{1}{2})$ such that

hence

We multiply (2.1) by $(-\Delta)^{-1}\dfrac{\partial u}{\partial t}$ and integrate over $\Omega$, we have

We then multiply (2.2) by $(-\Delta)^{-1}\theta$ and integrate over $\Omega$, we obtain

Summing (3.11) and (3.12), owing to (3.6)-(3.10), we find

where

Summing $\psi_{1}$(3.2) and (3.13), where $\psi_{1} > 0$ is small enough, we obtain

where

satisfies

In particular, we deduce from (3.14) and Gronwall's lemma the dissipative estimate

where we have used the continuous embedding $H^{2}(\Omega)\hookrightarrow C(\overline{\Omega})$ to deduce that

Furthermore,

Finally, more generally, for every $r > 0$

We now differentiate (2.1) with respect to time and have, noting that $\dfrac{\partial \theta}{\partial t} = -\dfrac{\partial u}{\partial t} +\Delta \theta$,

We multiply (3.18) by $t\dfrac{\partial u}{\partial t}$ and integrate over $\Omega$, owing (2.6), we find for $t\leqslant T_{0}$

We know that $H^{2}(\Omega)\subset L^{\infty}(\Omega)$ with continuous injection and $(-\Delta)^{-1}\dfrac{\partial u}{\partial t}\in H^{2}(\Omega)$. Then

Inserting (3.20) into (3.19), we have

Noting that,

hence

Using the estimates (3.17), (3.21) and Gronwall's lemma, we find

Multiplying then (3.18) by $\dfrac{\partial u}{\partial t}$ and integrate over $\Omega$, we obtain, proceeding as above,

It thus follows from (3.17), (3.23) and Gronwall's lemma

and it finally follows from (3.22) that

We now rewrite, for $t\geqslant T_{0}$ fixed, (2.1) in the form

where

We multiply (3.27) by $h_{u}(t)$ and integrate over $\Omega$, we obtain

owing to (3.15)-(3.25), we have

We multiply (3.26) by $u$ and integrate over $\Omega$, we find

which yields, owing to (2.7) and (2.8)

Then, multiplying (3.26) by $-\Delta u$ and integrate over $\Omega$, we obtain

which yields, owing to (2.6) and (2.8)

Summing (3.29) and $\psi_{2}$(3.30), where $\psi_{2} > 0$ is small enough, we find

We thus deduce from (3.28) and (3.31) that

The estimate (3.3) implies

It follows from (3.10), (3.23) and (3.25) that

We then deduce from (3.33) and (3.34) that

which gives

Combining (3.32) and (3.35), we obtain

Finally, we deduce from (3.10) and (3.36), that

We now note, it follows from (3.16) that

Furthermore, multiplying (2.1) by $u$ and integrate over $\Omega$, we find, thanks to (2.6)

Owing to (3.6) and (3.10), we have

We thus deduce from (3.39) that

Therefore, the estimates (3.38) and (3.40) allow to affirm, there exists $T\in(0, 1)$ such that

Actually, repeating the above estimates, starting from $t = T$ instead of t = 0, we see that (3.41) holds for $T = 1$, i.e. we have the smoothing property

In particular, having this smoothing property, it is not difficult to prove that we have, owing to (3.15), (3.37) and (3.42), the dissipative estimate

We multiply (2.2) by $-\Delta \dfrac{\partial \theta}{\partial t}$ and integrate over $\Omega$, we obtain

Thanks to (3.17) and (3.23), we have

Setting $y = \Vert \Delta \theta\Vert^{2}$, $g = 0$ and $h = \begin{Vmatrix}\nabla \dfrac{\partial u }{\partial t}\end{Vmatrix}^{2}$, we deduce from (3.44) that

where, owing to the above estimates, $y, g$ and $h$ satisfy the assumptions of the uniform Gronwall's lemma (for $t\geqslant t_{0}$), which yields that, for $t\geqslant t_{0}+r$,

hence

Combining (3.43) and (3.45), we obtain the dissipative estimate

From where the

Theorem 3.1. We assume that $(u_{0}, \theta_{0})\in (H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) \times L^{2}(\Omega)$. Then, the system (2.1)-(2.4) possesses at least solution $(u, \theta)$ such that $u \in L^{\infty}(0, T; H^{2}(\Omega)\cap H_{0}^{1}(\Omega))$, $\theta\in L^{\infty}(0, T; L^{2}(\Omega))$ and $\dfrac{\partial u}{\partial t} \in L^{2}(0, T; H^{-1}(\Omega))$, $\forall T > 0$.

The proof of existence is based on the estimates (3.15), (3.43) and a standard Galerkin scheme.

4.   The dissipative semigroup

We have the

Theorem 4.1. We assume that $(u_{0}, \theta_{0})\in (H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) \times L^{2}(\Omega)$. Then, the system (2.1)-(2.4) possesses a unique solution $(u, \theta)$ such that $u \in L^{\infty}(0, T; H^{2}(\Omega)\cap H_{0}^{1}(\Omega))$, $\theta\in L^{\infty}(0, T; L^{2}(\Omega))$ and $\dfrac{\partial u}{\partial t} \in L^{2}(0, T; H^{-1}(\Omega))$, $\forall T > 0$.

Proof. Let now $(u_{1}, \theta_{1})$ and $(u_{2}, \theta_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}\; , \; \theta_{1, 0})$ and $(u_{2, 0}\; , \; \theta_{2, 0})\in (H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) \times L^{2}(\Omega)$, respectively. We set $(u\; , \; \theta) = (u_{1}\; , \; \theta_{1})-(u_{2}\; , \; \theta_{2})$ and $(u_{0}\; , \; \theta_{0}) = (u_{1, 0}\; , \; \theta_{1, 0})-(u_{2, 0}\; , \; \theta_{2, 0})$. Then $(u\; , \; \theta)$ verifies the following problem

We multiply (4.1) by $(-\Delta)^{-1} \dfrac{\partial u}{\partial t}$ and (4.2) by $(-\Delta)^{-1}\theta$ integrate over $\Omega$, summing the two resulting equations, we have

We have thanks to lagrange theorem, the following estimate

Noting that $\ H^{1}(\Omega)\subset L^{4}(\Omega)$ with continuous injection and while using (3.43), we have

Besides

and owing to (3.43),

We insert the estimates (4.6) and (4.7) into (4.5), we find

Where $Q$ is monotone increasing with respect to both arguments. We deduce from

(4.8) and Gronwall's lemma that

hence the uniqueness, as well as the continuous depending with respect to the initial data.

We set $\Psi = (H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) \times L^{2}(\Omega))$. It follows from Theorem 3.1, that we have the continuous (with respect to the $H^{1}(\Omega)\times H^{-1}(\Omega)- \text{norm})$ of the following semigroup

(i.e, $S(0) = I, \ S(t)\circ S(s) = S(t+s), \ t, s\geqslant 0$). We then deduce from (3.43) the following theorem.

Theorem 4.2. The semigroup $S(t)$ is dissipative in $\Psi$, i.e., there exists a bounded set $\mathcal{B}_{0}\in \Psi$(called absorbing set) such that, for every bounded $B\in \Psi$, there exists $t_{0} = t_{0}(B)\geqslant 0$ such that $t\geqslant t_{0}$ implies $S(t)B\subset \mathcal{B}_{0}$.

Remark 4.1. It is easy to see that we can assume, without loss of generality, that $\mathcal{B}_{0}$ is positively invariant by $S(t)$, i.e., $S(t)\mathcal{B}_{0}\subset \mathcal{B}_{0}$, $\forall t\geqslant 0$. Furthermore, it follows from (3.46) that $S(t)$ is dissipative in $(H^{2}(\Omega))^{2}$ and it follows from (3.45) that we can take $\mathcal{B}_{0}$ in $(H^{2}(\Omega))^{2}$.

Corollary 4.1. The semigroup $S(t)$ possesses the global attractor $\mathcal{A}$ who is bounded in $(H^{2}(\Omega))^{2}$ and compact in $\Psi$.

The existence of the global attractor being established, one question is to know whether this attractor has a finite dimension in terms of the fractal or Hausdorff dimension. This is the aim of the final section.

5.   Existence of exponential attractors

The aim of this section is to prove the existence of exponential attractors for the semigroup $S(t), t\geqslant\; 0$, associated to the problem (2.1)-(2.4). To do so, we need the semigroup that has to be Lipschitz continuous, satisfying the smoothing property and checking a Hölder continuous with respect to time. This is enough to conclude on the existence of exponential attractors.

Lemma 5.1. Let $(u_{1}, \theta_{1})$ and $(u_{2}, \theta_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}\; , \; \theta_{1, 0})$ and $(u_{2, 0}\; , \; \theta_{2, 0})$, respectively, belonging to $\mathcal{B}_{0}$. Then, the corresponding solutions of the problem (2.1)-(2.4) satisfy the following estimate

where the constants only depend on $\mathcal{B}_{0}$.

Proof. We set $(u\; , \; \theta) = (u_{1}\; , \; \theta_{1})-(u_{2}\; , \; \theta_{2})$ and $(u_{0}\; , \; \theta_{0}) = (u_{1, 0}\; , \; \theta_{1, 0})-(u_{2, 0}\; , \; \theta_{2, 0})$, then $(u\; , \; \theta)$ satisfies

We first deduce from (4.8) that

and

where the constants only depend on $\mathcal{B}_{0}$.

We differentiate (5.2) with respect to time and have, owing to (5.3),

where $\varphi = \dfrac{\partial u}{\partial t}$.

We multiply (5.8) by $(t-T_{0})\varphi$ and integrate over $\Omega$, where $T_{0}$ is same as in one of previous section, owing to (2.6), we obtain

Noting that $u_{1}, u_{2}\in H^{2}(\Omega)$, then

proceeding as in (3.20), we find

and

where the constants only depend on $\mathcal{B}_{0}$.

By substituting (5.10), (5.11) and (5.12) into (5.9), we have, owing to the interpolation inequality,

We now multiply (5.3) by $-(t-T_{0}) \theta$ and integrate over $\Omega$, we obtain

Therefore, noting that it follows from (3.13), (3.15), (3.23) and (3.25) (for $(u, \theta) = (u_{2}, \theta_{2}))$ that

where the constants only depend on $\mathcal{B}_{0}$.

Combining (5.13) and (5.14), we find, owing to Gronwall's lemma over $(T_{0}, t)$; note that $T_{0} < 1$,

where the constants only depend on $\mathcal{B}_{0}$.

We rewrite (5.2) in the form

for $t\geqslant 1$ fixed, where

We multiply (5.17) by $\tilde{h}_{u}(t)$ and integrate over $\Omega$, we obtain

It then follows from (5.6) and (5.15) that

where the constants only depend on $\mathcal{B}_{0}$.

We multiply (5.16) by $-\Delta u$ and integrate over $\Omega$, we find

hence, owing to (5.18), we have

where the constants only depend on $\mathcal{B}_{0}$.

We finally deduce from (5.15) and (5.19), the estimate (5.1) which concludes the proof.

Lemma 5.2. Let $(u_{1}, \theta_{1})$ and $(u_{2}, \theta_{2})$ be two solutions to (2.1)-(2.4) with initial data $(u_{1, 0}\; , \; \theta_{1, 0})$ and $(u_{2, 0}\; , \; \theta_{2, 0})$, respectively, belonging to $\mathcal{B}_{0}$. Then, the semigroup $\{S(t)\}_{t\geqslant 0}$ is Lipschitz continuity with respect to space, i.e, there exists the constant $c > 0$ such that

where the constants only depend on $\mathcal{B}_{0}$.

Proof. The proof of the lemma 5.2 is a direct consequence of the estimate (5.6).

It just remains to prove the Hölder continuity with respect to time.

Lemma 5.3. Let $(u, \theta)$ be the solution of (5.2)-(5.5) with intial data $(u_{0}, \theta_{0})$ in $\mathcal{B}_{0}$. Then, the semigroup $\{S(t)\}_{t\geqslant 0}$ is Hölder continuous with respect to time, i.e, there exists the constant $c > 0$ such that

$\forall t_{1}, \ t_{2}\in[0, T]$

where the constants only depends on $\mathcal{B}_{0}$ and $T$.

Proof.

Noting that, thanks to (3.16), (3.23) and (3.25), we have

where the constant c depends only on $\mathcal{B}_{0}$ and $T \geqslant T_{0}$ such that $t_{1}, \ t_{2}\in[0, T]$.

Furthermore, multiplying (2.2) by $(-\Delta)^{-1} \dfrac{\partial \theta}{\partial t}$ and integrate over $\Omega$, we obtain

and it follows from (5.23) and (5.24) that

where c only depends on $\mathcal{B}_{0}$ and $T$ such that $t_{1}, \ t_{2}\in[0, T]$.

Finally, we obtain thanks to (5.23) and (5.25), the estimate (5.21). Thus, the lemma is proved.

We finally deduce from Lemma 5.1, Lemma 5.2 and Lemma 5.3 the following result (see, e.g, ^[9,10]).

Theorem 5.1. The semigroup $S(t)$ possesses an exponential attractor $\mathcal{M}\subset \mathcal{B}_{0}$, i.e,

(i) $\mathcal{M}$ is compact in $H^{1}(\Omega)\times H^{-1}(\Omega)$;

(ii) $\mathcal{M}$ is positively invariant, $S(t)\mathcal{M}\subset \mathcal{M}$, $\forall t\geqslant 0$;

(iii) $\mathcal{M}$ has finite fractal dimension in $H^{1}(\Omega)\times H^{-1}(\Omega)$;

(iv) $\mathcal{M}$ attracts exponentially fast the bounded subsets of $\Psi$

where the constant $c$ is independent of $B$ and $dist_{H^{1}(\Omega)\times H^{-1}(\Omega)}$ denotes the Hausdorff semidistance between sets defined by

Remark 5.1. Setting $\tilde{\mathcal{M}} = S(1)\mathcal{M}$, we can prove that $\tilde{\mathcal{M}}$ is an exponential attractor for $S(t)$, but now in the topology of $\Psi$.

Since $\mathcal{M}$ (or $\tilde{\mathcal{M}}$) is a compact attracting set, we deduce from Theorem 5.1 and standard results (see, e.g, ^[4,10]) the

Corollary 5.1. The semigroup $S(t)$ possesses the finite-dimensional global attractor $\mathcal{A}\subset \mathcal{B}_{0}.$

Remark 5.2. We note that the global attractor $\mathcal{A}$ is the smallest (for inlusion) compact set of the phase space which is invariant by the flow (i.e. $S(t)\mathcal{A} = \mathcal{A}, \forall t\geqslant 0)$ and attractors all bounded sets of initial data as time goes to infinity; thus, it appears as a suitable object in view of the study of the asymptotic behaviour of the system. Furthermore, the finite dimensionality means, roughy speaking, that, even though the initial phase space is infinite dimensional, the reduced dynamics is, in some proper sense, finite dimensional and can be described by a finite number of parameters.

Remark 5.3. Compared to the global attractor, an exponentiel attractor is expected to be more robust under perturbations. Indeed, the rate of attraction of trajectories to the global attractor may be slow and it is very difficult, if not impossible, to estimate this rate of attraction with respect to the physical parameters of the problem in general. As a consequence, global attractors may change drastically under small perturbations.

6.   Conclusion

This manuscript explains in a clear way, the context of dynamic system with a proliferation term, when the relative solution exists. The existence of exponential attractor, associated to the problem (2.1)-(2.4) that we have proved, allow to assert that the existing solution of the problem (2.1)-(2.4) that we have shown in this work, belongs to the finite-dimensional subset called global attractor, from a certain time.

Conflict of interest

All authors declare no conflicts of interest in this paper.