Long-Time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth

Abstract

We investigate the long-time dynamics and optimal control problem of a thermodynamically consistent diffuse interface model that describes the growth of a tumor in presence of a nutrient and surrounded by host tissues. The state system consists of a Cahn–Hilliard type equation for the tumor cell fraction and a reaction–diffusion equation for the nutrient. The possible medication that serves to eliminate tumor cells is in terms of drugs and is introduced into the system through the nutrient. In this setting, the control variable acts as an external source in the nutrient equation. First, we consider the problem of “long-time treatment” under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as $t\rightarrow +\infty$. Second, we consider the “finite-time treatment” that corresponds to an optimal control problem. Here we allow the objective cost functional to depend on a free time variable, which represents the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time. One of the main aim of the control problem is to realize in the best possible way a desired final distribution of the tumor cells, which is expressed by the target function $\phi _\Omega$. By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we show that $\phi _{\Omega }$ can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed.

Appendix

Let $m\in {\mathbb {R}}$ be a given constant. We consider the following nonlocal elliptic boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\displaystyle {-\Delta \phi +F'(\phi )=m-|\Omega |^{-1}\int _\Omega \phi \mathrm{{d}}x, \quad \text {in}\ \Omega ,}\\ &{}\partial _\nu \phi =0,\qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \;\text {on}\ \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

(6.1)

Problem (6.1) can be associated with the following functional

$$\begin{aligned} \Upsilon (\phi ) = \int _{\Omega } \left( \frac{1}{2} |\nabla \phi |^2 + F(\phi )\right) \mathrm{{d}}x+ \frac{1}{2}|\Omega |\left( m-|\Omega |^{-1}\int _\Omega \phi \mathrm{{d}}x\right) ^2,\quad \forall \, \phi \in H^1(\Omega ). \end{aligned}$$

(6.2)

The following result has been obtained in [63, Lemma 3.1, Lemma 3.2].

Lemma 6.1

Let assumption (F1) be satisfied.

• (1) Suppose that $\psi \in H^2_N(\Omega )$ is a (strong) solution to problem (6.1). Then $\psi$ is a critical point of the functional $\Upsilon (\phi )$ in $H^1(\Omega )$. Conversely, if $\psi$ is a critical point of the functional $\Upsilon (\phi )$ in $H^1(\Omega )$, then $\psi \in H^2_N(\Omega )$ and it is a strong solution to problem (6.1).

• (2) The functional $\Upsilon (\phi )$ has at least one minimizer $\psi \in H^1(\Omega )$ such that

  $$\begin{aligned} \Upsilon (\psi )= \displaystyle {\inf _{\phi \in H^1(\Omega )}}\Upsilon (\phi ). \end{aligned}$$

  (6.3)

Remark 6.1

By the elliptic regularity theory and a bootstrap argument, the minimizer $\psi$ is indeed a classical solution such that $\psi \in C^\infty ({\overline{\Omega }})$, provided that the domain boundary $\partial \Omega$ and the function F are smooth.

Associated with problem (6.1), the following Łojasiewicz–Simon type inequality has been proven in [63, Lemma 4.1] (see [65, Lemma 2.2] for a slightly weaker version).

Lemma 6.2

Let (F1) and (F2) be satisfied. Suppose that $\psi$ is a critical point of $\Upsilon (\phi )$ in $H^1(\Omega )$. Then there exist constants $\theta \in (0,\frac{1}{2})$ and $\beta >0$, depending on $\psi$, m and $\Omega$, such that, for any $\phi \in H^1(\Omega )$ with $\Vert \phi -\psi \Vert _{H^1(\Omega )}< \beta$, it holds

$$\begin{aligned} \left\| - \Delta \phi + F'(\phi )-\left( m - |\Omega |^{-1}\int _\Omega \phi \mathrm{{d}}x\right) \right\| _{(H^1(\Omega ))'} \ \ge \ | \Upsilon (\phi ) -\Upsilon (\psi )|^{1-\theta }. \end{aligned}$$

(6.4)

Now we are in a position to prove the Łojasiewicz–Simon type inequality (3.38) stated in Lemma 3.1, which plays a crucial role in the study of long-time behavior of problem (1.1)–(1.5).

Proof of Lemma 3.1

In Lemma 6.2, we take $m=m_\infty$ (see (3.35)) and $\psi =\phi _\infty$. Then it follows from (3.3)–(3.5) and (3.35) that $\phi _\infty$ satisfies the reduced elliptic problem (6.1). Hence, according to Lemma 6.1, we see that it is a critical point of $\Upsilon (\phi )$ (cf. (6.2) with $m=m_\infty$). As a consequence, Lemma 6.2 applies with constants $\theta \in (0, \frac{1}{2})$, $\beta >0$ depending on $\phi _\infty$, $m_\infty$ and $\Omega$. On the other hand, for any $\phi \in H^2_N(\Omega )$ we set

$$\begin{aligned} \mu =-\Delta \phi +F'(\phi ) \end{aligned}$$

and then using integration by parts, we get

$$\begin{aligned} \int _\Omega \mu \mathrm{{d}}x=\int _\Omega F'(\phi ) \mathrm{{d}}x. \end{aligned}$$

From the Łojasiewicz–Simon inequality (6.4) (applying to $\psi =\phi _\infty )$, Poincaré’s inequality and (3.37), we deduce that

$$\begin{aligned}&|\Upsilon (\phi ) -\Upsilon (\phi _\infty ) |^{1-\theta } \nonumber \\&\quad \le \left\| - \Delta \phi + F'(\phi )- \left( m_\infty - |\Omega |^{-1}\int _\Omega \phi \mathrm{{d}}x\right) \right\| _{(H^1(\Omega ))'}\nonumber \\&\quad \le \left\| - \Delta \phi + F'(\phi )-|\Omega |^{-1}\int _\Omega F'(\phi )\mathrm{{d}}x \right\| _{(H^1(\Omega ))'} \nonumber \\&\qquad + \left\| |\Omega |^{-1}\int _\Omega F'(\phi )\mathrm{{d}}x-\left( m_\infty -|\Omega |^{-1}\int _\Omega \phi \mathrm{{d}}x\right) \right\| _{(H^1(\Omega ))'}\nonumber \\&\quad = \left\| \mu - {\overline{\mu }} \right\| _{(H^1(\Omega ))'} + \left\| |\Omega |^{-1}\int _\Omega \mu \mathrm{{d}}x-|\Omega |^{-1}\int _\Omega \sigma \mathrm{{d}}x-m_u\right\| _{(H^1(\Omega ))'}\nonumber \\&\quad \le \left\| \mu - {\overline{\mu }} \right\| _{(H^1(\Omega ))'} + |\Omega |^{-1}\left| \int _\Omega (\mu - \sigma )\mathrm{{d}}x\right| + |m_u|\nonumber \\&\quad \le \left\| \mu - {\overline{\mu }} \right\| _{(H^1(\Omega ))'} + |\Omega |^{-1} \left( \int _\Omega \frac{1}{P(\phi )} \mathrm{{d}}x\right) ^\frac{1}{2}\left( \int _\Omega P(\phi )(\mu -\sigma )^2 \mathrm{{d}}x\right) ^\frac{1}{2}\nonumber \\&\qquad +|m_u|. \end{aligned}$$

(6.5)

By the Sobolev embedding $H^2(\Omega )\hookrightarrow C({\overline{\Omega }})$ ($n=2,3$), the continuity as well as the strictly positivity of P(s), then it holds

$$\begin{aligned} \int _\Omega \frac{1}{P(\phi )} \mathrm{{d}}x \le |\Omega |\left( \min _{x\in {\overline{\Omega }}} P(\phi (x))\right) ^{-1}\le C, \end{aligned}$$

where the constant $C>0$ depends on $\Omega$, $\Vert \phi \Vert _{H^2(\Omega )}$ and P.

On the other hand, on account of (1.6), (3.37), (6.2) and Poincaré’s inequality, since $\theta \in (0,\frac{1}{2})$ and $\sigma \in H^1(\Omega )$, we infer that

$$\begin{aligned}&|{\mathcal {E}}(\phi , \sigma )-\Upsilon (\phi )|^{1-\theta }&\quad \nonumber \\&\quad = \left| \frac{1}{2}\Vert \sigma \Vert _{L^2(\Omega )}^2-\frac{1}{2}|\Omega |( {\overline{\sigma }}+m_u)^2\right| ^{1-\theta }\nonumber \\&\quad \le \left( \frac{1}{2}\right) ^{1-\theta }\left( \int _{\Omega }(\sigma -{\overline{\sigma }})^2\mathrm{{d}}x +2|\Omega ||{\overline{\sigma }}||m_u| +|\Omega |m_u^2\right) ^{1-\theta }\nonumber \\&\quad \le C\Vert \nabla \sigma \Vert _{L^2(\Omega )}^{2(1-\theta )}+C\left( |m_u|^{1-\theta }+|m_u|^{2(1-\theta )}\right) \nonumber \\&\quad \le C\Vert \nabla \sigma \Vert _{L^2(\Omega )}+C|m_u|^\frac{1}{2}. \end{aligned}$$

(6.6)

Finally, since $\Upsilon (\phi _\infty )={\mathcal {E}}(\phi _\infty ,\sigma _\infty )$ (recalling that $\sigma _\infty$ is a constant satisfying (3.37)), we deduce from inequalities (6.5) and (6.6) that

$$\begin{aligned}&| {\mathcal {E}}(\phi , \sigma ) -{\mathcal {E}}(\phi _\infty , \sigma _\infty )|^{1-\theta } \\&\quad \le |{\mathcal {E}}(\phi , \sigma )-\Upsilon (\phi )|^{1-\theta }+ | \Upsilon (\phi ) -\Upsilon (\phi _\infty )|^{1-\theta }\\&\quad \le \left\| \mu - {\overline{\mu }} \right\| _{(H^1(\Omega ))'}+ C\Vert \nabla \sigma \Vert _{L^2(\Omega )} + C\Vert \sqrt{P(\phi )}(\mu -\sigma )\Vert _{L^2(\Omega )}+C|m_u|^\frac{1}{2}. \end{aligned}$$

The proof of Lemma 3.1 is complete. $\square$