Global Strong Solution and Periodic Dynamic Behavior to Chaplain–Lolas Model

\bf Abstract

In this paper, we study Chaplain–Lolas model in three dimensional bounded domain, which describes the invasion and diffusion process of solid tumors during the vascular growth stage. Although the model has received extensive attention since it was proposed, the existence of solutions in three-dimensional space is still missing, and only a global small solution is established by Pang and Wang (Math Models Methods Appl Sci 28(11):2211–2235, 2018) with sufficiently small proliferation coefficient. As for long time behavior, there are only corresponding stability result of constant steady-state for the case without ECM remodelling (Hillen et al. in Math Models Methods Appl Sci 25(1):165–198, 2013; Tao and Winkler in SIAM J Math Anal 47:4229–4250, 2016). If the remodeling effect of ECM is considered, the relevant research is still blank. In this paper, we first pay our attention to the study of existence of global strong solution and long time behavior. We prove that when the ratio of cell proliferation coefficient to chemotactic intensity $\frac{\mu }{\chi ^2}$ is large, there exists a unique global strong solution around the equilibrium state (1, 1, 0), and the global strong solution converges exponentially to the constant equilibrium point. In fact, such a largeness restriction on $\frac{\mu }{\chi ^2}$ is actually necessary to some extent, since that the constant equilibrium point (1, 1, 0) is actually linearly unstable when such a condition is not satisfied. Subsequently, we turn our attention to the study of dynamic behavior of solutions. We introduce a time periodic external force to this system, and prove that the solution will gradually show the same periodic behavior under the action of periodic external force, and become a time periodic solution. At last, we analysis the stability and instability of equilibrium points, and discuss the influence of spatial diffusion, chemotaxis and haptotaxis effect on the stability of solutions. In particular, we find that only chemotaxis can change the stability of the solution and make the originally stable equilibrium (1, 1, 0) unstable, while the haptotactic term has no effect on the stability.

Appendix A

This appendix is mainly used to prove lemma 2.1, which is also used to get lemma 3.1. We use three lemmas to complete the proof.

Lemma A.1

Let ${\mathcal {T}}({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}, \sigma )=({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})$. Then for any $M>0$, we have

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |{\tilde{u}}|^2+ |\nabla {\tilde{u}}|^2+\frac{\mu }{2}|{\tilde{v}}|^2+\frac{\mu }{2}|\nabla {\tilde{v}}|^2 +\frac{M}{2}|{{\tilde{\omega }}}|^2+\frac{M}{2}|\nabla {{\tilde{\omega }}}|^2+\frac{M}{2}|\Delta {{\tilde{\omega }}}|^2 \right) dx \nonumber \\&\qquad +\frac{1}{2}\int _\Omega |\Delta {\tilde{u}}|^2dx +\left( \frac{\mu }{2}+1\right) \int _\Omega |\nabla {\tilde{u}}|^2dx+\frac{\mu }{2}\int _\Omega |{\tilde{u}}|^2dx+ (\mu -2\chi ^2)\int _\Omega |\Delta {\tilde{v}}|^2 dx\nonumber \\&\qquad +\left( \frac{3\mu }{2}-2\chi ^2\right) \int _\Omega |\nabla {\tilde{v}}|^2dx+ \frac{\mu }{2}\int _\Omega |{\tilde{v}}|^2 dx\nonumber \\&\qquad +\left( \frac{M}{2}-\mu \right) \int _\Omega |{{\tilde{\omega }}}|^2dx\nonumber \\&\qquad +\left( \frac{M}{2}-2\xi ^2-\mu \right) \int _\Omega |\nabla {{\tilde{\omega }}}|^2dx\nonumber \\&\qquad +\left( \frac{M}{2}-2\xi ^2\right) \int _\Omega |\Delta {{\tilde{\omega }}}|^2dx\nonumber \\&\quad +M\varepsilon \int _\Omega \left( |\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2+|\nabla \Delta {{\tilde{\omega }}}|^2\right) dx\nonumber \\&\quad \le 2\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx+2\int _\Omega |g_1+\sigma g_2|^2 dx+\frac{M}{2}\int _\Omega \left( |f_2|^2+|\nabla f_2|^2+|\Delta f_2|^2\right) dx. \end{aligned}$$

(A.1)

Proof

By multiplying the first equation of (2.2) by ${\tilde{u}}$, and integrating it over $\Omega$, we see that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega |{\tilde{u}}|^2dx+\int _\Omega |\nabla {\tilde{u}}|^2dx+\mu \int _\Omega |{\tilde{u}}|^2dx\\&\quad =\chi \int _\Omega \nabla {\tilde{v}}\nabla {\tilde{u}}dx+\xi \int _\Omega \nabla {{\tilde{\omega }}}\nabla {\tilde{u}}dx-\mu \int _\Omega {\tilde{u}}{{\tilde{\omega }}} dx+\int _\Omega g_1({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}){\tilde{u}}dx\\&\qquad +\sigma \int _\Omega g_2({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}){\tilde{u}} dx\le \frac{1}{2}\int _\Omega \left( |\nabla {\tilde{u}}|^2+\mu |{\tilde{u}}|^2\right) dx\\&\qquad +\chi ^2\int _\Omega |\nabla {\tilde{v}}|^2dx+\xi ^2\int _\Omega |\nabla {{\tilde{\omega }}}|^2dx+\frac{\mu }{2}\int _\Omega |{{\tilde{\omega }}}|^2dx +\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx, \end{aligned}$$

that is

$$\begin{aligned}&\frac{d}{dt}\int _\Omega |{\tilde{u}}|^2dx+\int _\Omega |\nabla {\tilde{u}}|^2dx+\mu \int _\Omega |{\tilde{u}}|^2dx\nonumber \\&\quad \le 2\chi ^2\int _\Omega |\nabla {\tilde{v}}|^2dx+2\xi ^2\int _\Omega |\nabla {{\tilde{\omega }}}|^2dx\nonumber \\&\qquad +\mu \int _\Omega |{{\tilde{\omega }}}|^2dx +2\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx. \end{aligned}$$

(A.2)

We multiply the first equation of (2.2) by $-\Delta {\tilde{u}}$, integrate it over $\Omega$, and we conclude

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega |\nabla {\tilde{u}}|^2dx+\int _\Omega |\Delta {\tilde{u}}|^2dx+\mu \int _\Omega |\nabla {\tilde{u}}|^2dx \\&\quad =\chi \int _\Omega \Delta {\tilde{v}}\Delta {\tilde{u}}dx+\xi \int _\Omega \Delta {{\tilde{\omega }}}\Delta {\tilde{u}}dx-\mu \int _\Omega \nabla {\tilde{u}}\nabla {{\tilde{\omega }}} dx-\int _\Omega g_1\Delta {\tilde{u}}dx-\sigma \int _\Omega g_2\Delta {\tilde{u}} dx \\&\quad \le \int _\Omega \left( \frac{3}{4}|\Delta {\tilde{u}}|^2+\frac{\mu }{2}|\nabla {\tilde{u}}|^2\right) dx\\&\qquad +\chi ^2\int _\Omega |\Delta {\tilde{v}}|^2dx+\xi ^2\int _\Omega |\Delta {{\tilde{\omega }}}|^2dx+\frac{\mu }{2}\int _\Omega |\nabla {{\tilde{\omega }}}|^2dx +\int _\Omega |g_1+\sigma g_2|^2 dx, \end{aligned}$$

namely,

$$\begin{aligned}&\frac{d}{dt}\int _\Omega |\nabla {\tilde{u}}|^2dx+\frac{1}{2}\int _\Omega |\Delta {\tilde{u}}|^2dx+\mu \int _\Omega |\nabla {\tilde{u}}|^2dx\nonumber \\&\quad \le 2\chi ^2\int _\Omega |\Delta {\tilde{v}}|^2dx+2\xi ^2\int _\Omega |\Delta {{\tilde{\omega }}}|^2dx+\mu \int _\Omega |\nabla {{\tilde{\omega }}}|^2dx +2\int _\Omega |g_1+\sigma g_2|^2 dx. \end{aligned}$$

(A.3)

Similarly, multiplying the second equation of (2.2) by ${\tilde{v}}$, $-\Delta {\tilde{v}}$ respectively, integrating them over $\Omega$, and then adding the two equalities yields

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega (|{\tilde{v}}|^2+|\nabla {\tilde{v}}|^2)dx+\int _\Omega (|\Delta {\tilde{v}}|^2+2|\nabla {\tilde{v}}|^2+ |{\tilde{v}}|^2)dx \\&\quad =\sigma \int _\Omega {\tilde{u}}{\tilde{v}} dx-\sigma \int _\Omega {\tilde{u}}\Delta {\tilde{v}} dx \\&\quad \le \frac{1}{2}\int _\Omega (|{\tilde{v}}|^2+|\nabla {\tilde{v}}|^2)dx+\frac{1}{2}\int _\Omega (|{\tilde{u}}|^2+|\nabla {\tilde{u}}|^2)dx, \end{aligned}$$

which implies

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _\Omega (|{\tilde{v}}|^2+|\nabla {\tilde{v}}|^2)dx+\int _\Omega (|\Delta {\tilde{v}}|^2+\frac{3}{2}|\nabla {\tilde{v}}|^2+ \frac{1}{2}|{\tilde{v}}|^2)dx \le \frac{1}{2}\int _\Omega (|{\tilde{u}}|^2+|\nabla {\tilde{u}}|^2)dx. \end{aligned}$$

(A.4)

Multiplying the third equation of (2.2) by ${{\tilde{\omega }}}$, $\Delta {{\tilde{\omega }}}$ respectively, and integrating them over $\Omega$, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |{{\tilde{\omega }}}|^2+|\nabla {{\tilde{\omega }}}|^2\right) dx+\varepsilon \int _\Omega \left( |\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2\right) dx+\int _\Omega \left( |{{\tilde{\omega }}}|^2+|\nabla {{\tilde{\omega }}}|^2\right) dx\nonumber \\&\quad =\sigma \int _\Omega \left( f_2({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}){{\tilde{\omega }}}-\nabla f_2({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\nabla {{\tilde{\omega }}}\right) dx \nonumber \\&\quad \le \frac{1}{2}\int _\Omega \left( |{{\tilde{\omega }}}|^2+|\nabla {{\tilde{\omega }}}|^2\right) dx+ \frac{1}{2}\int _\Omega |f_2|^2dx+\frac{1}{2}\int _\Omega |\nabla f_2|^2dx. \end{aligned}$$

(A.5)

Different from u and v, the above estimation is not enough for $\Omega$, we need to make a higher-order estimation of $\omega$. Therefore, we further apply $\nabla$ to the third equation of (2.2), then multiply the resultant equation by $\nabla \Delta {{\tilde{\omega }}}$, and integrate it over $\Omega$, we arrive at

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |\Delta {{\tilde{\omega }}}|^2\right) dx+\varepsilon \int _\Omega \left( |\nabla \Delta {{\tilde{\omega }}}|^2\right) dx+\int _\Omega |\Delta {{\tilde{\omega }}}|^2 dx\nonumber \\&\quad =-\sigma \int _\Omega \Delta f_2({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Delta {{\tilde{\omega }}} dx\nonumber \\&\quad \le \frac{1}{2}\int _\Omega |\Delta {{\tilde{\omega }}}|^2 dx+\frac{1}{2}\int _\Omega |\Delta f_2|^2dx. \end{aligned}$$

(A.6)

Adding (A.5) to (A.6), we derive that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |{{\tilde{\omega }}}|^2+|\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2 \right) dx\nonumber \\&\qquad +\varepsilon \int _\Omega \left( |\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2+|\nabla \Delta {{\tilde{\omega }}}|^2\right) dx+\frac{1}{2}\int _\Omega \left( |{{\tilde{\omega }}}|^2+|\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2\right) dx\nonumber \\&\quad \le \frac{1}{2}\int _\Omega \left( |f_2|^2+|\nabla f_2|^2+|\Delta f_2|^2\right) dx. \end{aligned}$$

(A.7)

Then (A.1) is derived by letting (A.2)+(A.3)+$\mu \times$ (A.4)+$M\times$ (A.7). $\square$

However, the above estimation in Lemma A.1 is not enough. In order to get consistent energy estimation results, we need to make a higher-order estimation.

Lemma A.2

Let ${\mathcal {T}}({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}, \sigma )=({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})$. Then for any $M>0$,

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |{\tilde{u}}|^2+ |\nabla {\tilde{u}}|^2+|\Delta {\tilde{u}}|^2+\frac{\mu }{2}|{\tilde{v}}|^2+\frac{\mu }{2}|\nabla {\tilde{v}}|^2+ \mu |\Delta {\tilde{v}}|^2 \right. \nonumber \\&\qquad \left. +\frac{M}{2}|{{\tilde{\omega }}}|^2+\frac{M}{2}|\nabla {{\tilde{\omega }}}|^2+\frac{M}{2}|\Delta {{\tilde{\omega }}}|^2+4\xi ^2|\nabla \Delta {{\tilde{\omega }}}|^2 \right) dx\nonumber \\&\qquad +\int _\Omega \left( |\nabla \Delta {\tilde{u}}|^2 dx+\left( \mu +\frac{1}{2}\right) |\Delta {\tilde{u}}|^2+\left( \frac{\mu }{2}+1\right) |\nabla {\tilde{u}}|^2 +\frac{\mu }{2}|{\tilde{u}}|^2\right) dx \nonumber \\&\qquad +\int _\Omega \left( 2(\mu -\chi ^2) |\nabla \Delta {\tilde{v}}|^2+2(\mu -\chi ^2)|\Delta {\tilde{v}}|^2 +\left( \frac{3\mu }{2}-2\chi ^2\right) |\nabla {\tilde{v}}|^2+\frac{\mu }{2} |{\tilde{v}}|^2 \right) dx \nonumber \\&\qquad +\int _\Omega \left( 2\xi ^2 |\nabla \Delta {{\tilde{\omega }}}|^2+\left( \frac{M}{2}-2\xi ^2-\mu \right) |\nabla {{\tilde{\omega }}}|^2\right. \nonumber \\&\qquad \left. +\left( \frac{M}{2}-2\xi ^2-\mu \right) |\Delta {{\tilde{\omega }}}|^2+\left( \frac{M}{2}-\mu \right) |{{\tilde{\omega }}}|^2\right) dx \nonumber \\&\qquad +M\varepsilon \int _\Omega \left( |\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2+|\nabla \Delta {{\tilde{\omega }}}|^2\right) dx +8\xi ^2\varepsilon \int _\Omega |\Delta ^2{{\tilde{\omega }}}|^2 dx\nonumber \\&\quad \le 2\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx+2\int _\Omega |g_1+\sigma g_2|^2 dx+\frac{M}{2}\int _\Omega \left( |f_2|^2+|\nabla f_2|^2+|\Delta f_2|^2\right) dx\nonumber \\&\qquad +4\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2) dx+4\xi ^2\int _\Omega |\nabla \Delta f_2|^2 dx. \end{aligned}$$

(A.8)

Proof

Applying $\nabla$ to the first equation of (2.2), multiplying the resultant equation by $-\nabla \Delta {\tilde{u}}$, and integrating it over $\Omega$ gives

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |\Delta {\tilde{u}}|^2\right) dx+\int _\Omega |\nabla \Delta {\tilde{u}}|^2 dx+\mu \int _\Omega |\Delta {\tilde{u}}|^2 dx \\&\quad =\chi \int _\Omega \nabla \Delta {\tilde{v}}\nabla \Delta {\tilde{u}} dx+\xi \int _\Omega \nabla \Delta {{\tilde{\omega }}}\nabla \Delta {\tilde{u}} dx \\&\qquad +\mu \int _\Omega \Delta \omega \Delta {\tilde{u}} dx- \int _\Omega \nabla g_1\nabla \Delta {\tilde{u}} dx-\sigma \int _\Omega \nabla g_2\nabla \Delta {\tilde{u}} dx\\&\quad \le \frac{1}{2}\int _\Omega \left( |\nabla \Delta {\tilde{u}}|^2 +\mu |\Delta {\tilde{u}}|^2\right) dx \\&\qquad +\int _\Omega \left( \chi ^2|\nabla \Delta {\tilde{v}}|^2 +\xi ^2 |\nabla \Delta {{\tilde{\omega }}}|^2+\frac{\mu }{2}|\Delta \omega |^2\right) dx +2\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2) dx, \end{aligned}$$

which is equivalent to

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |\Delta {\tilde{u}}|^2\right) dx+\int _\Omega |\nabla \Delta {\tilde{u}}|^2 dx+\mu \int _\Omega |\Delta {\tilde{u}}|^2 dx\nonumber \\&\quad \le \int _\Omega \left( 2\chi ^2|\nabla \Delta {\tilde{v}}|^2 +2\xi ^2 |\nabla \Delta {{\tilde{\omega }}}|^2+\mu |\Delta \omega |^2\right) dx +4\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2) dx. \end{aligned}$$

(A.9)

Applying $\nabla$ to the second equation of (2.2), multiplying the resultant equation by $-\nabla \Delta {\tilde{v}}$, and integrating it over $\Omega$, we arrive at

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |\Delta {\tilde{v}}|^2\right) dx+\int _\Omega |\nabla \Delta {\tilde{v}}|^2 dx+\int _\Omega |\Delta {\tilde{v}}|^2 dx =\int _\Omega \Delta {\tilde{u}}\Delta {\tilde{v}} dx\nonumber \\&\quad \le \frac{1}{2}\int _\Omega |\Delta {\tilde{u}}|^2 dx+ \frac{1}{2}\int _\Omega |\Delta {\tilde{v}}|^2 dx. \end{aligned}$$

(A.10)

Noticing that

$$\begin{aligned} \left. \frac{\partial {\tilde{u}}}{\partial \mathbf{n}}\right| _{\partial \Omega }=\left. \frac{\partial {\tilde{v}}}{\partial \mathbf{n}}\right| _{\partial \Omega }=\left. \frac{\partial {{\tilde{\omega }}}}{\partial \mathbf{n}}\right| _{\partial \Omega }=0, \end{aligned}$$

and using the third equation of (2.2), we obtain that

$$\begin{aligned} \left. \frac{\partial \Delta {{\tilde{\omega }}}}{\partial \mathbf{n}}\right| _{\partial \Omega }=0. \end{aligned}$$

Applying $\nabla \Delta$ to the third equation of (2.2), multiplying the resultant equation by $\nabla \Delta {{\tilde{\omega }}}$, and integrating it over $\Omega$ yields

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega |\nabla \Delta {{\tilde{\omega }}}|^2 dx+\varepsilon \int _\Omega |\Delta ^2{{\tilde{\omega }}}|^2 dx+ \int _\Omega |\nabla \Delta {{\tilde{\omega }}}|^2 dx =\int _\Omega \nabla \Delta {{\tilde{\omega }}}\nabla \Delta f_2 dx\nonumber \\&\quad \le \frac{1}{2}\int _\Omega |\nabla \Delta {{\tilde{\omega }}}|^2 dx+ \frac{1}{2}\int _\Omega |\nabla \Delta f_2|^2 dx. \end{aligned}$$

(A.11)

Combining (A.9)–(A.11), we obtain that

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |\Delta {\tilde{u}}|^2+\mu |\Delta {\tilde{v}}|^2+4\xi ^2|\nabla \Delta {{\tilde{\omega }}}|^2\right) dx +8\xi ^2\varepsilon \int _\Omega |\Delta ^2{{\tilde{\omega }}}|^2 dx+2\xi ^2\int _\Omega |\nabla \Delta {{\tilde{\omega }}}|^2 dx\nonumber \\&\qquad +2(\mu -\chi ^2)\int _\Omega |\nabla \Delta {\tilde{v}}|^2 dx +\mu \int _\Omega |\Delta {\tilde{v}}|^2dx+ \int _\Omega |\nabla \Delta {\tilde{u}}|^2 dx+\mu \int _\Omega |\Delta {\tilde{u}}|^2 dx\nonumber \\&\quad \le \int _\Omega \mu |\Delta \omega |^2 dx +4\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2) dx+4\xi ^2\int _\Omega |\nabla \Delta f_2|^2 dx. \end{aligned}$$

(A.12)

Adding the two inequalities (A.1) and (A.12) gives (A.8). $\square$

Next, we need to supply the estimation on time derivative.

Lemma A.3

Let ${\mathcal {T}}({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}}, \sigma )=({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})$. Then for any $M>0$, we have

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( \left( \frac{\mu }{6}+1\right) |{\tilde{u}}|^2+ \frac{7}{6}|\nabla {\tilde{u}}|^2+|\Delta {\tilde{u}}|^2+\frac{3\mu }{4}|{\tilde{v}}|^2+\frac{3\mu }{4}|\nabla {\tilde{v}}|^2+\mu |\Delta {\tilde{v}}|^2\right) dx\nonumber \\&\quad + \frac{d}{dt}\int _\Omega \left( \left( \frac{M}{2}+1\right) |{{\tilde{\omega }}}|^2+\left( \frac{M}{2}+\varepsilon \right) |\nabla {{\tilde{\omega }}}|^2+\frac{M}{2}|\Delta {{\tilde{\omega }}}|^2+4\xi ^2|\nabla \Delta {{\tilde{\omega }}}|^2 \right) dx\nonumber \\&\quad +\int _\Omega \left( \frac{1}{12}|{\tilde{u}}_t|^2+|\nabla \Delta {\tilde{u}}|^2 dx+\left( \mu +\frac{1}{2}\right) |\Delta {\tilde{u}}|^2+\left( \frac{\mu }{2}+1\right) |\nabla {\tilde{u}}|^2 +\frac{\mu }{4}|{\tilde{u}}|^2\right) dx \nonumber \\&\quad +\int _\Omega \left( \frac{\mu }{4} |{\tilde{v}}_t|^2+ 2(\mu -\chi ^2) |\nabla \Delta {\tilde{v}}|^2+\left( 2\mu -\frac{7}{3}\chi ^2\right) |\Delta {\tilde{v}}|^2\right. \nonumber \\&\quad \left. +\left( \frac{3\mu }{2}-2\chi ^2\right) |\nabla {\tilde{v}}|^2+\frac{\mu }{2} |{\tilde{v}}|^2 \right) dx \nonumber \\&\quad +\int _\Omega \left( |{{\tilde{\omega }}}_t|^2+2\xi ^2 |\nabla \Delta {{\tilde{\omega }}}|^2+\left( \frac{M}{2}-2\xi ^2-\mu \right) |\nabla {{\tilde{\omega }}}|^2+\left( \frac{M}{2}-\frac{7}{3}\xi ^2-\mu \right) |\Delta {{\tilde{\omega }}}|^2\right) dx \nonumber \\&\quad +\int _\Omega \left( \frac{M}{2}-\mu -\frac{2\mu ^2}{3}\right) |{{\tilde{\omega }}}|^2 dx+M\varepsilon \int _\Omega \left( |\nabla {{\tilde{\omega }}}|^2+|\Delta {{\tilde{\omega }}}|^2+|\nabla \Delta {{\tilde{\omega }}}|^2\right) dx\nonumber \\&\quad +8\xi ^2\varepsilon \int _\Omega |\Delta ^2{{\tilde{\omega }}}|^2 dx \le I+II, \end{aligned}$$

(A.13)

where

$$\begin{aligned}&I=2\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx+\frac{8}{3}\int _\Omega |g_1+\sigma g_2|^2 dx+4\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2) dx, \end{aligned}$$

(A.14)

$$\begin{aligned}&II=\frac{M}{2}\int _\Omega \left( |f_2|^2+|\nabla f_2|^2+|\Delta f_2|^2\right) dx+\int _\Omega |f_2|^2 dx +4\xi ^2\int _\Omega |\nabla \Delta f_2|^2 dx. \end{aligned}$$

(A.15)

For I and II, we further have

$$\begin{aligned}&I+II\le {\tilde{C}}_1 \Vert ({\tilde{u}}, {{\tilde{\omega }}})\Vert _{2} \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{1}^2+{\tilde{C}}_2 \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{3}^2 \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{2}^2, \end{aligned}$$

(A.16)

where ${\tilde{C}}_1$, ${\tilde{C}}_2$ are constants depending only on $\chi$, $\xi$, $\mu$, and $\Omega$.

Proof

Multiplying the first equation of (2.2) by ${\tilde{u}}_t$, and integrating it over $\Omega$ yields

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \left( |\nabla {\tilde{u}}|^2+\mu |{\tilde{u}}|^2\right) dx+\int _\Omega |{\tilde{u}}_t|^2dx \\&\quad =-\chi \int _\Omega \Delta {\tilde{v}} {\tilde{u}}_tdx-\xi \int _\Omega \Delta {{\tilde{\omega }}} {\tilde{u}}_tdx-\mu \int _\Omega {\tilde{u}}_t{{\tilde{\omega }}} dx+\int _\Omega (g_1+\sigma g_2){\tilde{u}}_tdx\\&\quad \le \frac{3}{4}\int _\Omega |{\tilde{u}}_t|^2dx+\chi ^2\int _\Omega |\Delta {\tilde{v}}|^2dx+\xi ^2\int _\Omega |\Delta {{\tilde{\omega }}}|^2dx+2\mu ^2 \int _\Omega |{{\tilde{\omega }}}|^2dx\\&\qquad +2\int _\Omega (g_1+\sigma g_2)^2dx. \end{aligned}$$

That is

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |\nabla {\tilde{u}}|^2+\mu |{\tilde{u}}|^2\right) dx+\frac{1}{2}\int _\Omega |{\tilde{u}}_t|^2dx \nonumber \\&\quad \le 2\chi ^2\int _\Omega |\Delta {\tilde{v}}|^2dx+2\xi ^2 \int _\Omega |\Delta {{\tilde{\omega }}}|^2dx+4\mu ^2\int _\Omega |{{\tilde{\omega }}}|^2dx\nonumber \\&\qquad +4\int _\Omega (g_1+\sigma g_2)^2 dx. \end{aligned}$$

(A.17)

Similarly, multiplying the second and the third equation of (2.2) by ${\tilde{v}}_t$, ${{\tilde{\omega }}}_t$ respectively, and integrating them over $\Omega$ yields

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |{\tilde{v}}|^2+|\nabla {\tilde{v}}|^2\right) dx+\int _\Omega |{\tilde{v}}_t|^2dx\le \int _\Omega |{\tilde{u}}|^2dx, \end{aligned}$$

(A.18)

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \left( |{{\tilde{\omega }}}|^2+\varepsilon |\nabla {{\tilde{\omega }}}|^2\right) dx+\int _\Omega |{{\tilde{\omega }}}_t|^2dx\le \int _\Omega |f_2|^2dx. \end{aligned}$$

(A.19)

Letting (A.8)+$\frac{1}{6}\times$(A.17)+$\frac{\mu }{4}\times$(A.18)+(A.19), and (A.13) is derived.

Next, we estimate I and II.

Note that $\left( \sum _{k=1}^n a_k\right) ^2\le n\sum _{k=1}^n a_k^2$, and $H^2(\Omega )\hookrightarrow L^\infty (\Omega )$, then

$$\begin{aligned} I&=2\int _\Omega (g_1+\sigma g_2) {\tilde{u}} dx+\frac{8}{3}\int _\Omega |g_1+\sigma g_2|^2 dx+4\int _\Omega (|\nabla g_1|^2+|\nabla g_2|^2)dx\\&\le 2\chi \Vert {\tilde{u}}\Vert _{L^\infty }\Vert \nabla {\tilde{u}}\Vert _{L^2}\Vert \nabla {\tilde{v}}\Vert _{L^2}\\&\quad +2\xi \Vert {\tilde{u}}\Vert _{L^\infty }\Vert \nabla {\tilde{u}}\Vert _{L^2}\Vert \nabla {{\tilde{\omega }}}\Vert _{L^2}\\&\quad +2\mu (\Vert {\tilde{u}}\Vert _{L^\infty }+\Vert {{\tilde{\omega }}}\Vert _{L^\infty })\Vert {\tilde{u}}\Vert _{L^2}^2 \\&\quad +16\chi ^2\Vert \nabla {\tilde{u}}\Vert _{L^4}^2\Vert \nabla {\tilde{v}}\Vert _{L^4}^2 \\&\quad +16\left( \chi ^2\Vert {\tilde{u}}\Vert _{L^\infty }^2\Vert \Delta {\tilde{v}}\Vert _{L^2}^2+ \xi ^2\Vert \nabla {\tilde{u}}\Vert _{L^4}^2\Vert \nabla {{\tilde{\omega }}}\Vert _{L^4}^2\right. \\&\quad +\xi ^2\Vert {\tilde{u}}\Vert _{L^\infty }^2 \Vert \Delta {{\tilde{\omega }}}\Vert _{L^2}^2+\mu ^2\Vert {\tilde{u}}\Vert _{L^4}^4\\&\quad \left. +\mu ^2\Vert {\tilde{u}}\Vert _{L^4}^2\Vert {{\tilde{\omega }}}\Vert _{L^4}^2\right) \\&\quad +32\chi ^2\left( \Vert D^2{\tilde{u}}\Vert _{L^4}^2\Vert \nabla {\tilde{v}}\Vert _{L^4}^2 +2\Vert D^2{\tilde{v}}\Vert _{L^4}^2\Vert \nabla {\tilde{u}}\Vert _{L^4}^2+\Vert {\tilde{u}}\Vert _{L^\infty }^2 \Vert \nabla \Delta {\tilde{v}}\Vert _{L^2}^2\right) \\&\quad +32\xi ^2\left( \Vert D^2{\tilde{u}}\Vert _{L^4}^2\Vert \nabla {{\tilde{\omega }}}\Vert _{L^4}^2+2 \Vert D^2{{\tilde{\omega }}}\Vert _{L^4}^2\Vert \nabla {\tilde{u}}\Vert _{L^4}^2+\Vert {\tilde{u}}\Vert _{L^\infty }^2\Vert \nabla \Delta {{\tilde{\omega }}}\Vert _{L^2}^2\right) \\&\quad +12\mu ^2\left( \Vert {\tilde{u}}\Vert _{L^\infty }^2\Vert \nabla {\tilde{u}}\Vert _{L^2}^2 +\Vert {\tilde{u}}\Vert _{L^\infty }^2\Vert \nabla {{\tilde{\omega }}}\Vert _{L^2}^2+\Vert {{\tilde{\omega }}} \Vert _{L^\infty }^2\Vert \nabla {\tilde{u}}\Vert _{L^2}^2\right) \\&\le C_1\left( \Vert {\tilde{u}}\Vert _{2}+\Vert {{\tilde{\omega }}}\Vert _{2}\right) \left( \Vert {\tilde{u}}\Vert _{1}^2 +\Vert {\tilde{v}}\Vert _{1}^2+\Vert {{\tilde{\omega }}}\Vert _{1}^2\right) \\&\quad +C_2\left( \Vert {\tilde{u}}\Vert _{3}^2+\Vert {\tilde{v}}\Vert _{3}^2+\Vert {{\tilde{\omega }}}\Vert _{3}^2\right) \left( \Vert {\tilde{u}}\Vert _{2}^2+\Vert {\tilde{v}}\Vert _{2}^2+\Vert {{\tilde{\omega }}}\Vert _{2}^2\right) . \end{aligned}$$

That is

$$\begin{aligned} I\le C_1 \Vert ({\tilde{u}}, {{\tilde{\omega }}})\Vert _{2} \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{1}^2+C_2 \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{3}^2 \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{2}^2. \end{aligned}$$

(A.20)

Noticing that

$$\begin{aligned}&\left( \frac{M}{2}+1\right) \int _\Omega |f_2|^2 dx+\frac{M}{2}\int _\Omega |\nabla f_2|^2dx\nonumber \\&\quad \le C_3\left( \Vert {\tilde{u}}\Vert _{L^\infty }^2+\Vert {\tilde{v}}\Vert _{L^\infty }^2+\Vert {{\tilde{\omega }}}\Vert _{L^\infty }^2\right) \left( \Vert {\tilde{u}}\Vert _{1}^2+\Vert {\tilde{v}}\Vert _{1}^2+\Vert {{\tilde{\omega }}}\Vert _{1}^2\right) \nonumber \\&\quad \le C_4\left( \Vert {\tilde{u}}\Vert _{2}^2+\Vert {\tilde{v}}\Vert _{2}^2+\Vert {{\tilde{\omega }}}\Vert _{2}^2\right) ^2, \end{aligned}$$

(A.21)

$$\begin{aligned}&\frac{M}{2}\int _\Omega |\Delta f_2|^2dx \nonumber \\&\quad \le C_5\left( \Vert {\tilde{u}}\Vert _{L^\infty }^2+\Vert {\tilde{v}}\Vert _{L^\infty }^2+\Vert {{\tilde{\omega }}}\Vert _{L^\infty }^2\right) \left( \Vert {\tilde{u}}\Vert _{2}^2+\Vert {\tilde{v}}\Vert _{2}^2+\Vert {{\tilde{\omega }}}\Vert _{2}^2\right) \nonumber \\&\qquad +C_6\left( \Vert \nabla {\tilde{u}}\Vert _{L^4}^2+\Vert \nabla {\tilde{v}}\Vert _{L^4}^2+\Vert \nabla {{\tilde{\omega }}}\Vert _{L^4}^2\right) ^2\nonumber \\&\quad \le C_7\left( \Vert {\tilde{u}}\Vert _{2}^2+\Vert {\tilde{v}}\Vert _{2}^2+\Vert {{\tilde{\omega }}}\Vert _{2}^2\right) ^2, \end{aligned}$$

(A.22)

$$\begin{aligned}&4\xi ^2\int _\Omega |\nabla \Delta f_2|^2 dx\le C_8\left( \Vert {\tilde{u}}\Vert _{L^\infty }^2+\Vert {\tilde{v}}\Vert _{L^\infty }^2+\Vert {{\tilde{\omega }}}\Vert _{L^\infty }^2\right) \left( \Vert {\tilde{u}}\Vert _{3}^2+\Vert {\tilde{v}}\Vert _{3}^2+\Vert {{\tilde{\omega }}}\Vert _{3}^2\right) \nonumber \\&\quad +C_9\left( \Vert \nabla {\tilde{u}}\Vert _{L^4}^2+\Vert \nabla {\tilde{v}}\Vert _{L^4}^2+\Vert \nabla {{\tilde{\omega }}}\Vert _{L^4}^2\right) \left( \Vert D^2{\tilde{u}}\Vert _{L^4}^2+\Vert D^2{\tilde{v}}\Vert _{L^4}^2+\Vert D^2{{\tilde{\omega }}}\Vert _{L^4}^2\right) \nonumber \\&\quad \le C_{10}\left( \Vert {\tilde{u}}\Vert _{2}^2+\Vert {\tilde{v}}\Vert _{2}^2+\Vert {{\tilde{\omega }}}\Vert _{2}^2\right) \left( \Vert {\tilde{u}}\Vert _{3}^2+\Vert {\tilde{v}}\Vert _{3}^2+\Vert {{\tilde{\omega }}}\Vert _{3}^2\right) . \end{aligned}$$

(A.23)

Therefore,

$$\begin{aligned} II&=\left( \frac{M}{2}+1\right) \int _\Omega |f_2|^2 dx+\frac{M}{2}\int _\Omega \left( |\nabla f_2|^2+|\Delta f_2|^2\right) dx+4\xi ^2\int _\Omega |\nabla \Delta f_2|^2 dx\nonumber \\&\le C_{11}\Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{3}^2 \Vert ({\tilde{u}}, {\tilde{v}}, {{\tilde{\omega }}})\Vert _{2}^2. \end{aligned}$$

(A.24)

Then (A.16) is derived from (A.20) and (A.24). $\square$