Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow

In this paper, we consider the advective unstable Cahn-Hilliard equation in 2D with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon \Delta^2 u= \Delta(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} with an initial data $u_0 \in H_0^2(\mathbb T^2)$, where $\mathbb T^2$ is the two-dimensional torus, $A, \varepsilon>0$, $a<0$, $b \in \mathbb R$. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the $L^2$-energy of the solutions to such problems converges expotentially to zero, if in addition, both $|a|$ and $\left\| \int_{\mathbb T} u_0(x, \cdot ) dx \right\|_{L_y^2}$ are sufficiently small.