# On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations
## 1 Introduction  
In this paper, we investigate the well-posedness of the following Cauchy problem:  
\[ \begin{cases}\partial_{t}u+\kappa\partial_{x}u^{2}-\beta^{2}\partial_{x}^{6}u +\alpha\partial_{x}^{4}u+\delta^{2}\partial_{x}^{4}\left(u^{3}\right)=0,&t>0, \quad x\in\mathbb{R},\\ u(0,x)=u_{0}(x),&x\in\mathbb{R},\end{cases}  \tag{1}\]  
with  
\[\kappa,\,\beta,\,\alpha,\,\delta\in\mathbb{R},\quad\beta,\,\delta\
eq 0. \tag{2}\]  
On the initial datum, we assume  
\[u_{0}\in H^{3}(\mathbb{R}),\quad\int_{\mathbb{R}}u_{0}(x)dx=0. \tag{3}\]  
Inspired by [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], thanks to (3), we can define the following function:  
\[P_{0}(x)=\int_{-\infty}^{x}u_{0}(y)dy, \tag{4}\]  
on which we assume  
\[\|P_{0}\|_{L^{2}(\mathbb{R})}^{2}=\int_{\mathbb{R}}\left(\int_{-\infty}^{x}u_{ 0}(y)dy\right)^{2}dx<\infty. \tag{5}\]  
Equation (1) is derived in [13], with \(\beta=1\), \(\alpha=-1\), \(\delta=1\), to describe a growing crystalline surface with small slopes that undergoes faceting. The function \(u\) represents the slope of a surface, the constant \(\kappa\) is proportional to the deposition strength of an atomic flux, and the overall convective term \(\kappa\partial_{x}u^{2}\) stems from the normal impingement of the deposited atoms. The sixth order linear term \(\partial_{x}^{6}u\), in (1), comes from a curvaturedependent regularization, and all other terms represent the anisotropy of the surface energy under surface diffusion. These equations became popular objects of theoretical studies in the last decade, because they do play an important role in material modeling. Equations based on the Mullins surface diffusion formula, such as in (1), are interesting, as they describe the morphology of crystalline surfaces. As complex interfaces and surfaces are more and more important in highly technological applications such as photovoltaics, the study of analytical aspects also gains momentum (see, [14; 15; 16; 17; 18; 19; 20; 21; 22]). The existence and uniqueness of weak solutions of (1) with periodic boundary conditions is proven in [23], with \(\kappa>0\). A similar result is proven in [24], in two space dimensions. In [25], the authors derived stationary solutions of (1), while in [26], the problem for optimal control of (1) is analyzed. In [27], the problem of a global attractor is studied. Taking \(\kappa=0\), (1) reads  
\[\partial_{t}u-\beta^{2}\partial_{x}^{6}u+\alpha\partial_{x}^{4}u+\delta^{2} \partial_{x}^{4}\left(u^{3}\right)=0, \tag{6}\]  
which is a Cahn-Hilliard type equation [28; 29; 30]. It describes dynamics of phase transitions in ternary oil-water-surfactant systems. One part of the surfactant is hydrophilic and the other one is lipophilic (and termed amphiphile). In the system, almost pure oil, almost pure water, and microemulsion which consist of a homogeneous, isotropic mixture of oil and water can coexist in equilibrium. The function \(u\), in (6), represents the local difference between oil and water concentrations. In [31] the existence of global attractors is analyzed and in [32] the existence of weak solutions for the initial -boundary-value problem for (6) with degenerate mobility is proven. Assuming \(\beta=\delta=0\) and \(\alpha=a^{2}\
eq 0\), (1) reads  
\[\partial_{t}u+\kappa\partial_{x}u^{2}+a^{2}\partial_{x}^{4}u=0. \tag{7}\]  
(7) arises in interesting physical situations, for example as a model for long waves on a viscous fluid owing down an inclined plane [33] and to derive drift waves in a plasma [34]. Equation (7) was derived also independently by [PERSON] [35; 36; 37] as a model for phase turbulence in reaction-diffusion systems and by [PERSON] [38] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. Equation (7) also describes incipient instabilities in a variety of physical and chemical systems [39; 40; 41].