The Cauchy problem for the degenerate convective Cahn-Hilliard equation
The authors investigate the Cauchy problem associated with the following degenerate convective Cahn-Hilliard equation \[ \begin{cases} u_t+\Delta_{x'}^2u-\Delta_{x'}A(u)-{\mathbf r}\cdot \nabla B(u)=0, \;\;x'=(x_2,\dots,x_n)\in \mathbb{R}^{n-1}, \;t>0,& \\ u(x,0)=u_0(x),& \end{cases} \] where \(A=A(u)\), \(B=B(u)\) are sufficiently smooth functions, \({\mathbf r}=(r_1,\dots,r_n) \in \mathbb{R}^n\), \(\nabla =(\partial_{x_1},\dots, \partial_{x_n})\), \(\Delta_{x'}=\sum_{i=2}^n\partial_{x_i}^2\). In addition, \(A(u)={\mathcal{O}}(|u|^{\theta +1})\), \(B(u)={\mathcal{O}}(|u|^{\theta +1})\) with the same growth property and \(\theta \geq 1\) is an integer. They prove existence of a unique global classical solution for small \(u_0\) by establishing uniform estimates of the solution. Decay estimates are also addressed.
scientific article; zbMATH DE number 6999276
scientific article