On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth (Q2826335)

The authors consider the fourth order parabolic equation modeling epitaxial thin film growth NEWLINE\[NEWLINE \frac {\partial u}{\partial t}=-\nabla \cdot \Bigl [m(x,t)\bigl (k\nabla \Delta u-| \nabla u| ^{p-2}\nabla u\bigr)\Bigr],\qquad k>0,\quad p>2. NEWLINE\]NEWLINE They proved the existence and uniqueness of global weak solution for the initial and boundary problem in the space \(H^{4,1}(Q_T)\). Based on the framework of Campanato spaces, authors established the regularity of the solutions in two space dimensions. Using Temam's classical theorem, they proved that the problem admitted a global attractor in the space \(H^2(\Omega)\).