A vicinal surface model for epitaxial growth with logarithmic free energy

Figure 1.  (Top) A typical PDE simulation for (1.10) with $\alpha = 1$ on $0\le x\le 1$ and (bottom) the corresponding plot of $u(t,h)$ with boundary conditions (1.15) and $H = 2$, clearly showing the convergence of $h$ to a straight line, with the slope $u$ approaching to a spatially-uniform profile $u = 2$.

Figure 2.  A numerical simulation of PDE (1.6) plotted in semi-log coordinates starting from the initial condition (5.1) (plotted with the dashed line): (top) early stage near-rupture is approached as the global minimum decreases from $0.07$ to $0.007$ for $0<t< 0.0032$; (bottom) later stage behavior for $t>0.0032$ as the solution approaches a constant $u^{\star} = 0.27$.

Figure 3.  (Top) A numerical simulation of PDE (1.13) starting from identical initial conditions used in Fig. 2 showing convergence to a spatially-uniform solution $u = u^{\star}$ as $t \to \infty$. (Bottom) A plot showing that $u_{m}(t) = \min_h u(t,h)$ is bounded below by $\mathcal{J}(E(t))$ given by (3.32) which is in line with the conclusion of Theorem 1, and that the asymptotic lower bound $\mathcal{J}(E(t)) \to {1}/{(2L)}$ for $t \to \infty$ as in (3.8).

Figure 4.  Plots of corresponding energy $E$ in (2.5) and (5.7) for PDE simulations in Fig. 2 and Fig. 3. The energy $E(t)$ decays exponentially to zero following (5.6) with $k = 2\pi$.

Figure 5.  Evolution of the surface height $h(t,x)$ and slope $h_x(t,x)$ following equation (6.2) with $\alpha = 0$ starting from initial condition $h_0(x) = \sin(2\pi x)$ on $0\le x\le 1$, showing convergence to spatially-uniform solution $h \equiv 0$ as $t \to \infty$.

Figure 6.  Evolution of the surface height $h(t,x)$ and slope $h_x(t,x)$ for equation (6.2) with $\alpha = 1$ starting from identical initial data used in Fig. 5, showing convergence to a piece-wise constant profile in $h$ and jump in $h_x$.