Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

We study the following parabolic nonlocal 4-th order degenerate equation: u t = - [ 2 ⁢ π ⁢ H ⁢ ( u x ) + ln ⁡ ( u x ⁢ x + a ) + 3 2 ⁢ ( u x ⁢ x + a ) 2 ] x ⁢ x , u_{t}=-\Bigl{[}2\pi H(u_{x})+\ln(u_{xx}+a)+\frac{3}{2}(u_{xx}+a)^{2}\Bigr{]}_{% xx}, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a > 0 {a>0} is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when u x ⁢ x + a {u_{xx}+a} approaches zero. Thus we show that, if the initial datum u 0 {u_{0}} is such that ( u 0 ) x ⁢ x + a {(u_{0})_{xx}+a} is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.

Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.