Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity

Abstract

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearitywhere total energy E = |∇h| is the total variation of h. Using a logarithmic correction E = |∇h| ln |∇h| dx and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient hx which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity h + xx happens.

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

“…To the best of our knowledge, for arbitrarily large times, whether the solution to (1) remains strictly monotone is still an open question. We also refer to [11,12,13,14,22,10,9,7] and the references therein for some related 4th order degenerate equation but evolving only local derivatives coming from nearest-neighbor interactions between steps.…”

Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in $2$ space dimensions

“…For the former, we would like to mention [10] where the authors proved that there is a finite time extinction of solutions if p > 1, while in the latter case we refer the reader to [19] and the reference therein. The gradient flow theory is essential to the existence of a solution in the existing literature [1,4,9,10,8,13,19]. If p = 2 in (1.8) or D(∇u) = I in (1.8) or (1.10), the resulting equations have received much less consideration.…”

“…The gradient flow theory does not seem to be as effective here. In [4], the author dealt with a non-constant, singular D(∇u). However, the p-Laplace operator in the exponent in (1.8) had been modified there so that the resulting equation became a gradient flow.…”

“…Then there is a weak solution to (1.1)-(1.2). Furthermore, |∇u| ∈ L ∞ (Ω) when p > 4 3 . Even though our proof will be carried out under the additional assumption…”

Strong solutions to a fourth order exponential PDE describing epitaxial growth

“…It was first observed in [11] that one had to allow the possibility that the exponent be a measure-valued function. Later, the idea of "exponential singularity" was employed in [2,4,5,14,18]. However, measure exponents do not arise in the one-dimensional case.…”