We study numerically the one-dimensional Allen–Cahn equation with spectral fractional Laplacian (−Δ)α/2 on intervals homogeneous Neumann boundary conditions. In particular, we are interested in speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow case classical Laplacian. Here investigate how width change if vary exponent α For associ...

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the singlemultiplicity property of the limiting energy, that the Morse indic...

Abstract We analyze a phase field model for tumor growth consisting of Cahn–Hilliard–Brinkman system, ruling the evolution mass, coupled with an advection-reaction-diffusion equation chemical species acting as nutrient. The main novelty paper concerns discussion existence weak solutions to system covering all meaningful cases nonlinear potentials; in particular, typical choices given by regular...

A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidisCrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exac...