We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian
manifolds. For a general class of initial condition we show non positivity of the limiting …

We give a multiplicity result for solutions of the Van der Waals–Cahn–Hilliard two phase
transition equation with volume constraints on a closed Riemannian manifold. Our proof …

We prove the existence of multiple solutions to the Allen–Cahn–Hilliard (ACH) vectorial
equation (with two equations) involving a triple-well (triphasic) potential with a small volume …

We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed
Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show …

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion
by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our …

We investigate the equation (-Δ _ H^ n)^ γ w= f (w)\quad in H^ n,(-Δ H n) γ w= f (w) in H n,
where (-Δ _ H^ n)^ γ (-Δ H n) γ corresponds to the fractional Laplacian on hyperbolic space …

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn
equation in hyperbolic space:\[-\Delta _ {\mathbb {H}^ n} u+ F'(u)= 0;\] here $ F $ is a …

In this note, we discuss symmetry properties of solutions for simple scalar and vector-valued
systems of nonlinear elliptic partial differential equations (PDEs). The systems of interest are …

We investigate the existence of non-constant uniformly-bounded minimal solutions of the
Allen–Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace …

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic
graph. Under some natural conditions on the graph, we show the existence of non-constant …