The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures amphiphilic molecules. We study existence nonnegative weak solutions gradient flow subject to degenerate mobility M(u) that is zero u≤0. Assuming initial data u0(x) positive, we construct solution limit corresponding non-degenerate mobilities and verify it satisfies an d...

In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1/H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp ...

We introduce the explosion problem for semilinear elliptic and parabolic PDEs. The explosion problem is a question of existence of nonnegative solutions to −Lφ = λg(x, φ) and nonnegative global solutions to ∂tφ−Lφ = λg(x, t, φ). Under certain conditions there is a λ∗ > 0, called the explosion threshold, for which λ < λ∗ implies a solution exists, and λ > λ∗ implies no solution exists. In this p...