On a critical nonlinear problem involving the fractional Laplacian

Fractional Yamabe-type equations of the form [Formula: see text] in [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded domain of [Formula: see text], [Formula: see text] is a given function on [Formula: see text] and [Formula: see text], is the fractional Laplacian are considered. Bahri’s estimates in the fractional setting will be proved and used to establish a global existence result through an index-counting formula.

Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

In this article, we deal with an inhomogeneous fast diffusive polytropic filtration equation. By using the energy estimate approach, Hardy–Littlewood–Sobolev inequality, and a series of ordinary differential inequalities, we prove the global existence result and obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.

Continuous solution for a non-linear eikonal system

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and <inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> estimates. A uniqueness result is also proved through a comparison principle property.</p>

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $$H^2$$ H 2 -close to a multiply covered circle and is sufficiently rotationally symmetric.