Search results for " 35R11"
showing 5 items of 5 documents
Four solutions for fractional p-Laplacian equations with asymmetric reactions
2020
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.
Representation of solutions and large-time behavior for fully nonlocal diffusion equations
2017
Abstract We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal L 2 -decay of mild solutions in all dimensions, (iv) L 2 -decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii). Next we use Fourier analysis techniques to obtain the optimal dec…
Landis-type conjecture for the half-Laplacian
2023
In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage’s Liouville-type theorem. peerReviewed
The higher order fractional Calderón problem for linear local operators : Uniqueness
2020
We study an inverse problem for the fractional Schr\"odinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the order of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth and first order perturbations to higher order perturbations.
Short time existence of the classical solution to the fractional mean curvature flow
2019
Abstract We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C 1 , 1 -regular. We provide the same result also for the volume preserving fractional mean curvature flow.