Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

Numerical solutions for fractional reaction-diffusion equations

Fractional diffusion equations are useful for applications where a cloud of particles spreads faster than the classical equation predicts. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fr...

Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential

Positive solutions for asymptotically periodic Kirchhoff-type equations with critical growth

In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive ...

Periodic solutions of forced Kirchhoff equations

We consider Kirchhoff equations for vibrating strings and elastic membranes under the action of an external forcing of period 2π/ω and small amplitude ε. We prove existence, regularity and local uniqueness of 2π/ω-periodic solutions of order ε by means of a Nash-Moser iteration scheme; the results hold for parameters (ω, ε) in a Cantor-like set which has asymptotically full measure for ε→ 0.

Discontinuous Galerkin Methods for Fractional Diffusion Equations

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems, characterized by having fractional derivatives, parameterized by β ∈ [1, 2]. We show through analysis that one can construct a numerical flux which results in a scheme that exhibit optimal order of convergence O(hk+1) in the continuous range between pure advection (β = 1) and pure...