Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ${\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}$ , ${(\alpha\in(0,2], \;p > 1)}$ posed on ${\mathbb{R}^N}$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

The Decay of mass for a nonlinear fractional reaction–diffusion equation

We study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ?_tu = ? t^σ( ? Δ)^{α ∕ 2}u ? h(t)u^p (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows:

• if , then there exists M _∞ ∈ (0, ∞ ) such that ;
• if , then .

Copyright © 2014 John Wiley & Sons, Ltd.     

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a “mean field” equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the “elementary vortex patch”.     

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].     

On a nonlinear equation with fractional derivative

We consider a nonlinear equation with fractional derivative in which the nonlinearity has the form of a linear combination of convective and nonconvective types. We prove the time-global existence of solutions of the Cauchy problem and find their asymptotics at large times uniformly with respect to the space variable. The proof method is based on a detailed investigation of the behavior of the Green function, which permits one to drop the restriction of smallness of the initial data in the case of a nonlinearity of special form.     

Periodic solutions of a semilinear elliptic equation with a fractional Laplacian

We consider periodic solutions of the following problem associated with the fractional Laplacian

$$(-\partial _{xx})^s u(x) + F'(u(x))=0,\quad u(x)=u(x+T),\quad \text{ in } \, \mathbb {R}, $$

where \((-\partial _{xx})^s\) denotes the usual fractional Laplace operator with \(0<s<1\). The primitive function F of the nonlinear term is a smooth double-well potential. We prove the existence of periodic solutions with large period T using variational methods. An estimate of the energy of the periodic solutions is also established.

     

Decay estimates for the wave equation of p‐Laplacian type with dissipation of m‐Laplacian type

In this paper, we study decay properties of solutions to the wave equation of p‐Laplacian type with a weak dissipation of m‐Laplacian type. Copyright © 2006 John Wiley & Sons, Ltd.     

Blow-up of solutions for a nonlinear beam equation with fractional feedback

A nonlinear beam equation describing the transversal vibrations of a beam with boundary feedback is considered. The boundary feedback involves a fractional derivative. We discuss the asymptotic behavior of solutions. In fact, we prove that solutions blow up in finite time under certain assumptions on the nonlinearity.     

Decay and growth for a nonlinear parabolic difference equation

We prove a difference equation analogue of the decay-of-mass result for the nonlinear parabolic equation when and a new growth result when 0$">.

     

Liouville-type theorems for a nonlinear fractional Choquard equation

In this paper, we are concerned with the fractional Choquard equation on the whole space ${\mathbb{R}}^N$ $${(-\mathrm{\Delta })}^{s}u=\left(\frac{1}{{|x|}^{N-2s}}\ast u^p\right)u^{p-1}$$ with , $N>2s$ and $p\in \mathbb{R}$. We first prove that the equation does not possess any positive solution for $p\le 1$. When $p>1$, we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.     

An efficient difference scheme for the coupled nonlinear fractional Ginzburg–Landau equations with the fractional Laplacian

In this article, an efficient difference scheme for the coupled fractional Ginzburg–Landau equations with the fractional Laplacian is studied. We construct the discrete scheme based on the implicit midpoint method in time and a weighted and shifted Grünwald difference method in space. Then, we prove that the scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent in the norm. Finally, numerical tests are given to confirm the theoretical results and show the effectiveness of the scheme.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Nodal bound state of nonlinear problems involving the fractional Laplacian

This paper is concerned with the following nonlinear fractional Schrödinger equation where ε>0 is a small parameter, V(x) is a positive function, 0<s<1, and . Under some suitable conditions, we prove that for any positive integer k, one can construct a nonradial sign‐changing (nodal) solutions with exactly k maximum points and k minimum points near the local minimum point of V(x).     

奇异分数阶Laplacian方程正弱解的存在性及正则性

因为奇异项使得分数阶Laplacian方程没有变分结构,所以临界点理论不能直接使用,成为研究此类方程弱解存在性的本质困难.本文首次运用闭锥上的临界点理论,得到奇异分数阶Laplacian方程的正弱解及其正则性.而且,此方法适用于其他奇异分数阶问题.     

The finite element method for fractional diffusion with spectral fractional Laplacian

This paper focuses on the finite element method for Caputo-type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite-part integral (or the finite-part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full-discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.     

Computational method for solving space fractional Fisher's nonlinear equation

This paper aims to present a general framework of the quadratic spline functions to develop a numerical method for solving the nonlinear space fractional Fisher's equation. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm. The results reveal that the proposed approach is very effective, convenient, and quite accurate to such considered problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Riquier problem for a nonlinear equation resolved with respect to the iterated Levi Laplacian

Solutions are found for the nonlinear equation Δ _L ² U(x) = f(U(x)) (here, Δ _L is an infinite-dimensional Laplacian) which is solved with respect to the iterated infinite-dimensional Laplacian. The Riquier problems are stated for an equation of this sort. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1574–1577, November, 1998.