We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla ·\left(u\nabla {(-\Delta )}^{-s}u\right),0<s<1$. The problem is posed in $\{x\in {ℝ}^n,t\in ℝ\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions. Finally, we extend...

We consider the problem of determining the unknown source term $f=f(x,t)$ in a space fractional diffusion equation from the measured data at the final time $u(x,T)=\psi \left(x\right)$. In this way, a methodology involving minimization of the cost functional $J\left(f\right)={\int }_0^l{(u(x,t;f){|}_{t=T}-\psi \left(x\right))}^2\mathrm{d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and...

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,I_{\alpha }]:L^p\left({\mu }^p\right)\to L^q\left({\lambda }^q\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{A_1,A_2,...,A_k},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{A_1,A_2,...,A_k},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p,{\varphi }_1}\left(w^p\right)\mathrm{to}{M}_{p,{\varphi }_2}\left(w^q\right)$ for 1 < p < q < ∞. In all cases the conditions...

The nonlinear heat equation with a fractional Laplacian $[u_t+{(-\Delta )}^{\alpha /2}u=u^2,0<\alpha \le 2]$, is considered in a unit ball $B$. Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space $C⁰([0,\infty ),H{₀}^{\kappa }\left(B\right))$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form $u(x,t)=t^{–\alpha }f\left(\right|x|t^{–\beta })$ with suitable $$ and $\beta$. As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ${\left|\right|f\left|\right|}_{W^{\sigma ,2}}$ of a function f ∈ L²(E,μ) have the property $1/Cℰ(f,f)\le lim in{f}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le lim su{p}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{\alpha ₙ}-{\sum }_{i=1}^{n-1}{a}_i{D}^{{\alpha }_i})x\left(t\right)\in F(t,x\left(\phi \left(t\right)\right))$, a.e. on (0,1), $I^{1-\alpha ₙ}x\left(0\right)=c$, αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${aⁿ/n}_{n=1}^{\infty }$ is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length $c{N}^{-0.475}$ contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma$ in time, where $0<\gamma <1$ is the order of the Caputo fractional derivative...

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $-d/dt(1/2_0{D}_t^{-\sigma }\left(u^{\text{'}}\left(t\right)\right)+1/2_t{D}_T^{-\sigma }\left(u^{\text{'}}\left(t\right)\right))-\lambda \beta \left(t\right)f\left(u\left(t\right)\right)-\mu \gamma \left(t\right)g\left(u\left(t\right)\right)=0$, a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where ${}_0{D}_t^{-\sigma }$ and ${}_t{D}_T^{-\sigma }$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

We introduce function spaces $B^{p,\lambda }$ with Morrey-Campanato norms, which unify $B^{p,\lambda }$, $CM{O}^{p,\lambda }$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{\alpha }$ on these spaces.

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1$ and $g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\gamma }_{fg}\left(G\right)$ is defined as follows: ${\gamma }_{fg}\left(G\right)$ = min|g|:g is an MGDF of G where $\left|g\right|={\sum }_{v\in V}g\left(v\right)$. In this paper we initiate a study of this parameter.

Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, ${(X,\varrho ,\mu )}_{d,\theta }$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B_{pq}^s\left(X\right)$ and Triebel-Lizorkin spaces $F_{pq}^s\left(X\right)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional...

Let $\mu$ be a nonnegative Borel measure on ${ℝ}^d$ satisfying that $\mu \left(Q\right)\le l{\left(Q\right)}^n$ for every cube $Q\subset {ℝ}^n$, where $l\left(Q\right)$ is the side length of the cube $Q$ and $0<n\le d$. We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function $B$ in the context of non-homogeneous spaces related to the measure $\mu$. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result...

We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{\kappa }\left(t\right),t\ge 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{\kappa }\left(t\right)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{\kappa }\left(t\right)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...

We introduce the generalized fractional integrals $I{̃}_{\alpha ,d}$ and prove the strong and weak boundedness of $I{̃}_{\alpha ,d}$ on the central Morrey spaces $B^{p,\lambda }\left(ℝⁿ\right)$. In order to show the boundedness, the generalized λ-central mean oscillation spaces ${\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ and the generalized weak λ-central mean oscillation spaces $W{\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ play an important role.

rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p),\theta }(X,\mu )$ to $L^{q),q\theta /p}(X,\nu )$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...

The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

This article deals with some results about the existence of solutions and bounded solutions and the attractivity for a class of fractional $q$-difference equations. Some applications are made of Schauder fixed point theorem in Banach spaces and Darbo fixed point theorem in Fréchet spaces. We use some technics associated with the concept of measure of noncompactness and the diagonalization process. Some illustrative examples are given in the last section.

For α ∈ (1,2) we consider the equation ${\partial }_{t}u={\Delta }^{\alpha /2}u+b·\nabla u$, where b is a time-independent, divergence-free singular vector field of the Morrey class $M{₁}^{1-\alpha }$. We show that if the Morrey norm ${\left|\right|b\left|\right|}_{M{₁}^{1-\alpha }}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

We consider the fractional Laplacian $-{(-\Delta )}^{\alpha /2}$ on an open subset in ${ℝ}^d$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1,1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $\alpha$-stable process killed upon leaving a $C^{1,1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

We prove that a function belonging to a fractional Sobolev space $L^{\alpha ,p}\left(ℝⁿ\right)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,\lambda }\left(ℝⁿ\right)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

In this paper, we consider the following boundary value problem $$\left\{\begin{array}{c}{D}_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }\left(D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)\right)=f(t,u\left(t\right)),t\in [0,T],\hfill \\ u\left(0\right)=u\left(T\right)=0\hfill \\ D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(0\right)\right)=D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(T\right)\right)=0,\hfill \end{array}\right.$$ where $0<\alpha \le 1$ and $f:[0,T]\times ℝ\to ℝ$ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.