Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez

Displaying similar documents to “Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type”

Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

Vladimir Varlamov (2000)

Studia Mathematica

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The nonlinear heat equation with a fractional Laplacian $[u_{t} + {(- Δ)}^{α / 2} u = u^{2}, 0 < α \leq 2]$ , 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 $C ⁰ ([0, \infty), H ₀^{κ} (B))$ 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.

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

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We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $- d / d t (1 / 2_{0} D_{t}^{- σ} (u^{'} (t)) + 1 / 2_{t} D_{T}^{- σ} (u^{'} (t))) - λ β (t) f (u (t)) - μ γ (t) g (u (t)) = 0$ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where $_{0} D_{t}^{- σ}$ and $_{t} D_{T}^{- σ}$ 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].

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

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We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{κ} (t), t \geq 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{κ} (t)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{κ} (t)$ 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...

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_{t} = \nabla \cdot (u \nabla {(- Δ)}^{- s} u), 0 < s < 1$ . 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 $C^{α}$ regularity of such weak solutions. Finally, we extend...

A spatially sixth-order hybrid $L 1$ -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng (2021)

Applications of Mathematics

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We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L 1$ 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 $L 1$ -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2 - γ$ in time, where $0 < γ < 1$ is the order of the Caputo fractional derivative...

Fractional $q$ -difference equations on the half line

Saïd Abbas, Mouffak Benchohra, Nadjet Laledj, Yong Zhou (2020)

Archivum Mathematicum

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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.

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{α ₙ} - \sum_{i = 1}^{n - 1} a_{i} D^{α_{i}}) x (t) \in F (t, x (φ (t)))$ , a.e. on (0,1), $I^{1 - α ₙ} x (0) = c$ , 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.

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

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Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

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In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

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Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${a ⁿ / n}_{n = 1}^{\infty}$ 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.

Nonoscillatory solutions of discrete fractional order equations with positive and negative terms

Jehad Alzabut, Said Rezk Grace, A. George Maria Selvam, Rajendran Janagaraj (2023)

Mathematica Bohemica

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This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form $\begin{matrix} Δ^{γ} u (κ) & + Θ [κ + γ, w (κ + γ)] \\ = & Φ (κ + γ) + Υ (κ + γ) w^{ν} (κ + γ) + Ψ [κ + γ, w (κ + γ)], κ \in ℕ_{1 - γ}, \\ u_{0} = & c_{0}, \end{matrix}$ where $ℕ_{1 - γ} = {1 - γ, 2 - γ, 3 - γ, \dots}$ , $0 < γ \leq 1$ , $Δ^{γ}$ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut, Amil Hasanov (2016)

Open Mathematics

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In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ 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, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p, ϕ_{1}} (w^{p}) to M_{p, ϕ_{2}} (w^{q})$ for 1 < p < q < ∞. In all cases the conditions...

Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ, λ \in A_{p, q}$ and α/n + 1/q = 1/p, the norm $| | [b, I_{α}] : L^{p} (μ^{p}) \to L^{q} (λ^{q}) | |$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μ λ^{- 1}$ . 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,...

Fractional integral operators on $B^{p, λ}$ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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We introduce function spaces $B^{p, λ}$ with Morrey-Campanato norms, which unify $B^{p, λ}$ , $C M O^{p, λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.

On the variation of certain fractional part sequences

Michel Balazard, Leila Benferhat, Mihoub Bouderbala (2021)

Communications in Mathematics

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Let $b > a > 0$ . We prove the following asymptotic formula $\sum_{n ⩾ 0} | {x / (n + a)} - {x / (n + b)} | = \frac{2}{π} ζ (3 / 2) \sqrt{c x} + O (c^{2 / 9} x^{4 / 9}),$ with $c = b - a$ , uniformly for $x ⩾ 40 c^{- 5} {(1 + b)}^{27 / 2}$ .

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

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

Existence Results for a Fractional Boundary Value Problem via Critical Point Theory

A. Boucenna, Toufik Moussaoui (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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

Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case

Krzysztof Dębicki, Grzegorz Sikora (2011)

Applicationes Mathematicae

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Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $ℙ (s u p_{t \in [0, T]} (\sum_{i = 1}^{n} λ_{i} B_{H_{i}} (t) - c t) > u)$ , where $B_{H_{i}} (t) : t \geq 0$ , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_{i} \in (0, 1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $m i n_{i = 1, . . ., n} H_{i}$ .