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Displaying similar documents to “Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball”

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

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We consider the fractional Laplacian - ( - Δ ) α / 2 on an open subset in 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 α -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.

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 D T - α ( D 0 + α ( D T - α ( D 0 + α u ( t ) ) ) ) = f ( t , u ( t ) ) , t [ 0 , T ] , u ( 0 ) = u ( T ) = 0 D T - α ( D 0 + α u ( 0 ) ) = D T - α ( D 0 + α u ( T ) ) = 0 , where 0 < α 1 and f : [ 0 , T ] × 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.

Results of nonexistence of solutions for some nonlinear evolution problems

Medjahed Djilali, Ali Hakem (2019)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), u t t + f ( x ) u t + ( - Δ ) α / 2 ( u m ) = h ( t , x ) | u | p , posed in ( 0 , T ) × N , where ( - Δ ) α / 2 , 0 < α 2 is α / 2 -fractional power of - Δ . Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a 2 × 2 system of the same type.

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 = · ( u ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x n , t } 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...

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.

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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

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

Similarity:

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 global domination in graphs

Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid (2010)

Discussiones Mathematicae Graph Theory

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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 ( N [ v ] ) = u N [ v ] g ( u ) 1 and g ( N ( v ) ¯ ) = u N ( v ) g ( u ) 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 γ f g ( G ) is defined as follows: γ f g ( G ) = min|g|:g is an MGDF of G where | g | = v V g ( v ) . In this paper we initiate a study of this parameter.

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Gladis Pradolini, Jorgelina Recchi (2018)

Czechoslovak Mathematical Journal

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Let μ be a nonnegative Borel measure on d satisfying that μ ( Q ) l ( Q ) n for every cube Q n , where l ( Q ) is the side length of the cube Q and 0 < n 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 μ . Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result...

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D d and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by 𝒫 . We associate to Problem 𝒫 an optimal control problem, denoted by 𝒬 . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K ˜ , we provide results concerning the well-posedness...

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

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

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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 α f ( | x | t β ) with suitable and β . 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...

Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces

Katsuo Matsuoka (2014)

Banach Center Publications

Similarity:

We introduce the generalized fractional integrals I ̃ α , d and prove the strong and weak boundedness of I ̃ α , d on the central Morrey spaces B p , λ ( ) . In order to show the boundedness, the generalized λ-central mean oscillation spaces Λ p , λ ( d ) ( ) and the generalized weak λ-central mean oscillation spaces W Λ p , λ ( d ) ( ) play an important role.

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Given a two-dimensional fractional multiplicative process ( F t ) t [ 0 , 1 ] determined by two Hurst exponents H 1 and H 2 , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of [ 0 , 1 ] by F if and only if H 1 = H 2 .

Inverse source problem in a space fractional diffusion equation from the final overdetermination

Amir Hossein Salehi Shayegan, Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie (2019)

Applications of Mathematics

Similarity:

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 ) = ψ ( x ) . In this way, a methodology involving minimization of the cost functional J ( f ) = 0 l ( u ( x , t ; f ) | t = T - ψ ( x ) ) 2 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...

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 n 0 | { x / ( n + a ) } - { x / ( n + b ) } | = 2 π ζ ( 3 / 2 ) c x + O ( c 2 / 9 x 4 / 9 ) , with c = b - a , uniformly for x 40 c - 5 ( 1 + b ) 27 / 2 .

L p - L q boundedness of analytic families of fractional integrals

Valentina Casarino, Silvia Secco (2008)

Studia Mathematica

Similarity:

We consider a double analytic family of fractional integrals S z γ , α along the curve t | t | α , introduced for α = 2 by L. Grafakos in 1993 and defined by ( S z γ , α f ) ( x , x ) : = 1 / Γ ( z + 1 / 2 ) | u - 1 | z ψ ( u - 1 ) f ( x - t , x - u | t | α ) d u | t | γ d t / t , where ψ is a bump function on ℝ supported near the origin, f c ( ² ) , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that S z γ , α maps L p ( ² ) to L q ( ² ) boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel K - 1 + i θ i ϱ , α is a product kernel on ℝ², adapted to the curve t | t | α ; as a consequence, we show...