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  Subjects -> PHYSICS (Total: 736 journals)
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    - OPTICS (84 journals)
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PHYSICS (537 journals)                  1 2 3 4 5 6 | Last

Acta Acustica united with Acustica     Full-text available via subscription   (Followers: 7)
Acta Mechanica     Hybrid Journal   (Followers: 15)
Acta Physica Slovaca     Open Access   (Followers: 3)
Advanced Composite Materials     Hybrid Journal   (Followers: 10)
Advanced Functional Materials     Hybrid Journal   (Followers: 31)
Advanced Materials     Hybrid Journal   (Followers: 281)
Advances in Condensed Matter Physics     Open Access   (Followers: 6)
Advances in Exploration Geophysics     Full-text available via subscription   (Followers: 3)
Advances in Geophysics     Full-text available via subscription   (Followers: 4)
Advances in High Energy Physics     Open Access   (Followers: 12)
Advances in Imaging and Electron Physics     Full-text available via subscription   (Followers: 1)
Advances in Materials Physics and Chemistry     Open Access   (Followers: 11)
Advances in Natural Sciences: Nanoscience and Nanotechnology     Open Access   (Followers: 13)
Advances in OptoElectronics     Open Access   (Followers: 3)
Advances In Physics     Hybrid Journal   (Followers: 6)
Advances in Physics Theories and Applications     Open Access   (Followers: 4)
Advances in Remote Sensing     Open Access   (Followers: 7)
Advances in Synchrotron Radiation     Hybrid Journal   (Followers: 1)
AIP Advances     Open Access   (Followers: 4)
AIP Conference Proceedings     Full-text available via subscription  
American Journal of Applied Sciences     Open Access   (Followers: 27)
American Journal of Condensed Matter Physics     Open Access   (Followers: 2)
Analysis and Mathematical Physics     Hybrid Journal   (Followers: 1)
Annalen der Physik     Hybrid Journal   (Followers: 2)
Annales Geophysicae (ANGEO)     Open Access   (Followers: 3)
Annales Henri PoincarĂ©     Hybrid Journal   (Followers: 2)
Annales UMCS, Physica     Open Access  
Annals of Nuclear Medicine     Hybrid Journal   (Followers: 3)
Annals of Physics     Hybrid Journal   (Followers: 2)
Annual Reports on NMR Spectroscopy     Full-text available via subscription   (Followers: 1)
Annual Review of Analytical Chemistry     Full-text available via subscription   (Followers: 9)
Annual Review of Condensed Matter Physics     Full-text available via subscription   (Followers: 1)
Annual Review of Materials Research     Full-text available via subscription   (Followers: 4)
APL Materials     Open Access   (Followers: 2)
Applied Composite Materials     Hybrid Journal   (Followers: 8)
Applied Physics A     Hybrid Journal   (Followers: 9)
Applied Physics Frontier     Open Access   (Followers: 1)
Applied Physics Letters     Hybrid Journal   (Followers: 23)
Applied Physics Research     Open Access   (Followers: 6)
Applied Physics Reviews     Hybrid Journal   (Followers: 7)
Applied Radiation and Isotopes     Hybrid Journal   (Followers: 5)
Applied Remote Sensing Journal     Open Access   (Followers: 8)
Applied Spectroscopy     Full-text available via subscription   (Followers: 12)
Applied Spectroscopy Reviews     Hybrid Journal   (Followers: 2)
Archive for Rational Mechanics and Analysis     Hybrid Journal   (Followers: 3)
Astronomy & Geophysics     Hybrid Journal   (Followers: 1)
Astrophysical Journal Letters     Full-text available via subscription   (Followers: 2)
Astrophysical Journal Supplement Series     Full-text available via subscription   (Followers: 2)
Atoms     Open Access  
Attention, Perception & Psychophysics     Full-text available via subscription   (Followers: 4)
Autonomous Mental Development, IEEE Transactions on     Hybrid Journal   (Followers: 5)
Axioms     Open Access  
Bangladesh Journal of Medical Physics     Open Access  
Bauphysik     Hybrid Journal   (Followers: 1)
Biomaterials     Hybrid Journal   (Followers: 25)
Biomedical Engineering, IEEE Reviews in     Full-text available via subscription   (Followers: 14)
Biomedical Engineering, IEEE Transactions on     Hybrid Journal   (Followers: 11)
Biomedical Imaging and Intervention Journal     Open Access   (Followers: 5)
Biophysical Reviews     Hybrid Journal  
Biophysical Reviews and Letters     Hybrid Journal   (Followers: 3)
BMC Biophysics     Open Access   (Followers: 7)
BMC Nuclear Medicine     Open Access   (Followers: 5)
Brazilian Journal of Physics     Hybrid Journal  
Broadcasting, IEEE Transactions on     Hybrid Journal   (Followers: 5)
Bulletin of Materials Science     Open Access   (Followers: 35)
Bulletin of the Atomic Scientists     Full-text available via subscription   (Followers: 4)
Bulletin of the Lebedev Physics Institute     Hybrid Journal   (Followers: 1)
Bulletin of the Russian Academy of Sciences: Physics     Hybrid Journal  
Caderno Brasileiro de Ensino de FĂ­sica     Open Access  
Canadian Journal of Physics     Full-text available via subscription   (Followers: 1)
Cells     Open Access  
Central European Journal of Physics     Hybrid Journal   (Followers: 1)
Chinese Journal of Astronomy and Astrophysics     Full-text available via subscription  
Chinese Journal of Chemical Physics     Hybrid Journal   (Followers: 1)
Chinese Physics B     Full-text available via subscription  
Chinese Physics C     Full-text available via subscription  
Chinese Physics Letters     Full-text available via subscription  
Cohesion and Structure     Full-text available via subscription   (Followers: 1)
Colloid Journal     Hybrid Journal   (Followers: 2)
Communications in Mathematical Physics     Hybrid Journal   (Followers: 2)
Communications in Numerical Methods in Engineering     Hybrid Journal   (Followers: 3)
Communications in Theoretical Physics     Full-text available via subscription   (Followers: 1)
Composites Part A : Applied Science and Manufacturing     Hybrid Journal   (Followers: 26)
Composites Part B : Engineering     Hybrid Journal   (Followers: 24)
Computational Materials Science     Hybrid Journal   (Followers: 15)
Computational Mathematics and Mathematical Physics     Hybrid Journal   (Followers: 1)
Computational Particle Mechanics     Hybrid Journal  
Computational Science and Discovery     Full-text available via subscription  
Computer Physics Communications     Hybrid Journal  
Contemporary Concepts of Condensed Matter Science     Full-text available via subscription  
Contemporary Physics     Hybrid Journal   (Followers: 9)
Continuum Mechanics and Thermodynamics     Hybrid Journal   (Followers: 3)
Contributions to Plasma Physics     Hybrid Journal   (Followers: 2)
COSPAR Colloquia Series     Full-text available via subscription  
Cryogenics     Hybrid Journal   (Followers: 11)
Current Applied Physics     Full-text available via subscription   (Followers: 4)
Diamond and Related Materials     Hybrid Journal   (Followers: 12)
Differential Equations and Nonlinear Mechanics     Open Access   (Followers: 4)
Doklady Physics     Hybrid Journal   (Followers: 1)
Dynamical Properties of Solids     Full-text available via subscription  

        1 2 3 4 5 6 | Last

Journal Cover Archive for Rational Mechanics and Analysis
   [5 followers]  Follow    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
     ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527
     Published by Springer-Verlag Homepage  [2209 journals]   [SJR: 3.43]   [H-I: 51]
  • Front Quenching in the G-equation Model Induced by Straining of Cellular
           Flow
    • Abstract: Abstract We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton–Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed s T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as s T ≧ O(A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and s T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), s T is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.
      PubDate: 2014-10-01
       
  • Metastability and Dynamics of Discrete Topological Singularities in Two
           Dimensions: A Γ-Convergence Approach
    • Abstract: Abstract This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.
      PubDate: 2014-10-01
       
  • A Polyconvex Integrand; Euler–Lagrange Equations and Uniqueness of
           Equilibrium
    • Abstract: Abstract In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy \({{\rm lim}_{\det \xi \rightarrow 0^+} W(\xi)=\infty.}\) We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form \({E(\mathbf{u})=\int_\Omega (W(\nabla \mathbf{u}) -\mathbf{F} \cdot \mathbf{u}) {\rm d}x}\) , as well as their Euler–Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form \({W(\xi)=f(\xi) +h( {\rm det} \xi)}\) – such that \({{\rm lim}_{t \rightarrow 0^+} h(t)=\infty}\) – which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler–Lagrange equations.
      PubDate: 2014-10-01
       
  • Decaying Turbulence in the Generalised Burgers Equation
    • Abstract: Abstract We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0, \,\, t \geqq 0, \,\, x \in S^1,$$ where f is strongly convex and ν is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for the dissipation length scale and the small-scale quantities which characterise the decaying Burgers turbulence, i.e., the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell et al. (J Fluid Mech 238:467–486, 1992). Note that we are dealing with decaying, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval [T 1, T 2], where T 1 and T 2 depend only on f and the initial condition, and do not depend on the viscosity. These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov’s 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form k −2. These results remain valid in the inviscid limit.
      PubDate: 2014-10-01
       
  • Local and Global Well-Posedness of Strong Solutions to the 3D Primitive
           Equations with Vertical Eddy Diffusivity
    • Abstract: Abstract In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with H 2 initial data.
      PubDate: 2014-10-01
       
  • Quasilinear and Hessian Type Equations with Exponential Reaction and
           Measure Data
    • Abstract: Abstract We prove existence results concerning equations of the type \({-\Delta_pu=P(u)+\mu}\) for p > 1 and F k [−u] = P(u) + μ with \({1 \leqq k < \frac{N}{2}}\) in a bounded domain Ω or the whole \({\mathbb{R}^N}\) , where μ is a positive Radon measure and \({P(u)\sim e^{au^\beta}}\) with a > 0 and \({\beta \geqq 1}\) . Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form \({u={\bf W}_{\alpha, p}^R[P(u)]+f}\) in \({\mathbb{R}^N}\) , where \({0 < R \leqq \infty}\) and f is a positive integrable function.
      PubDate: 2014-10-01
       
  • The Fokker–Planck Equation with Absorbing Boundary Conditions
    • Abstract: Abstract We study the initial-boundary value problem for the Fokker–Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker–Planck equation with absorbing boundary conditions, as well as the Hölder continuity of the solutions up to the singular set.
      PubDate: 2014-10-01
       
  • Variational Proof of the Existence of the Super-Eight Orbit in the
           Four-Body Problem
    • Abstract: Abstract Using the variational method, Chenciner and Montgomery (Ann Math 152:881–901, 2000) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver numerically found a similar periodic solution called “super-eight” in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the scaling technique established by Tanaka (Ann Inst H Poincaré Anal Non Linéaire 10:215–238, 1993), (Proc Am Math Soc 122:275–284, 1994) and investigate the asymptotic behavior of a binary collision.
      PubDate: 2014-10-01
       
  • Intrinsic Geometry and Analysis of Diffusion Processes and        class="a-plus-plus">L        
    • Abstract: Abstract The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L ∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C 1. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.
      PubDate: 2014-10-01
       
  • Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional
           Compressible Euler Equations with or without Self-Gravitation
    • Abstract: Abstract This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.
      PubDate: 2014-09-01
       
  • Optimal Continuous Dependence Estimates for Fractional Degenerate
           Parabolic Equations
    • Abstract: Abstract We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, \({\triangle^{\alpha/2}}\) for \({\alpha \in (0,2)}\) . Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on α in the BV-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits \({\alpha \downarrow 0}\) and \({\alpha \uparrow 2}\) . In the limit \({\alpha \uparrow 2}\) , \({\triangle^{\alpha/2}}\) converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231–251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).
      PubDate: 2014-09-01
       
  • Existence and Concentration Result for the Kirchhoff Type Equations with
           General Nonlinearities
    • Abstract: Abstract In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations $$- \varepsilon^2 M \left(\varepsilon^{2-N}\!\!\int_{\mathbf{R}^N} \nabla u ^2\,\mathrm{d} x \right) \Delta u \!+\! V(x) u \!=\! f(u) \quad{\rm in}\ \mathbf{R}^N, \quad u \!\in\! H^1(\mathbf{R}^N), \ N \!\geqq\!1,$$ where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions \({(u_\varepsilon)_{\varepsilon \in (0,\tilde{\varepsilon}]}}\) which concentrates at a local minimum of V after extracting a subsequence (ε k ).
      PubDate: 2014-09-01
       
  • Existence and Regularity of the Reflector Surfaces in        class="a-plus-plus inline-equation id-i-eq1">        class="a-plus-plus equation-source
           format-t-e-x">\({\mathbb{R}^{n+1}}\)
    • Abstract: Abstract In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain \({\mathcal{D}}\) in the ambient space such that the weak solution is smooth if and only if its graph lies in \({\mathcal{D}}\) .
      PubDate: 2014-09-01
       
  • Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body
           Problem via Index Theory
    • Abstract: Abstract It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter \({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\) and the eccentricity \({e \in [0, 1)}\) . We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed \({e\in [0, 1)}\) , we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are −1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail.
      PubDate: 2014-09-01
       
  • Busemann Functions for the N-Body Problem
    • Abstract: Abstract Following ideas in Maderna and Venturelli (Arch Ration Mech Anal 194:283–313, 2009), we prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton–Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion.
      PubDate: 2014-09-01
       
  • Existence and Stability of a Screw Dislocation under Anti-Plane
           Deformation
    • Abstract: Abstract We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model.
      PubDate: 2014-09-01
       
  • Hydrodynamic Limit for a Hamiltonian System with Boundary Conditions and
           Conservative Noise
    • Abstract: Abstract We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is attached to a point on the left and there is a force (tension) τ acting on the right. In order to provide good ergodic properties to the system, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distributions of the elongation, momentum and energy converge to the solution of the Euler system of equations in the smooth regime.
      PubDate: 2014-08-01
       
  • Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating
           Obstacle
    • Abstract: Abstract We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L 3 spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L 3 spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.
      PubDate: 2014-08-01
       
  • Higher Integrability for Minimizers of the Mumford–Shah Functional
    • Abstract: Abstract We prove higher integrability for the gradient of local minimizers of the Mumford–Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55–62, 1991).
      PubDate: 2014-08-01
       
  • A Widder’s Type Theorem for the Heat Equation with Nonlocal
           Diffusion
    • Abstract: Abstract The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi- \xi ^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.
      PubDate: 2014-08-01
       
 
 
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