1,720,966 research outputs found
Long-time behavior for the porous medium equation with small initial energy
No full textWe study the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We point out that the initial datum is allowed to be sign-changing. We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one
Comparison and regularity results for the fractional Laplacian via symmetrization methods
No full textAnisotropic p-Laplacian Evolution of Fast Diffusion Type
Full text linkWe study an anisotropic, possibly non-homogeneous version of the evolution p-Laplacian equation
when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results
from which we derive sharp L1-L∞ estimates. We prove the existence of a self-similar fundamental solution
of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the
asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as
well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not
common in the related literature. It is however essential in our analysis and creates mathematical difficulties
that are solved for fast diffusions
Anisotropic fast diffusion equations
No full textWe prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite-mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods
SYMMETRIZATION FOR FRACTIONAL NONLINEAR ELLIPTIC PROBLEMS
No full textIn this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result
The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity
No full textFor 0 < s < 1, we consider the Dirichlet problem for the fractional nonlocal Ornstein– Uhlenbeck equation ((−∆ + x · ∇)^s u = f in Ω, u = 0 on ∂Ω, where Ω is a possibly unbounded open subset of Rn, n ≥ 2. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel L^p and L^p(log L)^α regularity estimates in terms of the datum f are obtained by comparing u with half-space solutions
Long-time asymptotics for nonlocal porous medium equation with absorption or convection
Full text linkIn this paper, the long-time asymptotic behaviours of nonlocal porous medium
equations with absorption or convection are studied. In the parameter regimes
when the nonlocal diffusion is dominant, the entropy method is adapted in this
context to derive the exponential convergence of relative entropy of solutions
in similarity variables
Symmetrization results for general nonlocal linear ellipitic and parabolic problems
Full text linkWe establish a Talenti-type symmetrization result in the form of mass concentration (i.e. integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions. In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing the previous results in [21]. Finally, using an implicit time discretization techniques, similar results are obtained for the solutions of Cauchy-Dirichlet nonlocal linear parabolic problems
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
- …
