Archivio della ricerca di Napoli "Parthenope"https://ricerca.uniparthenope.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Wed, 11 Dec 2019 16:06:59 GMT2019-12-11T16:06:59Z10171Ground States for Diffusion Dominated Free Energies with Logarithmic Interactionhttp://hdl.handle.net/11367/30194Titolo: Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction
Abstract: Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/11367/301942015-01-01T00:00:00ZSymmetrization for fractional elliptic
and parabolic equations and
an isoperimetric applicationhttp://hdl.handle.net/11367/47977Titolo: Symmetrization for fractional elliptic
and parabolic equations and
an isoperimetric application
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11367/479772017-01-01T00:00:00ZBourgain-Brézis-Mironescu formula for magnetic operatorshttp://hdl.handle.net/11367/53800Titolo: Bourgain-Brézis-Mironescu formula for magnetic operators
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11367/538002016-01-01T00:00:00ZFractional semilinear Neumann problems arising from a fractional
Keller--Segel modelhttp://hdl.handle.net/11367/30174Titolo: Fractional semilinear Neumann problems arising from a fractional
Keller--Segel model
Abstract: We consider the following fractional semilinear Neumann problem on a smooth
bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases}
(-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu
u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where
$\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the
semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The
problem arises by considering steady states of the Keller--Segel model with
nonlocal chemical concentration diffusion. Using the semigroup language for the
extension method and variational techniques, we prove existence of nonconstant
smooth solutions for small $\varepsilon$, which are obtained by minimizing a
suitable energy functional. In the case of large $\varepsilon$ we obtain
nonexistence of nonconstant solutions. It is also shown that as
$\varepsilon\to0$ the solutions $u_\varepsilon$ tend to zero in measure on
$\Omega$, while they form spikes in $\overline{\Omega}$. The regularity
estimates of the fractional Neumann Laplacian that we develop here are
essential for the analysis. The latter results are of independent interest.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/11367/301742015-01-01T00:00:00ZSymmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Typehttp://hdl.handle.net/11367/30042Titolo: Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type
Abstract: We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and its elliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u
+(-\Delta)^{\sigma/2}A(u)=f$, but only when the nondecreasing function$A:\re_+\to\re_+$ is concave. In the elliptic case, complete symmetrization results are proved for $\,B(v) +(-\Delta)^{\sigma/2}v=f$ \ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results hold when $B$ is concave. Remarkable counterexamples are constructed for the parabolic equation when $A$ is convex, resp. for the elliptic equation when $B$ is concave. Such counterexamples do not exist in the standard diffusion case $\sigma=2$.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11367/300422014-01-01T00:00:00ZOptimal estimates for Fractional Fast diffusion equationshttp://hdl.handle.net/11367/21544Titolo: Optimal estimates for Fractional Fast diffusion equations
Abstract: We obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation ut+(-δ)σ/2um=0, posed in the whole space with 0<σ<2, 0<m≤1. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the L1-norm and the Marcinkiewicz norm. The estimates contain exact exponents and best constants. We also obtain optimal estimates for the extinction time of the solutions in the range m near 0 where solutions may vanish completely in finite time. Actually, our results apply to equations with a more general nonlinearity. Our main tools are symmetrization techniques and comparison of concentrations. Classical results for σ=2 are recovered in the limit.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/11367/215442015-01-01T00:00:00ZSymmetrization for fractional Neumann problemshttp://hdl.handle.net/11367/53799Titolo: Symmetrization for fractional Neumann problems
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11367/537992016-01-01T00:00:00ZComparison and regularity results for the fractional Laplacian via symmetrization methodshttp://hdl.handle.net/11367/24264Titolo: Comparison and regularity results for the fractional Laplacian via symmetrization methods
Abstract: In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well-known ones for the classical Laplacian. (C) 2012 Elsevier Inc. All rights reserved.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11367/242642012-01-01T00:00:00ZSemilinear elliptic equations with degenerate and singular weights related to Caffarelli-Kohn-Nirenberg inequalitieshttp://hdl.handle.net/11367/17247Titolo: Semilinear elliptic equations with degenerate and singular weights related to Caffarelli-Kohn-Nirenberg inequalities
Abstract: In this note we give some existence and nonexistence results of solutions to a problem of the type -div(vertical bar x vertical bar(-2 gamma) del u) = lambda/vertical bar x vertical bar(2(gamma+1)) u + u(p)/vertical bar x vertical bar(alpha) + f/vertical bar x vertical bar(2 gamma) in Omega u >= 0, u not equivalent to 0 in Omega (P-t,P-p) u = 0 on partial derivative Omega, where Omega is an open bounded subset of R-N containing the origin, the constants p, t. alpha, gamma, lambda satisfy suitable conditions and f not equivalent to 0 is a nonnegative, smooth bounded function on Omega. The results that will be given generalize some known results in Brezis et al. (2005) [1] and Dupaigne (2002) [2]. (C) 2012 Elsevier Inc. All rights reserved.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11367/172472012-01-01T00:00:00ZThe fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularityhttp://hdl.handle.net/11367/65254Titolo: The fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularity
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11367/652542018-01-01T00:00:00Z