1,721,009 research outputs found
The fractional Laplacian in power-weighted Lp spaces: Integration-by-parts formulas and self-adjointness
No full textWe consider the fractional Laplacian operator (−Δ)s (let s∈(0,1)) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the L2(Rd) scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space H ̇s(Rd). More precisely, we focus on functions belonging to some weighted L2 space whose fractional Laplacian belongs to another weighted L2 space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight ρ(x) at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by ρ−1(−Δ)s. The generality of the techniques developed allows us to deal with weighted Lp spaces as well
Some recent advances in nonlinear diffusion on negatively-curved Riemannian manifolds: from barriers to smoothing effects
Full text linkIn this survey paper we discuss a series of recent results concerning nonnegative solutions to nonlinear diffusion equations of porous-medium type on Cartan–Hadamard manifolds, a special class of negatively-curved Riemannian manifolds that generalize the Euclidean space. We focus on sharp barrier estimates, asymptotic convergence and smoothing effects, describing quantitatively how the curvature behavior at infinity affects the way solutions depart from having a Euclidean-like structure
Sharp asymptotics for the porous media equation in low dimensions via Gagliardo-Nirenberg inequalities
No full textWe prove sharp asymptotic bounds for solutions to the porous media equation with homogeneous Dirichlet or Neumann boundary conditions on a bounded Euclidean domain, in dimension and . This is achieved by making use of appropriate Gagliardo-Nirenberg inequalities only. The generality of the discussion allows to prove similar bounds for \emph{weighted} porous media equations, provided one deals with weights for which suitable Gagliardo-Nirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions
Moduli of Continuity and Absolute Continuity: Any Relation?
No full textWe construct a monotone, continuous, but not absolutely continuous function whose minimal modulus of continuity is absolutely continuous. In particular, we establish that there is no equivalence between the absolute continuity of a function and the absolute continuity of its modulus of continuity, in contrast with a well-known property of Lipschitz functions
Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds
Full text linkWe prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan–Hadamard manifold
whose sectional curvatures are bounded above by a strictly negative constant. This
bound includes as a special case the sharp smoothing effect recently proved by
V ́azquez on the hyperbolic space in V ́azquez (2015), which is similar to the absolute
bound valid in the case of bounded Euclidean domains but has a logarithmic
correction. Secondly we prove a bound which interpolates between such smoothing
effect and the Euclidean one, supposing a suitable quantitative Sobolev inequality
holds, showing that it is sharp by means of explicit examples. Finally, assuming a
stronger functional inequality of sub-Poincar ́e type, we prove that the above mentioned
(sharp) absolute bound holds, and provide examples of weighted porous media
equations on manifolds of infinite volume in which it holds, in contrast with the nonweighted
Euclidean situation. It is also shown that sub-Poincar ́e inequalities cannot
hold on Cartan–Hadamard manifolds
Radial fast diffusion on the hyperbolic space
Full text linkWe consider positive radial solutions to the fast diffusion equationon the hyperbolic
space. By radial, we mean solutions depending only on the geodesic distance r from a given point o ∈ H^N. We investigate their fine asymptotics near the extinction time T in terms of a separable solution defined in terms of the unique positive energy solution, radial with respect to
o, to a semilinear elliptic problem thoroughly studied in [G.
Mancini and K. Sandeep, ‘On a semilinear elliptic equation in Hn’, Ann. Sc. Norm. Super. Pisa
Cl. Sci. 7 (2008) 635–671; M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, ‘Classification
of radial solutions to the Emden–Fowler equation on the hyperbolic space’, Calc. Var. Partial
Differential Equations 46 (2013) 375–401]. We show that u converges to V in relative error. Solutions
are smooth, and bounds on derivatives are given as well. In particular, sharp convergence results
as t → T are shown for spatial derivatives, again in the form of convergence in relative error
Fast diffusion on noncompact manifolds: Well-posedness theory and connections with semilinear elliptic equations
Full text linkWe investigate the well-posedness of the fast diffusion equation (FDE) on noncompact Riemannian manifolds. Existence and uniqueness of solutions for integrable initial data was established in Bonforte, Grillo, and Vazquez [J. Evol. Equ. 8 (2008), pp. 99–128]. However, in the Euclidean space, it is known from Herrero and Pierre [Trans. Amer. Math. Soc. 291 (1985), pp. 145–158], that the Cauchy problem associated with the FDE is well posed for initial data that are merely locally integrable. We establish here that such data still give rise to
global solutions on general manifolds. If, moreover, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to minus infinity at
spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, assuming in addition that the initial
datum is locally square integrable and nonnegative, a minimal solution is shown to exist, and we establish uniqueness of purely (nonnegative) distributional solutions, a fact
that to our knowledge was not known before even in the Euclidean space. The required curvature bound is sharp, since on model manifolds it is equivalent to stochastic completeness, and it was shown in Grillo, Ishige, and Muratori [J. Math. Pures Appl. (9) 139 (2020), pp. 63–82] that uniqueness for the FDE fails even in the class of bounded solutions when stochastic completeness does not hold. A crucial ingredient of the uniqueness result is the proof of nonexistence of nonnegative, nontrivial distributional subsolutions to certain
semilinear elliptic equations with power nonlinearities, of independent interest
Blow-up and global existence for the porous medium equation with reaction on a class of Cartan–Hadamard manifolds
Full text linkWe consider the porous medium equation with power-type reaction terms up on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If p>m, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If p<m, large data blow up at worst in infinite time, and under the stronger restriction p∈(1,(1+m)/2] all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation
Uniqueness of very weak solutions for a fractional filtration equation
Full text linkWe prove existence and uniqueness of distributional, bounded solutions to a fractional filtration equation in . With regards to uniqueness, it was shown even for more general equations in cite{TJJ} that if two bounded solutions of eqref{eq6} satisfy , then . We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions. For nonnegative initial data, we first show that a minimal solution exists and then that any other solution must coincide with it. A similar procedure is carried out for sign-changing solutions. As a consequence, distributional solutions have locally-finite energy
Porous medium equations on manifolds with critical negative curvature: unbounded initial data
No full textWe investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of CartanâHadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity
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