104 research outputs found
Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients
We investigate existence and uniqueness of solutions to a class of fractional
parabolic equations satisfying prescribed pointwise conditions at infinity (in
space), which can be time- dependent. Moreover, we study the asymptotic
behaviour of such solutions. We also consider solutions of elliptic equations
satisfying appropriate conditions at infinity
On the Cauchy problem for a general fractional porous medium equation with variable density
We study the well-posedness of the Cauchy problem for a fractional porous
medium equation with a varying density. We establish existence of weak energy
solutions; uniqueness and nonuniqueness is studied as well, according with the
behavior of the density at infinity
Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations
We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions
to a class of fractional parabolic and elliptic equations
Porous medium equations on manifolds with critical negative curvature: unbounded initial data
We investigate existence and uniqueness of solutions of the Cauchy problem
for the porous medium equation on a class of Cartan-Hadamard 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
Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds I, uniform estimates
We are concerned with solutions to the parabolic Allen-Cahn equation in
Riemannian manifolds. For a general class of initial condition we show non
positivity of the limiting energy discrepancy. This in turn allows to prove
almost monotonicity formula (a weak counterpart of Huisken's monotonicity
formula) which gives a local uniform control of the energy densities at small
scales.
Such results will be used in [40] to extend previous important results from
[31] in Euclidean space, showing convergence of solutions to the parabolic
Allen-Cahn equations to Brakke's motion by mean curvature in space forms
Uniqueness of very weak solutions for a fractional filtration equation
We prove existence and uniqueness of distributional, bounded, nonnegative
solutions to a fractional filtration equation in . With regards
to uniqueness, it was shown even for more general equations in [19] that if two
bounded solutions of (1.1) satisfy , then . We obtain here that this extra assumption can in
fact be removed and establish uniqueness in the class of merely bounded
solutions, provided they are nonnegative. Indeed, we show that a minimal
solution exists and that any other solution must coincide with it. As a
consequence, distributional solutions have locally-finite energy.Comment: Final version. To appear on Adv. Mat
Global Solutions of Semilinear Parabolic Equations with Drift Term on Riemannian Manifolds
We study existence and non-existence of global solutions to the semilinear
heat equation with a drift term and a power-like source term, on
Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional
curvatures, we show that global solutions cannot exists if the initial datum is
large enough. Furthermore, under appropriate conditions on the drift term,
global existence is obtained, if the initial datum is sufficiently small. We
also deal with Riemannian manifolds whose Ricci curvature tends to zero at
infinity sufficiently fast
Fractional porous media equations: existence and uniqueness of weak solutions with measure data
We prove existence and uniqueness of solutions to a class of porous media
equations driven by the fractional Laplacian when the initial data are positive
finite Radon measures on the Euclidean space. For given solutions without a
prescribed initial condition, the problem of existence and uniqueness of the
initial trace is also addressed. By the same methods we can also treat weighted
fractional porous media equations, with a weight that can be singular at the
origin, and must have a sufficiently slow decay at infinity (power-like). In
particular, we show that the Barenblatt-type solutions exist and are unique.
Such a result has a crucial role in [24], where the asymptotic behavior of
solutions is investigated. Our uniqueness result solves a problem left open,
even in the non-weighted case, in [42]Comment: Further results on initial traces added. Some proofs shortene
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