1,235 research outputs found
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity
The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017]
classified the behaviour near zero for all positive solutions of the perturbed
elliptic equation with a critical Hardy--Sobolev growth
where denotes the open unit ball centred at in for
, , , and . For
with , it was shown in the op. cit. that
the positive solutions with a non-removable singularity at could exhibit up
to three different singular profiles, although their existence was left open.
In the present paper, we settle this question for all three singular profiles
in the maximal possible range. As an important novelty for , we prove
that for every there exist infinitely many
positive solutions satisfying as , using a dynamical system approach.
Moreover, we show that there exists a positive singular solution with
and
if (and only if) .Comment: Mathematische Annalen, to appea
On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat
equation with inverse singular potential, is proved for 2mth-order linear
parabolic equations with Hardy-supercritical singular potentials. Extensions to
other linear and nonlinear singular PDEs are discussed.Comment: 22 page
Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces
This paper is motivated by the characterization of the optimal symmetry
breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence,
optimal functions and sharp constants are computed in the symmetry region. The
result solves a longstanding conjecture on the optimal symmetry range.
As a byproduct of our method we obtain sharp estimates for the principal
eigenvalue of Schr\"odinger operators on some non-flat non-compact manifolds,
which to the best of our knowledge are new.
The method relies on generalized entropy functionals for nonlinear diffusion
equations. It opens a new area of research for approaches related to carr\'e du
champ methods on non-compact manifolds. However key estimates depend as much on
curvature properties as on purely nonlinear effects. The method is well adapted
to functional inequalities involving simple weights and also applies to general
cylinders. Beyond results on symmetry and symmetry breaking, and on optimal
constants in functional inequalities, rigidity theorems for nonlinear elliptic
equations can be deduced in rather general settings.Comment: 33 pages, 1 figur
Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M
This paper concludes the series begun in [M. Dafermos and I. Rodnianski,
Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the
cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof
of definitive boundedness and decay results for the scalar wave equation on
Kerr backgrounds in the general subextremal |a| < M case without symmetry
assumptions. The essential ideas of the proof (together with explicit
constructions of the most difficult multiplier currents) have been announced in
our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for
linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann
Meeting on General Relativity, T. Damour et al (ed.), World Scientific,
Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the
quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative
Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to
appear, Ann. Henri Poincare], together with a streamlined continuity argument
in the parameter a, appearing here for the first time. While serving as Part
III of a series, this paper repeats all necessary notations so that it can be
read independently of previous work.Comment: 84 pages, 2 figures, v2: ode estimates strengthened so as to admit
scattering theory application
Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation
inequalities (CKN), with two radial power law weights and exponents in a
subcritical range. We address the question of symmetry breaking: are the
optimal functions radially symmetric, or not ? Our intuition comes from a
weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy
- entropy production inequality which governs the intermediate asymptotics is
indeed equivalent to (CKN), and the self-similar profiles are optimal for
(CKN). We establish an explicit symmetry breaking condition by proving the
linear instability of the radial optimal functions for (CKN). Symmetry breaking
in (CKN) also has consequences on entropy - entropy production inequalities and
on the intermediate asymptotics for (WFD). Even when no symmetry holds in
(CKN), asymptotic rates of convergence of the solutions to (WFD) are determined
by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a
linearized entropy - entropy production inequality. All our results rely on the
study of the bottom of the spectrum of the linearized diffusion operator around
the self-similar profiles, which is equivalent to the linearization of (CKN)
around the radial optimal functions, and on variational methods. Consequences
for the (WFD) flow will be studied in Part II of this work
Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3
This paper is devoted to a collection of results on nonlinear interpolation
inequalities associated with Schr\"odinger operators involving Aharonov-Bohm
magnetic potentials, and to some consequences. As symmetry plays an important
role for establishing optimality results, we shall consider various cases
corresponding to a circle, a two-dimensional sphere or a two-dimensional torus,
and also the Euclidean spaces of dimensions two and three. Most of the results
are new and we put the emphasis on the methods, as very little is known on
symmetry, rigidity and optimality in presence of a magnetic field. The most
spectacular applications are new magnetic Hardy inequalities in dimensions 2
and 3
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