4,374 research outputs found
Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction
We perform a thorough study of the blow up profiles associated to the
following second order reaction-diffusion equation with non-homogeneous
reaction: in the range
of exponents . We classify blow up solutions in
self-similar form, that are likely to represent typical blow up patterns for
general solutions. We thus show that the non-homogeneous coefficient
has a strong influence on the qualitative aspects related to the
finite time blow up. More precisely, for , blow up profiles have
similar behavior to the well-established profiles for the homogeneous case
, and typically \emph{global blow up} occurs, while for
sufficiently large, there exist blow up profiles for which blow up \emph{occurs
only at space infinity}, in strong contrast with the homogeneous case. This
work is a part of a larger program of understanding the influence of unbounded
weights on the blow up behavior for reaction-diffusion equations
Aggregation-diffusion equations: dynamics, asymptotics, and singular limits
Given a large ensemble of interacting particles, driven by nonlocal
interactions and localized repulsion, the mean-field limit leads to a class of
nonlocal, nonlinear partial differential equations known as
aggregation-diffusion equations. Over the past fifteen years,
aggregation-diffusion equations have become widespread in biological
applications and have also attracted significant mathematical interest, due to
their competing forces at different length scales. These competing forces lead
to rich dynamics, including symmetrization, stabilization, and metastability,
as well as sharp dichotomies separating well-posedness from finite time blowup.
In the present work, we review known analytical results for
aggregation-diffusion equations and consider singular limits of these
equations, including the slow diffusion limit, which leads to the constrained
aggregation equation, as well as localized aggregation and vanishing diffusion
limits, which lead to metastability behavior. We also review the range of
numerical methods available for simulating solutions, with special attention
devoted to recent advances in deterministic particle methods. We close by
applying such a method -- the blob method for diffusion -- to showcase key
properties of the dynamics of aggregation-diffusion equations and related
singular limits
A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
We propose a positivity preserving entropy decreasing finite volume scheme
for nonlinear nonlocal equations with a gradient flow structure. These
properties allow for accurate computations of stationary states and long-time
asymptotics demonstrated by suitably chosen test cases in which these features
of the scheme are essential. The proposed scheme is able to cope with
non-smooth stationary states, different time scales including metastability, as
well as concentrations and self-similar behavior induced by singular nonlocal
kernels. We use the scheme to explore properties of these equations beyond
their present theoretical knowledge
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