109 research outputs found
Blowup in diffusion equations: A survey
AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems
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
Spikes and diffusion waves in one-dimensional model of chemotaxis
We consider the one-dimensional initial value problem for the viscous
transport equation with nonlocal velocity with a given kernel . We show the existence
of global-in-time nonnegative solutions and we study their large time
asymptotics. Depending on , we obtain either linear diffusion waves ({\it
i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion
waves (the fundamental solution of the viscous Burgers equation) in asymptotic
expansions of solutions as . Moreover, for certain aggregation
kernels, we show a concentration of solution on an initial time interval, which
resemble a phenomenon of the spike creation, typical in chemotaxis models
Local existence of solutions and comparison principle for initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory
We consider an initial value problem for a nonlinear parabolic equation with
memory under nonlinear nonlocal boundary condition. In this paper we study
classical solutions. We establish the existence of a local maximal solution. It
is shown that under some conditions a supersolution is not less than a
subsolution. We find conditions for the positiveness of solutions. As a
consequence of the positiveness of solutions and the comparison principle of
solutions, we prove the uniqueness theorem
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