## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

DCDS

We study heteroclinic connections in a nonlinear heat equation that
involves blow-up. More precisely we discuss the existence of $L^1$ connections among
equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium
$\phi^+$ we mean a function $u$($.,t$)
which is a classical solution on the interval
$(-\infty,T)$
for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a
certain weak sense for $t\in [T,\infty)$
and satisfies $u$($.,t$)$\to \phi^\pm$
as $t\to\pm\infty$ in a suitable sense.
The main tool in our analysis is the zero number argument;
namely to count the number of intersections between the graph of a given solution and that of various
specific solutions.

keywords:
zero number.
,
connecting orbits
,
blowup
,
Semilinear parabolic equation
,
nonlinear heat equation

CPAA

We study the asymptotic behaviour near extinction of positive
solutions of the Cauchy problem for the fast diusion equation with a critical
exponent. We improve a previous result on slow convergence to Barenblatt
proles.

NHM

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy
problem for a semilinear heat equation with Sobolev critical nonlinearity.
The conjectures say that, depending on the decay rate of initial data and
the space dimension, the
threshold solutions may grow up, stabilize, or decay to zero as $t→∞$.
The rates of grow up or decay are computed formally using matched asymptotics.

DCDS-S

We find a continuum of extinction rates of solutions of the Cauchy problem for
the fast diffusion equation
$u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension.
The extinction rates
depend explicitly on the
spatial decay rates of initial data and contain a logarithmic term.

CPAA

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of
the fast diffusion equation
$u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is
posed in $R^n$ for times up to
the extinction time $T>0$. The rates take the form
$\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a whole
interval of $\theta>0$. These extinction rates depend explicitly on the
spatial decay rates of initial data.

CPAA

We study the large time behavior of positive solutions for
the Laplace equation on the half-space with a nonlinear dynamical boundary condition.
We show the convergence to the Poisson kernel in a suitable sense provided initial data
are sufficiently small.

DCDS

This paper examines the following question: Suppose that we have a
reaction-diffusion equation or system such that some solutions
which are homogeneous in space blow up in finite time. Is it
possible to inhibit the occurrence of blow-up as a consequence of
imposing Dirichlet boundary conditions, or other effects where
diffusion plays a role? We give examples of equations and systems
where the answer is affirmative.

DCDS

We study the behavior of solutions of the Cauchy problem for
a parabolic equation with power nonlinearity. Our concern is
the rate of convergence of solutions to forward self-similar solutions.
We determine the exact rate of convergence which turns out to
depend on the spatial decay rate of initial data.

## Year of publication

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