**SMS scnews item created by Daniel Daners at Fri 13 May 2016 2038**

Type: Seminar

Distribution: World

Expiry: 16 May 2016

**Calendar1: 16 May 2016 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@d58-110-201-128.mas800.nsw.optusnet.com.au (ddan2237) in SMS-WASM

### PDE Seminar

# Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions

### Maolin Zhou

Maolin Zhou

University of New England, Armidale

16th May 2016 14:00-15:00, Carslaw Room 829 (AGR)

## Abstract

We study the long time behavior, as
$t\to \infty $, of
solutions of

$$\begin{array}{llllllll}\hfill {u}_{t}& ={u}_{xx}+f\left(u\right),\phantom{\rule{2em}{0ex}}& \hfill & x>0,\phantom{\rule{1em}{0ex}}t>0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill u\left(0,t\right)& =b{u}_{x}\left(0,t\right),\phantom{\rule{2em}{0ex}}& \hfill & t>0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill u\left(x,0\right)& ={u}_{0}\left(x\right)\ge 0,\phantom{\rule{2em}{0ex}}& \hfill & x\ge 0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
where $b\ge 0$ and
$f$ is an
unbalanced bistable nonlinearity. By investigating families of initial data of the type
${\left\{\sigma \varphi \right\}}_{\sigma >0}$, where
$\varphi $ belongs
to an appropriate class of nonnegative compactly supported functions, we exhibit the
sharp threshold between vanishing and spreading. More specifically, there exists some
value ${\sigma}^{*}$
such that the solution converges uniformly to 0 for any
$0<\sigma <{\sigma}^{*}$,
and locally uniformly to a positive stationary state for any
$\sigma >{\sigma}^{*}$. In the
threshold case $\sigma ={\sigma}^{*}$,
the profile of the solution approaches the symmetrically decreasing ground state with
some shift, which may be either finite or infinite. In the latter case, the shift evolves
as $Clnt$ where
$C$ is a
positive constant we compute explicitly, so that the solution is traveling with a
pulse-like shape albeit with an asymptotically zero speed. Depending on
$b$, but
also in some cases on the choice of the initial datum, we prove that one or both of
the situations may happen.

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