PDE Seminar Abstracts

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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