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| Propagating terrace for semilinear diffusion equations in higher dimensions
In this talk we will consider semilinear diffusion equations of the form
\[
u_t = \Delta u + f(u) \ \ \ \hbox{on}\ \; {\bf R}^N,
\]
where $f$ is a multi-stable nonlinearity satisfying $f(0)=0$ and discuss the long-time behavior of solutions with compactly supported non-negative intial data. For this type of equations, under some mild non-degeneracy assumptions, it was proved recently by P. Polacik and Y. Du that the solution converges to a stationary solution as $t\to\infty$. What we are interested in is the transition process, that is, the behavior of the solutions in the intermediate time range. More precisely, we will show that every such solution behaves asymptotically like what we call a ``radially symmetric propagating terrace", which roughly means a multiple layer of radially symmetric spreading fronts that expand at different speeds. |
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