| Abstract: |
| I will discuss the generation of interface for solutions of the mass conserved Allen-Cahn equation of the form
\[
u_t=\Delta u +\varepsilon^{-2} \left( f(u) - \big\langle f (u) \big\rangle \right)
\]
on a bounded domain $\Omega$ in ${\mathbb R}^N$, where$f$ is an Allen--Cahn type nonlinearity such as $u-u^3$, while $\varepsilon$ is a small parameter and $\langle\,\cdot\,\rangle$ denotes the avarage over $\Omega$. The main goal is to show that, given a virtually arbitrary initial function $u(x,0)$, the solution generally develops a steep transition layer of thickness $O(\varepsilon^\gamma)$ for some $0< \gamma \leq 1$ at a very early intial stage of $t=O(\varepsilon^2 |\ln \varepsilon|)$. The location of this interface is in an $O(\varepsilon)$ neighborhood of the set
\[
\Gamma_0=\{x\in \Omega\mid f(u_0(x))=\big\langle f (u_0) \big\rangle,\,f'(u_0(x))>0\}.
\]
No interface develops if this set is empty. |
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