Stability and quenching-profile conjectures for self-similar solutions of the MEMS wave equation

Let $u=u(r,t)$ be a radial solution of the semi-linear wave equation in spatial dimension $n$, let $T_{\max}$ be its quenching time, and introduce the transformed variables

$$\tau=-\ln(T_{\max}-t),\qquad \xi=\frac{r}{(T_{\max}-t)^{1/2}}.$$

Writing $u(r,t)=(T_{\max}-t)^{2/3}v(\xi,\tau)$, the self-similar profiles $v_j$ are equilibria of the corresponding similarity PDE, and $c^*$ is the constant profile $v_0=c^*$. A perturbation is spatially uniform when it depends only on $\tau$.

Stability and quenching-profile conjecture. The following statements are conjectured:

1. The trivial self-similar solution $v_0=c^*$ is stable as a solution of the similarity PDE, except for spatially uniform perturbations that grow as $\exp(\tau)$ and correspond to solutions quenching at different times.
2. The generic local behaviour of a strictly increasing radial quenching solution near $T_{\max}$ is

$$u(r,t)=(T_{\max}-t)^{2/3}V(\xi,t),\qquad \xi=\frac{r}{(T_{\max}-t)^{1/2}},$$

with

$$V(\xi,t)\sim C\xi^{4/3}\quad\text{as }\xi\to\infty,\qquad V(0,t)\to c^*$$

for $t\to T_{\max}$. Consequently,

$$u(r,T_{\max})\sim Cr^{4/3}\quad\text{as }r\to0,\qquad u(0,t)\sim c^*(T_{\max}-t)^{2/3}\quad\text{as }t\to T_{\max},$$

provided $u$ is strictly increasing in $r$ near the quenching time. Here $C$ depends on $n$ and on the initial and boundary conditions. 3. For $2\le n\le7$, the smooth self-similar solutions $v_j$ are unstable as solutions of the similarity PDE for every $j\in\mathbb N$.

These claims describe the expected stability mechanism and generic local profile of quenching, and relate the distinguished constant profile to the nontrivial self-similar solutions. The evidence presented is formal and numerical; the conjectures remain unproved in the source.