Some remarks on stability for a phase field model with memory

(PFM) $ u_t + \frac{l}{2} \phi_t =\int_{0}^t a_1(t-s) $Δ$ u(s) ds$,
$\tau \phi_t = \int_{0}^t a_2(t-s)[\xi^2 $Δ$ \phi + \frac{1}{\eta}(\phi - \phi^3) + u](s) ds$,

for $(x, t) \in \Omega \times (0, T)$, $0 < T < \infty$, with the boundary conditions

n $\cdot \nabla u$= n $\cdot \nabla \phi=0, (x, t) \in \partial\Omega \times (0, T)$,

and initial conditions $u(x, 0)=u_0(x)$, $\phi(x, 0)=\phi_0(x)$, $x \in \Omega$, which was proposed in [36] to model phase transitions taking place in the presence of memory effects which arise as a result of slowly relaxing internal degrees of freedom, although in [36] the effects of past history were also included. This system has been shown to exhibit some intriguing effects such as grains which appear to rotate as they shrink [36]. Here the set of steady states of (PFM) and of an associated classical phase field model are shown to be the same. Moreover, under the assumption that $a_1$ and $a_2$ are both proportional to a kernel of positive type, the index of instability and the number of unstable modes for any given stationary state of the two systems can be compared and spectral instability is seen to imply instability. By suitably restricting further the memory kernels, the (weak) $\omega-$limit set of any initial condition can be shown to contain only steady states and linear stability can be shown to imply nonlinear stability.