Stability for stationary solutions of a nonlocal Allen-Cahn equation (Q828308)

The authors investigate the dynamics of a nonlocal Allen-Cahn equation with Neumann boundary conditions on an interval of the form \[ \begin{cases} -du''=(1-u^2)(u-\lambda), \quad x\in I:=(-1,1), \\ u'(-1)=0,\quad u'(1)=0, \\ u'(x)\geq 0,\quad x\in I \end{cases} \] with the nonlocal constraint \[ \lambda=\frac{\mu}{2}\int_{-1}^{1}u(x)\,\mathrm{d}x. \] At first, the authors use the exact representation of symmetric solutions and show the distribution of eigenvalues of the linearized eigenvalue problem around these solutions. Next, the authors derive the stability/instability of all symmetric solutions and instability of a part of asymmetric solutions. Here, the proof of the instability of the asymmetric solution is mainly based on the SLEP method. Finally, the results with respect to stability are supported by some numerical simulations.