Pinning and unpinning in nonlocal systems (Q330531)

The article deals with the nonlocal evolution equation \[ u_t = d(-u + K * u) + f_a(u), \qquad x \in {\mathbb R}, \] where \(u = u(t,x)\) is an unknown, \(K\) a kernel, \(f_a(u)\) is a bistable nonlinearity in \(u\) with parameter \(a\) detuning the energy level of stable equilibria, \(d> 0\). Really, the authors consider some special examples. In the case \(f_1(u) = u(1 - u)(u - a)\) it is proved that the pinning region is described by an interval \(a \in (a_-(d),a_+(d))\) where \[ a_\pm(d) \sim a_1 + a_2(d - d_*)^2, \] provided that \(0< d< d_*\) (\(d^*\), \(a_1\), \(a_2\) are some constants). In the cases \(K(x) = \dfrac12 e^{-|x|}\) and \(K(x) = e^{-x} \chi_{[0,\infty)}(x)\) it is proved that the unpinning asymptotic is \[ c = k_1|a - a_\pm|^\frac32 (1 + o(1)) \] with some constant \(k_1\); in the limit case \(d = d_*\) the unpinning asymptotic is different: \[ c = k_1|a - a_\pm|^\frac54 (1 + o(1)). \] The authors also present some comments for other nonlinearities and kernels. They present some numerical illustrations.