On moving spike type internal layers in nonlinear singularly perturbed problems (Q1269556)

The author considers \(T\)-periodic solutions of the scalar singularly perturbed differential problem: \[ \varepsilon^2\Biggl({\partial^2u\over\partial x^2}- {\partial u\over\partial t}\Biggr)= f(u, x,t,\varepsilon),\;0<x<1,\;-\infty< t<+\infty, \] \[ u(0,t,\varepsilon)= g_0(t),\;u(1,t,\varepsilon)= g_1(t), \] where \(f\), \(g_0\), \(g_1\) are smooth, \(T\)-periodic functions of \(t\). He is interested in the solution \(u(x,t,\varepsilon)\), which is close to some solution of \(f(u,x,t,0)= 0\) in \(Q_T\) except in a small neighborhood of some \(T\)-periodic smooth curves \(x= h(t,\varepsilon)= \sum^\infty_{i= 0}\varepsilon^i h_i(t)\), in which a spike solution originates. He sets conditions under which he can determine all functions \(h_i\) and construct an asymptotic expansion of any order for \(u\).