“…Second, a distinct type of phasespace oriented, topological computer-assisted approach (see [19,20,21,22] and the references therein), can be applied to prove the existence of the connections. Such methods have been used successfully to find connecting orbits in a variety of nonlinear systems [23,24,25,26,27], and they are especially adept in low regularity settings. In the present paper, on the other hand, by combining a functional analytic setting and a parameterization method, we exploit the high regularity of solutions in the (parabolic) pattern formation problem.…”