A geometric approach to pinned pulses in a class of non-autonomous reaction–diffusion equations

Abstract This paper develops a geometric and analytical framework for studying the existence and stability of pinned pulse solutions in a class of non-autonomous reaction–diffusion equations. The analysis relies on geometric singular perturbation theory, matched asymptotic method and nonlocal eigenvalue problem method. First, we derive the general criteria on the existence and spectral (in)stability of pinned pulses in slowly varying heterogeneous media. Then, as a specific example, we apply our theory to a heterogeneous Gierer–Meinhardt (GM) equation, where the nonlinearity varies slowly in space. We identify the conditions on parameters under which the pulse solutions are spectrally stable or unstable. It is found that when the heterogeneity vanishes, the results for the heterogeneous GM system reduce directly to the known results on the homogeneous GM system. This demonstrates the validity of our approach and highlights how the spatial heterogeneity gives rise to richer pulse dynamics compared to the homogeneous case.