Vast multiplicity of very singular self-similar solutions of a semilinear higher-order diffusion equation with time-dependent absorption

Abstract

It is shown that a fourth-order semilinear parabolic equation with time-dependent absorption admit a vast multiplicity of the so-called very singular self-similar solutions (VSSs), which can bifurcate from some eigenfunctions of the linearized operators and also are generated by branching from other critical point. For the second-order semilinear heat equations, such VSSs have been known since the beginning of the 1980s and were constructed in papers by Brezis, Friedman, Galaktionov, Kyrdyumov, Samarskii, Peletier, Kamin, Veron, and many others.