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

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