  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.
