Applications of three types are considered:
parabolic, hyperbolic, and nonlinear dispersion, is dealt with. General local, global, and
blow-up features of such PDEs are studied on the basis of their blow-up similarity or
travelling wave (for the last one) solutions.
In [11, 12], Lusternik–Schnirel’man category theory of variational calculus and fibering
methods were applied. The case m = 2 and n > 0 was studied in greater detail
analytically and numerically. Here, more attention is paid to a combination of a Cartesian
approximation and fibering to get new compactly supported similarity patterns.
Using numerics, such compactly supported solutions constructed for m = 3 and for
higher orders. The “smother” case of negative n < 0 is included, with a typical “fast
diffusion-absorption” parabolic PDE:
ut = (−1)m+1m(|u|nu) − |u|nu, where n 2 (−1, 0),
which admits finite-time extinction rather than blow-up. Finally, a homotopy approach is
developed for some kind of classification of various patterns obtained by variational and
other methods. Using a variety of analytic, variational, qualitative, and numerical methods
allows to justify that the above PDEs admit an infinite countable set of countable
families of compactly supported blow-up (extinction) or travelling wave solutions.
