Variational approach to complicated similarity solutions of higher-order nonlinear PDES II

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.