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

This paper continues the study that began in [1] and [2] of the Cauchy problem for (x,t)∈RN×R+(x,t)∈RN×R+ for three higher-order degenerate quasilinear partial differential equations (PDEs), as basic models, ut=(−1)m+1Δm(|u|nu)+|u|nu,ut=(−1)m+1Δm(|u|nu)+|u|nu,utt=(−1)m+1Δm(|u|nu)+|u|nu,utt=(−1)m+1Δm(|u|nu)+|u|nu,ut=(−1)m+1[Δm(|u|nu)]x1+(|u|nu)x1,ut=(−1)m+1[Δm(|u|nu)]x1+(|u|nu)x1, where n>0n>0 is a fixed exponent and ΔmΔm is the (m≥2)(m≥2)th iteration of the Laplacian. A diverse class of degenerate PDEs from various areas of applications of three types: 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 traveling wave (for the last one) solutions.In [1] and [2], the Lusternik–Schnirel’man category theory of variational calculus and fibering methods were applied. The case m=2m=2 and n>0n>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 are constructed for m=3m=3 and for higher orders. The “smother” case of negative n<0n<0 is included, with a typical “fast diffusion–absorption” parabolic PDE: ut=(−1)m+1Δm(|u|nu)−|u|nu,where n∈(−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 us to justify that the above PDEs admit an infinite countable set of countable families of compactly supported blow-up (extinction) or traveling wave solutions.