Variational approach to complicated similarity solutions of higher order nonlinear evolution partial differential equations,

We consider the Cauchy problem for three higher order degenerate quasilinear partial differential equations, as basic models, ut = (−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 , where (x,t) ∈ RN ×R+, n > 0, and Δm is the (m �� 1)th iteration of the Laplacian. Based on the blow-up similarity and travelling wave solutions, we investigate general local, global, and blow-up properties of such equations. The nonexistence of global in time solutions is established by different meth- ods. In particular, for m = 2 and m = 3 such similarity patterns lead to the semilinear 4th and 6th order elliptic partial differential equations with noncoercive operators and non-Lipschitz nonlinearities n+1 n+1 −Δ2F+F−|F|− n F=0andΔ3F+F−|F|− n F=0inRN, (1) which were not addressed in the mathematical literature. Using analytic vari- ational, qualitative, and numerical methods, we prove that Eqs. (1) admit an infinite at least countable set of countable families of compactly supported solutions that are oscillatory near finite interfaces. This shows typical prop- erties of a set of solutions of chaotic structure.