Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations

  • Victor Galaktionov
  • Enzo Mitidieri
  • Stanislav Pokhozhaev
Part of the International Mathematical Series book series (IMAT, volume 9)


We consider the Cauchy problem for three higher order degenerate quasilinear partial differential equations, as basic models,
$$\eqalign{ & u_t {\rm{ = ( - 1)}}^{m{\rm{ + 1}}} \Delta ^m {\rm{(|}}u{\rm{|}}^n u{\rm{) + |}}u{\rm{|}}^n u{\rm{,}} \cr & u_{tt} {\rm{ = ( - 1)}}^{m{\rm{ + 1}}} \Delta ^m {\rm{(|}}u{\rm{|}}^n u{\rm{) + |}}u{\rm{|}}^n u{\rm{,}} \cr & u_t {\rm{ = ( - 1)}}^{m{\rm{ + 1}}} \left[ {\Delta ^m {\rm{(|}}u{\rm{|}}^n u{\rm{)}}} \right]x_1 {\rm{ + }}\left( {{\rm{|}}u{\rm{|}}^n u} \right)x_1 \cr}$$
where (x,t) ϵ RN × R+, n > 0, and Δ m 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 methods. 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
$${\rm{ - }}\Delta ^{\rm{2}} F{\rm{ + }}F{\rm{ - |}}F{\rm| }^{\rm - {n \over {n{\rm{ + 1}}}}}{\rm{ }}F{\rm{ = 0\ and\ }}\Delta ^{\rm{3}} F{\rm{ + }}F{\rm{ - |}}F{\rm |}^{\rm - {n \over {n{\rm{ + 1}}}}}{\rm{ }}F{\rm{ = 0\ in\ R}}^N$$
which were not addressed in the mathematical literature. Using analytic variational, 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 properties of a set of solutions of chaotic structure.


Periodic Solution Periodic Orbit Travel Wave Solution Countable Family Nonlinear Dispersion Equation 
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Victor Galaktionov
    • 1
  • Enzo Mitidieri
    • 2
  • Stanislav Pokhozhaev
    • 3
  1. 1.University of BathBathUK
  2. 2.Università di TriesteItaly
  3. 3.Steklov Mathematical InstituteRussia

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