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

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Sobolev Spaces in Mathematics II

Part of the book series: International Mathematical Series ((IMAT,volume 9))

Abstract

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

$${\rm{ - }}\Delta ^{\rm{2}} F{\rm{ + }}F{\rm{ - |}}F{\rm| }^{\rm - {n \over {n{\rm{ + 1}}}}}{\rm{ }}F{\rm{ = 0\ and\ }}\Delta ^{\rm{3}} F{\rm{ + }}F{\rm{ - |}}F{\rm |}^{\rm - {n \over {n{\rm{ + 1}}}}}{\rm{ }}F{\rm{ = 0\ in\ R}}^N$$
((1))

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.

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Author information

Authors and Affiliations

  1. University of Bath, Bath, BA2 7AY, UK

    Victor Galaktionov

  2. Università di Trieste, Via Valerio 12/1, 34127 Trieste, Italy

    Enzo Mitidieri

  3. Steklov Mathematical Institute, Gubkina St. 8, 119991 Moscow, Russia

    Stanislav Pokhozhaev

Authors
  1. Victor Galaktionov
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  2. Enzo Mitidieri
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  3. Stanislav Pokhozhaev
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Correspondence to Victor Galaktionov .

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Editors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, USA

    Vladimir Maz'ya

  2. Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

    Vladimir Maz'ya

  3. Department of Mathematics, Linköping University, Linköping, Sweden

    Vladimir Maz'ya

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Galaktionov, V., Mitidieri, E., Pokhozhaev, S. (2009). Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations. In: Maz'ya, V. (eds) Sobolev Spaces in Mathematics II. International Mathematical Series, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85650-6_8

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