Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

  • Panizzi S
  • 2


    Mendeley users who have this article in their library.
  • 4


    Citations of this article.


We study the global solvability of the Cauchy-Dirichlet problem for two second order in time nonlinear integro-differential equations: 1)the extensible beam/plate equationut t+ Δ2u - m (under(∫, Ω) | ∇ u |2d x) Δ u = 0 (x ∈ Ω, t ∈ R) ;2)a special case of the Kirchhoff equationut t- (a + b under(∫, Ω) | ∇ u |2d x)-2Δ u = 0 (x ∈ Ω, t ∈ R) . By exploiting the I-method of J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, we prove that both equations admit global-in-time infinite energy solutions. In case 1), the energy is the mechanical energy; in case 2), it is a second order invariant introduced by S.I. Pokhozhaev. For the extensible beam equation 1), we also consider the effect of linear dissipation on such low regularity solutions, and we prove their exponential decay as t → + ∞. © 2006 Elsevier Inc. All rights reserved.

Author-supplied keywords

  • Extensible beam equation
  • Global solutions
  • Kirchhoff equation
  • Mixed problem

Get free article suggestions today

Mendeley saves you time finding and organizing research

Sign up here
Already have an account ?Sign in

Find this document


  • Stefano Panizzi

Cite this document

Choose a citation style from the tabs below

Save time finding and organizing research with Mendeley

Sign up for free