Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k₁, k₂>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sⁿ (where n?2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C^∞((0, T]; H^∞( R ))∩C(0, T]; H³( R ))∩C¹(0, T]; H^?1( R )) as long as initial data u₀∈W^{4, 1}( R )∩H³( R ), u₁∈L¹( R )∩H^?1( R ). Moreover, when (u₀, u₁)∈H²( R ) × L²( R ), g∈C²( R ) satisfy certain conditions, the Cauchy problem has no global solution in space C(0, T]; H²( R ))∩C¹(0, T]; L²( R ))∩H¹(0, T; H²( R )). Copyright © 2009 John Wiley & Sons, Ltd.