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

Abstract

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∈W4, 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.