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 u(tt)+k(1)u(xxxx) + k(2)u(xxxxt) = g(u(xx))(xx) where k(1),k(2) > 0 are real numbers, g(s) is a given nonlinear function. When g(s) = s(n) (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 is an element of C-infinity((0,T]; H-infinity(R))boolean AND C([0, T]; H-3(R))boolean AND C-1([0,T]; H-1(R)) as long as initial data u(0) is an element of W-4,W-1(R)boolean AND H-3(R), u(1)is an element of L-1(R)boolean AND H-1(R). Moreover, when (u(0), u(1))is an element of H-2(R) x L-2(R), g is an element of C-2(R) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H-2(R))boolean AND C-1([0,T]; L-2(R))boolean AND H-1(0, T; H-2(R)). Copyright (C) 2009 John Wiley & Sons, Ltd.