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 utt +k1 uxxxx + k2uxxxxt= g(uxx)xx where k1, k2 > 0 are real numbers, g(s) is a given nonlinear function. When g(s)=s (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 (uo, u1) ∈ H2(R) x 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)).