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

✍ Scribed by Changming Song; Zhijian Yang

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 207 KB
• Volume: 33
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1175

✦ Synopsis

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 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 ∈ C ∞ ((0,T]; H ∞ (R))∩C([0,T]; H 3 (R))∩C 1 ([0,T]; H -1 (R)) as long as initial data u 0 ∈ W 4,1 (R)∩H 3 (R), u 1 ∈ L 1 (R)∩H -1 (R). Moreover, when (u 0 ,u 1 ) ∈ H 2 (R)×L 2 (R),g∈ C 2 (R) satisfy certain conditions, the Cauchy problem has no global solution in space C([0,T]; H 2 (R))∩C 1 ([0,T]; L 2 (R))∩H 1 (0,T; H 2 (R)).