The existence of solutions of a nonlinear suspension bridge equation

  • Park, Q-heung (Department of Mathematics, Inha University, Incheon 402-751) ;
  • Park, Kyeongpyo (Department of Mathematics, Inha University, Incheon 402-751) ;
  • Tacksun Jung (Department of Mathematics, Kunsan National University, Kunsan 573-360)
  • Published : 1996.11.01

Abstract

In this paper we investigate a relation between the multiplicity of solutions and source terms in a nonlinear suspension bridge equation in the interval $(-\frac{2}{\pi}, \frac{2}{\pi})$, under Dirichlet boundary condition $$ (0.1) u_{tt} + u_{xxxx} + bu^+ = f(x) in (-\frac{2}{\pi}, \frac{2}{\pi}) \times R, $$ $$ (0.2) u(\pm\frac{2}{\pi}, t) = u_{xx}(\pm\frac{2}{\pi}, t) = 0, $$ $$ (0.3) u is \pi - periodic in t and even in x and t, $$ where the nonlinearity - $(bu^+)$ crosses an eigenvalue $\lambda_{10}$. This equation represents a bending beam supported by cables under a load f. The constant b represents the restoring force if the cables stretch. The nonlinearity $u^+$ models the fact that cables expansion but do not resist compression.

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