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We cope with a free boundary fluid-structure interaction model. In the model,the viscous incompressible fluid interacts with elastic body via the commonboundary. The motion of the fluid is governed by Navier-Stokes equations whilethe displacement of elastic structure is described by variable coefficient waveequations. The dissipation is placed on the common boundary between fluid andelastic body. Given small initial data, the global existence of the solutionsof this system is proved and the exponential decay of solutions are obtained.

Energy decay and global solutions for a damped free boundary fluid-elastic structure interface model with variable coefficients in elasticity