SHARP DECAY ESTIMATES FOR SEMIGROUPS ASSOCIATED WITH SOME ONE-DIMENSIONAL FLUID-STRUCTURE INTERACTIONS INVOLVING DEGENERACY

摘要

In this paper, we consider two one-dimensional fluid-structure models where the parabolic component is degenerate. The first model that we consider involves a wave equation and a degenerate parabolic equation with natural transmissions. We discuss stability issues for this system under the Dirichlet boundary conditions at the outer endpoints. We show that when the degeneracy is weak (power is less than one), the underlying semigroup is polynomially stable, and provide sharp decay rates. The second model involves a beam equation and a weakly degenerate parabolic equation. For this model, we prove that the semigroup is exponentially stable. Our proofs are constructive, and combine the frequency domain method, boot strapping arguments, Hardy inequalities, and multipliers technique. In particular for the hyperbolic/parabolic model, our stability result generalizes and improves in some sense existing ones in the one-dimensional setting, while providing a simpler proof. As for the beam/parabolic model, our result seems to be the first even in the nondegenerate case.