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Roberto Triggiani (University of Virginia)

Title: Backward uniqueness of parabolic-hyperbolic coupled systems of PDEs: thermo-elastic equations, structural acoustic chamber, fluid-structure interaction

Abstract: Backward uniqueness of a group (hyperbolic case) is true at once; of an analytic semigroup (parabolic case) is proved in less than one-line. But a nilpotent semigroup (which may arise also from a 1-d wave with boundary damping) is the supreme counterexample.
What then if the PDE-dynamics is mixed, with a parabolic-hyperbolic coupling? Then backward uniqueness cannot be taken for granted. Here, an abstract semigroup result (circa 2001) provides a strategy to establish backward uniqueness. Applications include: thermo-elastic PDEs (2001; coupling a hyperbolic plate eqt and a heat eqt across the entire domain); a structural acoustic chamber (2003; coupling a wave in the chamber with a "parabolic" plate-equation on one of its walls); and fluid-structure interaction (2008; coupling an elastic wave -structure- surrounded by a parabolic fluid, with coupling taking place at the boundary interface). It should be emphasized that application of the abstract result to "concrete" PDEs-applications is far more challenging than the abstract result itself.