A Generalized Interpolation Inequality and its Application to the Stabilization of Damped Equations

In this paper, we establish a generalized H{ö}lder's or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for $1-d$ models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point.

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