Decay of solutions of wave equations in a bounded region with boundary dissipation

The exponential stability of a coupled hyperbolic/parabolicsystem arising in structural acoustics

“…Chen in [7] proved the exponential stability of solutions (24) under the geometrical conditions that Ω be "star-shaped". J. Lagnese in [11], and subsequently, R. Triggiani in [15] through an alternate proof, showed the uniform stabilization of (24) under the lessened constraint that there exist…”

“…Remark 3. In estimating the energy contribution of (1), one could also proceed as in [11] and [15] to eventually arrive at the uniform stabilization of (1)- (2) under all the geometrical conditions (j)-(jjj). To reiterate, it is the abstract trace estimate (25) which helps to yield the stronger result by eliminating outright the condition (jj).…”

Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation

“…It should be noted that this type of observability inequality is reminiscent of inequalities governing boundary controllability and stabilizability theory of unforced dissipative wave equation and goes back to [La1] (see also [La2,). In our case this inequality needs to be established for difference of two solutions which leads to an analysis of a non-dissipative system.…”

Stabilization of second-order evolution equations with time delay

“…From Theorem 1 and Remark 1 of [13] (see also [5,6,7,16]), the observability estimate (4.25) holds for the damped wave equation (4.21) − (4.24) if the boundary of Ω is of class C 2 , if T is large enough and if Γ 0 is given by (4.14) for some x 0 ∈ R n . 2.…”