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A memory type boundary stabilization of a mildly damped wave equation. (English) Zbl 0945.35053
We consider the wave equation with a mild internal dissipation of the type \[ u_{tt}(t,x)+ \alpha u_t(t,x)= \Delta u(t,x)+ g(t,x), \quad t> 0, \;x\in \Omega, \] \[ \frac{\partial u}{\partial\nu} (t,x)+ \int_0^t k(t-s,x) u_s(s,x) dx= h(t,x), \quad t>0,\;x\in \Gamma_0, \] \[ u(t,x)= 0, \quad t>0,\;x\in \Gamma_1, \qquad u(0,x)= u_0(x), \;u_t(x)= u_1(x), \qquad x\in \Omega. \] It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.

MSC:
35L20 Initial-boundary value problems for second-order hyperbolic equations
93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions to PDEs
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