Evolution Equations and Control Theory (EECT)

A backward uniqueness result for the wave equation with absorbing boundary conditions

Pages: 347 - 353, Volume 4, Issue 3, September 2015      doi:10.3934/eect.2015.4.347

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Michael Renardy - Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, United States (email)

Abstract: We consider the wave equation $u_{tt}=\Delta u$ on a bounded domain $\Omega\subset{\mathbb R}^n$, $n>1$, with smooth boundary of positive mean curvature. On the boundary, we impose the absorbing boundary condition ${\partial u\over\partial\nu}+u_t=0$. We prove uniqueness of solutions backward in time.

Keywords:  Backward uniqueness, wave equation, Phragmen-Lindelöf theorem, $C_0$-semigroups, absorbing boundary conditions.
Mathematics Subject Classification:  47D06, 93B99.

Received: February 2015;      Revised: June 2015;      Available Online: September 2015.