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September  2015, 4(3): 347-353. doi: 10.3934/eect.2015.4.347

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

1. 

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123

Received  February 2015 Revised  June 2015 Published  September 2015

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.
Citation: Michael Renardy. A backward uniqueness result for the wave equation with absorbing boundary conditions. Evolution Equations & Control Theory, 2015, 4 (3) : 347-353. doi: 10.3934/eect.2015.4.347
References:
[1]

G. Avalos and T. Clark, Backward uniqueness for a PDE fluid-structure interaction,, preprint, ().   Google Scholar

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G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction,, J. Diff. Eq., 245 (2008), 737.  doi: 10.1016/j.jde.2007.10.036.  Google Scholar

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G. Avalos and R. Triggiani, Backwards uniqueness of the $C_0$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system,, Trans. Amer. Math. Soc., 362 (2010), 3535.  doi: 10.1090/S0002-9947-10-04851-8.  Google Scholar

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M. Renardy, Backward uniqueness for linearized compressible flow,, Evol. Eqns. Control Th., 4 (2015), 107.  doi: 10.3934/eect.2015.4.107.  Google Scholar

show all references

References:
[1]

G. Avalos and T. Clark, Backward uniqueness for a PDE fluid-structure interaction,, preprint, ().   Google Scholar

[2]

G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction,, J. Diff. Eq., 245 (2008), 737.  doi: 10.1016/j.jde.2007.10.036.  Google Scholar

[3]

G. Avalos and R. Triggiani, Backwards uniqueness of the $C_0$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system,, Trans. Amer. Math. Soc., 362 (2010), 3535.  doi: 10.1090/S0002-9947-10-04851-8.  Google Scholar

[4]

I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness for thermoelastic plates with rotational forces,, Semigroup Forum, 62 (2001), 217.  doi: 10.1007/s002330010035.  Google Scholar

[5]

M. Renardy, Backward uniqueness for linearized compressible flow,, Evol. Eqns. Control Th., 4 (2015), 107.  doi: 10.3934/eect.2015.4.107.  Google Scholar

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