In this entry, the spatial behavior of solutions for the backward-in-time problem of the linear theory of thermoelasticity is studied. In this type of problem final data are assigned, usually at the time t=0, instead of initial data, and then we are interested in extrapolating to previous times. We associate with a solution of the considered problem an appropriate time-weighted volume measure, for which we get a spatial estimate describing a spatial exponential decay of the solution. The backward-in-time problems have been initially considered by Serrin [1] who established uniqueness results for the Navier–Stokes equations. Explicit uniqueness and stability criteria for classical Navier–Stokes equations backward in time have been further established by Knops and Payne [2] and Galdi and Straughan [3] (see also Payne and Straughan [4] for a class of improperly posed problems for parabolic partial differential equations). Such backward-in-time problems have been considered also by Ames and Payne [5] in order to obtain stabilizing criteria for solutions of the boundary-final value problem. It is well known that this type of problem is ill posed. In [6], Ciarletta established uniqueness and continuous dependence results upon mild require- ments concerning the thermoelastic coefficients; in particular the author considers hypotheses not real- istic from the physical point of view, such as a positive semidefinite elasticity tensor or a nonpositive heat capacity. Moreover, introducing an appropriate time-weighted volume measure, Ciarletta and Chiria [7] established the spatial estimate describing the spatial exponential decay of the thermoelastic process backward in time.

Spatial Behavior Backward in Time

TIBULLO, VINCENZO
2014

Abstract

In this entry, the spatial behavior of solutions for the backward-in-time problem of the linear theory of thermoelasticity is studied. In this type of problem final data are assigned, usually at the time t=0, instead of initial data, and then we are interested in extrapolating to previous times. We associate with a solution of the considered problem an appropriate time-weighted volume measure, for which we get a spatial estimate describing a spatial exponential decay of the solution. The backward-in-time problems have been initially considered by Serrin [1] who established uniqueness results for the Navier–Stokes equations. Explicit uniqueness and stability criteria for classical Navier–Stokes equations backward in time have been further established by Knops and Payne [2] and Galdi and Straughan [3] (see also Payne and Straughan [4] for a class of improperly posed problems for parabolic partial differential equations). Such backward-in-time problems have been considered also by Ames and Payne [5] in order to obtain stabilizing criteria for solutions of the boundary-final value problem. It is well known that this type of problem is ill posed. In [6], Ciarletta established uniqueness and continuous dependence results upon mild require- ments concerning the thermoelastic coefficients; in particular the author considers hypotheses not real- istic from the physical point of view, such as a positive semidefinite elasticity tensor or a nonpositive heat capacity. Moreover, introducing an appropriate time-weighted volume measure, Ciarletta and Chiria [7] established the spatial estimate describing the spatial exponential decay of the thermoelastic process backward in time.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4669724
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