Spatial Behavior Backward in Time

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.