This work will be concerned principally with the analysis of classes of second order partial differential equations or partial integrodifferential equations describing time-dependent phenomena. In certain cases, good approximations to the solutions can be obtained by discarding the highest order time derivative in the equation. Work will be done in justifying such approximations by considering the effects of sending the highest time derivative to zero. For the short time behavior, one obtains a singular perturbation problem in which transient phenomena governed by a related equation should occur on a fast time scale. For the long time behavior, regular dependence of the time-asymptotic limit or - in cases of severe non-uniqueness of stationary states - selection effects from the transient behavior are expected. In this project, efforts will be made to place some of these heuristic conjectures on a mathematically sound basis. Work will also be done on questions of existence of solutions for the reduced problems. Sources for the equations under study include mathematical models describing shear flows for viscoelastic materials and polymerrheology.
