This mathematical research is concerned with a class of partial differential equations of second order arising in the description of viscoelastic solids or liquids. The problems occupy an intermediate position between the basic parabolic partial differential equations for diffusive phenomena and hyperbolic equations modeling propagative phenomena. The goal is to establish the existence of generalized solutions for arbitrary initial data under suitable assumptions and to analyze their properties. Specifically, the asymptotic spacial and temporal behavior of such solutions is to be studied. The question of whether the damping mechanisms that are inherent in the equations can be powerful enough to prevent certain discontinuities of the solutions (corresponding to material instabilities) from appearing, will also be taken up. Specific tasks will include the construction of weak solutions in unbounded domains and reflecting problems of shear flows of viscoelastic liquids between two plates. Also, solutions of quasilinear equations with nonintegrable kernels will be sought. A passage to the limit of approximate solutions appears tractable in this setting but has yet to be confirmed. The actual smoothness of the resulting weak solution also would require study. A third specific task concerns detailed analysis of linear equations with variable coefficients. There is evidence suggesting that solutions may actually gain in smoothness over their data. Standard techniques from partial differential equations (maximum principle, variational characterizations) do not apply, so that new tools will have to be developed before progress can be expected.
