We consider some initial-boundary value problems for non-linear equations of the three dimensional viscoelasticity. We examine the Dirichlet and the Neumann boundary conditions. We assume that the stress tensor is a nonlinear tensor valued function depending on the strain tensor fulfilling the rules of the continuum mechanics. We consider the initial-boundary value problems in a ball B R with radius R. Since, we are interested in proving global existence the spherically symmetric solutions are considered. Therefore we have to examine the spherically symmetric viscoelasticity system in spherical coordinates. Applying the energy method implies estimates in weighted anisotropic Sobolev spaces, where the weight is a power function of radius. Hence the origin of coordinates becomes a singular point. First the existence of weak solutions is proved. Next having appropriate estimates the weak solutions appear bounded and continuous. We have to emphasize that non-small data problem is considered.

We consider the homogeneous heat equation in a domain Ω in R n with vanishing initial data and the Dirichlet boundary condition. We are looking for solutions in W r,s p,q (Ω × (0, T )), where r < 2, s < 1, 1 ≤ p < ∞, 1 ≤ q ≤ ∞. Since we work in the Lp,q framework any extension of the boundary data and integration by parts are not possible. Therefore, the solution is represented in integral form and is referred as very weak solution. The key estimates are performed in the halfspace and are restricted to Lq(0, T ; W α p (Ω)), 0 ≤ α < 1 p and Lq(0, T ; W α p (Ω)), α ≤ 1. Existence and estimates in the bounded domain Ω follow from a perturbation and a fixed point arguments.2000 Mathematics Subject Classification. 35K05, 35K20.