Global Non-Small Data Existence of Spherically Symmetric Solutions to Nonlinear Viscoelasticity in a Ball

Abstract

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.

On Global Existence of Solutions of the Neumann Problem for Spherically Symmetric Nonlinear Viscoelasticity in a Ball

“…Mainly, we define the anisotropic Sobolev spaces with weights. Section 3 is devoted to the proof of energy-type estimates to solutions of problems (11)- (13).…”

“…Lemma 4 (see [13,Lemma 3.3]). We consider problems (11)- (13). Assume that there exist positive constants 1 , 2 , 1 , 1 , and 2 such that…”

“…Let us introduce a smooth function = ( ) such that ( ) = 0 for < 0 /2 and ( ) = 1 for > 0 , where 0 < . First, we obtain a local estimate in , 0 for solutions to problems (11)- (13). Multiplying (11) by 2 , 2 and integrating over yields…”

“…Lemma 8. Let be a solution to problems (11)- (13). Assume that there exist constants , , = 1, 2, 1 such that (41) holds, where…”

“…To calculate the above expression we need problems for and . From Lemma 3.4 from [13], we have the following problems for and :…”