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

We examine spherically symmetric solutions to the viscoelasticity system in a ball with the Neumann boundary conditions. Imposing some growth restrictions on the nonlinear part of the stress tensor, we prove the existence of global regular solutions for large data in the weighted Sobolev spaces, where the weight is a power function of the distance to the centre of the ball. First, we prove a global a priori estimate. Then existence is proved by the method of successive approximations and appropriate time extension. 1. Introduction First, we recall some important facts from the nonlinear theory of viscoelasticity. Among the papers devoted to nonlinear viscoelasticity, we mention below some of them. The global solution (in time) for sufficiently small and smooth data are proved by Ponce (cf. [1]) and Kawashima and Shibata (cf. [2]) for quasilinear hyperbolic system of second-order with viscosity. In the paper of [3], Kobayashi, Pecher, and Shibata proved global in time solution to a nonlinear wave equation with viscoelasticity under the special assumption about nonlinearity. In the paper of [4], Paw？ow and Zaj？czkowski showed the existence and uniqueness of global regular solutions to the Cahn-Hilliard system coupled with viscoelasticity. Finally, in [5], global existence of regular solutions to one-dimensional viscoelasticity is proved. Moreover, many facts on elasticity and viscoelasticity theory can be found in [6–10]. Some existence results in the linear and nonlinear thermoviscoelasticity are shown in [11, 12]. In our paper, we consider a more general nonlinear system of viscoelasticity because the stress tensor is a general nonlinear function depending on a strain. We assume that the stress tensor is a function of a strain at a given instant of time , but it does not depend on strains at time . It is worth to emphasize that our constitutive relation for the stress tensor and any another constitutive relation satisfy the rules of continuum mechanics. In order to prove the global (in time) solution for nonsmall data for nonlinear system of viscoelasticity (cf. formulae (1), (2), and (3)), we consider the spherically symmetric case and use the anisotropic Sobolev spaces with weights. Speaking more precisely, we consider the motion of viscoelastic medium described by the following system of equations (cf. [7, 11]): where is the displacement vector, is a given system of the Cartesian coordinates, , is the mass density, is the stress tensor, and is the external force field. We examine system (1) in a bounded domain with the Neumann boundary conditions