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.


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], Pawłow and Zajączkowski showed the existence and uniqueness of global regular solutions to the Cahn-Hilliard system coupled with viscoelasticity.
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.
We examine system (1) in a bounded domain Ω ⊂ R 3 with the Neumann boundary conditions where  = Ω,  is the unit outward vector normal to .
Our aim is to prove the global existence of solutions to problems (1)-( 4) for nonsmall data.
To simplify the notation, we introduce  =   .
Assuming  = 1 and transforming (1) to the spherical coordinates, we obtain where Let us introduce the quantity where  = /.
Then, (8) takes the form and in view of (3), we have the initial conditions and in view of ( 2) and ( 9), the boundary condition where   =   .
Global existence of regular solutions to problems ( 11) and ( 12) with the Dirichlet boundary condition is proved in [13].To formulate the main results of this paper we need the following.
In Section 4 the existence of the global solution for nonsmall data of problems ( 11)-( 13) is proved.

Notation and Auxiliary Results
By  we denote the generic constant which changes from formula to formula.By (),  > 0, we denote a generic function which is always positive and increasing.
We replace forms of right-hand side (left-hand side) by the abbreviation r.h.s.(l.h.s.).We mark  , =   ,  , =   , and so on.
By () we denote the space of bounded functions on the interval .
The following result is valid.
The weighted Sobolev spaces with fractional derivatives are introduced in [16].
In the nonstationary case, Lemma 1 follows from the methods described in [18] in the case  = 2.For the general  Lemma 1 results from considerations in [18][19][20].
Finally, we introduce spaces used in this paper.We will define them by introducing finite norms.
(  × (0, )) is the space of bounded functions.  (  ), We introduce also the Sobolev spaces where where Moreover, (   ), and so on.

Estimates
To prove the Main Theorem we have to recall some estimates proved in [13].
Then, solutions to problems (11)-( 13) satisfy the estimate Next, we need the following.
Continuing, we have the following.
We need to obtain an estimate from (44).For this purpose, we examine the last two norms on the r.h.s. of (44).We express them in the form where Now, we examine the terms from the r.h.s. of (58).

ISRN Mathematical Analysis
The first norm on the r.h.s. of (58) equals Applying the Hölder inequality, the second integral on the r.h.s. of ( 58) is bounded by where the last inequality holds in virtue of Lemma 3 and under the assumption where  ∈ (0, 3/2) is introduced in Lemma 3. Hence, The third integral on the r.h.s. of (58) we express in the form Assuming ] < 1, setting  = 1 − ] > 0, and recalling imbedding (22) and Lemma 3, we obtain for  ≤ 6 the estimate Finally, the last term on the r.h.s. of (58) is bounded by Using the above estimates in the r.h.s. of (58) yields (56).
Using (56) and ( 57 ) +  1 . (66) Applying the equality in the first term on the r.h.s. of (66) and using the Gronwall lemma, we get Now, we have to estimate the norm on the r.h.s. of (68).For the purpose we need the following.
Proof.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 Introducing the notation we obtain Continuing, we have Applying the Hölder and the Young inequalities to the r.h.s. of the above equality and using (38) yields Using that  2 ,  2 = (w , −  ,  , ) 2 in the above inequality implies Integrating the above inequality with respect to time and using (37), we have Since w2 , ≤ 2(( , ) 2 +  2  2 , ), the above inequality yields  (79) The nonhomogeneous Dirichlet boundary condition is not convenient so we introduce the new function which is a solution to the problem We have to calculate  1 .For this purpose, we need the expressions which follow from (11) From ( 98), we obtain (69).This concludes the proof.
Finally, we recall some local properties of solutions to problems (11)-( 13) which are proved in Lemma 3.11 in [13].

Existence
We prove the existence of solutions to problems ( 11)-( 13) by the following method of successive approximations: , where  () =  ()  ,  () =  () /, and we assume that the zero approximation is defined by The boundary condition (113) Looking for solutions that  ,  is finite at  = 0, the last but one term on the l.h.s. of (118) vanishes for  > 0. Vanishing of the term can be guaranteed by choosing an appropriate base functions.
Integrating by parts in the third term on the l.h.s. of (118) implies For  ∈ (0, 1), the last term on the l.h.s. is positive so it can be omitted in the estimate implied by (119).Then we show the existence of solutions such that Hence,  ∼   near  = 0, where  > − + 1/2.Then the last term on the l.h.s. of (119) vanishes.For  > 1, the last term on the l.h.s. of (119) vanishes under the assumption that we are looking for solutions such that (, ) ∼   near  = 0,  > − + 1/2.
Similarly, the inequalities hold for   −2 = −1 so   −1 = .Hence, the same restriction on   as before is satisfied.Since the existence follows from the Galerkin method, Lemma 12 is proved.
In view of Lemma 12, we have the following.
Then there exists a solution to problems (11)-( 13) such that  , ∈ Hence, for sufficiently small , we have for any  ∈ N.
To show convergence of the sequence { () } ∈N we introduce the differences which are solutions to the problems To show convergence we multiply (140) by  (+1) , 2] and integrate the result over   .Then we have  , 2]−1  ≡ where   0 <  0 and the same considerations as in (125)-(127) are repeated.Integrating (147) with respect to time and employing the formula  () () = ∫  0  ()  , ,      2  2]−2 ) . (149) Hence, we have that the sequence { () } ∈N converges and then we show the existence of solutions to problems ( 11)-( 13) on a small interval (0, ).Since we have the estimate on the interval (0, ) with arbitrary , the above considerations can be repeated step by step.

ISRN Mathematical Analysis 13
Proof.Proof of the Main Theorem.In view of assumptions of Lemmas 8 and 10, Remark 9, and also by Lemmas 12 and 13, we have the first assertion of the theorem.In view of Lemma 3 and the Galerkin method, we prove the second assertion.In view of Lemmas 3 and 11, the third assertion follows.This concludes the proof.