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

Authors: Jerzy A. Gawinecki; Wojciech M. Zajączkowski
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

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.


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## Body

## 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], Pawłow and Zajączkowski 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 timet, but it does not depend on strains at time t′<t. 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]):
(1)ϱu,tt=divσ+ϱf,
where u=u(x,t)=(u1(x,t),u2(x,t),u3(x,t))∈ℝ3 is the displacement vector, x=(x1,x2,x3)∈ℝ3 is a given system of the Cartesian coordinates, t∈ℝ+∪{0}, ϱ is the mass density, σ=σ(x,t)∈ℝ9 is the stress tensor, and f=(f1(x,t),f2(x,t),f3(x,t))∈ℝ3 is the external force field.We examine system (1) in a bounded domain Ω⊂ℝ3 with the Neumann boundary conditions
(2)n-·σ|S=0,
where S=∂Ω, n- is the unit outward vector normal to S.Moreover, we add the initial conditions(3)u|t=0=u0,ut|t=0=u1.We will assume that(4)σ=∂F∂ε(ε)+μ0εt,
where ε=(1/2)(∇u+(∇u)T) is the linearized strain tensor, F=F(ε) is some function which will be specified later, and μ0 is a positive constant.Our aim is to prove the global existence of solutions to problems (1)–(4) for nonsmall data.Since we do not know how to show the existence in a general case, we restrict our considerations to the spherically symmetric case. We assume thatΩ is a ball BR with radius R centered at the origin of the introduced Cartesian coordinates. We introduce the spherical coordinates r,φ,ϑ by the relations
(5)x1=rcosφsinϑ,x2=rsinφsinϑ,x3=rcosϑ.With these coordinates, we connect the orthonormal vectors(6)e-r=(cosφsinϑ,sinφsinϑ,cosϑ),e-ϑ=(cosφcosϑ,sinφcosϑ,-sinϑ),e-φ=(-sinφ,cosφ,0).Then, we defineur=u·e-r, uϑ=u·e-ϑ, uφ=u·e-φ, εrr=e-r·ε·e-r=ur,r, εφφ=ur/r, εϑϑ=ur/r. Since the spherically symmetric case is considered, we have uϑ=uφ=0.To simplify the notation, we introduce(7)w=ur.Assumingϱ=1 and transforming (1) to the spherical coordinates, we obtain
(8)w,tt=1r2(σrrr2),r-1r(σϑϑ+σφφ)+fr,
where
(9)σrr=∂F∂εrr+μ0εrr,t,σϑϑ=∂F∂εϑϑ+μ0εϑϑ,tσφφ=∂F∂εφφ+μ0εφφ,t.Let us introduce the quantity(10)F(ε)=ψ(w,r,η),
where η=w/r.Then, (8) takes the form
(11)w,tt=1r2[(∂ψ∂w,r+μ0w,rt)r2],r-1r(∂ψ∂η+μ0w,tr),and in view of (3), we have the initial conditions
(12)w|t=0=w0,w,t|t=0=w1,and in view of (2) and (9), the boundary condition
(13)(∂ψ∂w,r+μ0w,rt)|SR=0,
where SR=∂BR.Global existence of regular solutions to problems (11) and (12) with the Dirichlet boundary condition
(14)w|SR=0is proved in [13]. To formulate the main results of this paper we need the following.Assumptions. Let us introduce the notation ϑ=w,r, η=w/r. Assume that (1)
ψ(ϑ,η)=ψ1(ϑ)+ψ2(η),(2)
there exist positive constantsα1, α2, β1, β2 such that
(15)∂2ψ1∂ϑ2≥α1,∂2ψ2∂η2≥α2,|∂ψ1∂ϑ|≤β1|ϑ|,|∂ψ2∂η|≤β2|η|,(3)
w0∈Hμ12(0,R),w1∈Hμ12(0,R), μ1∈(2,1+5/2),(4)
w1∈L2,μ2(0,R),w0∈Hμ21(0,R), μ2∈(0,3/2), ψ1≥α1ϑ2, ψ2≥α2η2, |ψ1,ϑ|≤β1|ϑ|, |ψ2,η|≤β2|η|,(5)
w1∈L2,μ3(0,R), w0∈Hμ31(0,R), μ3∈(0,1/2).Main Theorem. (1) Let Assumptions (1)–(3) hold. Then there exists a solution to problems (11)–(13) such that
(16)w,tt∈B(0,T;L2,μ1(0,R)),w,t∈B(0,T;Hμ11(0,R)),w∈B(0,T;Hμ12(0,R)),w,tt∈L2(0,T;Hμ11(0,R)).(2) Let Assumptions (1) and (4) hold, then(17)w,t∈B(0,T;L2,μ2(0,R))∩L2(0,T;Hμ21(0,R)),w∈B(0,T;Hμ21(0,R)).(3) Let Assumptions (1), (4), and (5) be satisfied. Then
(18)w∈L∞((0,R)×(0,T)),w∈B(0,T;C1/2-μ3/2(0,R)),w∈Lβ(0,T;C(β-1)/β(0,T)),β∈(1,22μ3+1),w,t∈L2(0,T;C1/2-μ3/2(0,R)).Our paper is organized as follows. In Section1, the formulation of the considered problem and the main results are presented. In Section 2, the notation is introduced. 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).In Section4 the existence of the global solution for nonsmall data of problems (11)–(13) is proved.


## 2. Notation and Auxiliary Results

Byc we denote the generic constant which changes from formula to formula. By c(σ), σ>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 markw,t=∂tw, w,r=∂rw, and so on.ByB(I) we denote the space of bounded functions on the interval I.ByHμk(0,R), μ∈ℝ, k∈ℕ∪{0}, we denote a weighted Sobolev space with the finite norm
(19)∥u∥Hμk(0,R)=(∑α=0k‍∫0R|∂rαu|2r2(μ-k+|α|)dr)1/2andL2,μ(0,R)=Hμ0(0,R).ByCα(I), α∈(0,1) we denote the Hölder space with the finite norm
(20)∥u∥Cα(I)=supτ∈I|u(τ)|+supτ′,τ′′∈I|u(τ′)-u(τ′′)||τ′-τ′′|α.Next, we recall the Hardy inequality (see [14, Chapter 1, Section  2.15])
(21)|1p′-μ|p∫0∞rp(μ-1)|f|pdr≤∫0∞rpμ|f,r|pdr,
where 1/p+1/p′=1, μ∈ℝ, andμ≠1/p′.The inequality holds also for functions with compact support. Assuming thatsuppf⊂[0,R], we introduce F(x)=∫x∞f(y)dy and repeat the proof from [14, Chapter 1, Section  2.15].From [15, Chapter 2, Section  3], we have the imbedding
(22)∥u∥Lq(0,T;Lp(0,R))≤c(∥u∥L∞(0,T;L2(0,R))+∥∇ru∥L2(0,T;L2(0,R))),
where 1/p+2/q≥1/2.Finally, we consider the problem(23)u,t-u,rr=f,u|t=0=u0,u,r|r=R=0.To examine nonstationary problem (23), we need anisotropic weighted Sobolev spaces Vp,ν2,1((0,R)×(0,T)), p∈(1,∞), ν∈ℝ, of functions with the finite norm
(24)∥u∥Vp,ν2,1((0,R)×(0,T))=(∑α+2a≤2‍∫0T∫0R|∂rα∂tau|prp(ν-2+α+2a)drdt)1/p.SpacesVp,ν2(0,R) appropriate for elliptic problems were introduced in [16]. Moreover, we assume that Lp,ν(0,R)=Vp,ν0(0,R).The following result is valid.Lemma 1.
We assume thatf∈Lp,ν((0,R)×(0,T)), u0∈Vp,ν2-2/p(0,R). Then there exists a solution to problem (23) such that u∈Vp,ν2,1((0,R)×(0,T)) and
(25)∥u∥Vp,ν2,1((0,R)×(0,T))≤c(∥f∥Lp,ν((0,R)×(0,T))+∥u0∥Vp,ν2-2/p(0,R)).In the case of elliptic equations, a similar result was proved in [17] for p=2 and in [16] for any p∈(1,∞).The weighted Sobolev spaces with fractional derivatives are introduced in [16].In the nonstationary case, Lemma1 follows from the methods described in [18] in the case p=2. For the general p Lemma 1 results from considerations in [18–20].Finally, we introduce spaces used in this paper. We will define them by introducing finite norms.Besov spaceBp,νl(BR), l∈ℝ+, p∈(1,∞), ν∈ℝ,
(26)∥u∥Bp,νl(BR)=∥u∥Bp,ν•l-[l](BR)+∥u∥Vp,ν[l](BR),
where [l] is the integer part of l,
(27)∥u∥Bp,ν•α(BR)=(∫0R∫0R|u(r1)r1ν-u(r2)r2ν|p|r1-r2|1+pαdr1dr2)1/p,
where α∈(0,1) and
(28)∥u∥Vp,νk(BR)=(∑α≤k‍∫0R|∂rαu|prp(ν-k+α)dr)1/p,fork integer.B
(
B
R
×
(
0
,
T
)
) is the space of bounded functions.C
α
(
B
-
R
), Cα,β(B-R×[0,T]), α,β∈(0,1), are the Hölder spaces with the finite norms
(29)∥u∥Cα(B-R)=∥u∥B(B-R)+supr1,r2∈B-R|u(r1)-u(r2)||r1-r2|α,∥u∥Cα,β(B-R×[0,T])=∥u∥B(B-R×[0,T])∥u∥Cα,β(B-R×[0,T])=+supt∈[0,T]supr1,r2∈B-R|u(r1,t)-u(r2,t)||r1-r2|α∥u∥Cα,β(B-R×[0,T])=+supr∈B-Rsupt′,t′′∈[0,T]|u(r,t′)-u(r,t′′)||t′-t′′|β,
where B-R is the closure of BR.ByLql(0,T;Wpk(BR)), l,k∈ℕ∪{0}, p,q∈[1,∞], we denote a space of functions with the finite norm
(30)∥∂tlu∥Lq(0,T;Wpk(BR)).We introduce also the Sobolev spaces(31)Wpk(BR)={u=u(x):∥u∥Wpk(BR)=∑l≤k‍∥∂rlu∥Lp(BR)<∞},
where k∈ℤ+∪{0}, p∈[1,∞],
(32)Wpk,k/2(BRT)={∑l+2a≤k‍u=u(r,t):∥u∥Wpk,k/2(BRT)v=∑l+2a≤k‍∥∂rl∂tau∥Lp(BRT)<∞},
where k,k/2∈ℤ+∪{0}, p∈[1,∞], BRT=BR×(0,T), BR={r∈ℝ+:r<R},
(33)Wpk(0,T)={∑l≤k‍u=u(t):∥u∥Wpk(0,T)v=∑l≤k‍∥∂tlu∥Lp(0,T)<∞},k
∈
ℤ
+
∪
{
0
}, p∈[0,1]. For k=0, we have Wp0(BR)=Lp(BR), Wp0,0(BRT)=Lp(BRT), Wp0(0,T)=Lp(0,T), Bpα(BR)=Bp,pα(BR) is the Besov space introduced in [14, Chapter 4, Section  18] and Wpα(BR) is the Sobolev-Slobodetsky space, where α∉ℤ+,p∈[1,∞].Moreover,(34)Vp,0k(BR)⊂Wpk(BR),Vp,0k,k/2(BRT)⊂Wpk,k/2(BRT).FinallyHk(BR)=W2k(BR), Hk,k/2(BRT)=W2k,k/2(BRT), and so on.


## 3. Estimates

To prove the Main Theorem we have to recall some estimates proved in [13].Lemma 2 (see [13, Lemma  3.1]).
We consider problems (11)–(13). Assume that w0, w1 are initial data defined by (12). Let c0 be a constant such that
(35)∫BR[12w12+ψ(w0,r,w0r)]r2dr≡c02<∞,
where ψ is a positive function differentiable with respect to its arguments introduced by (10).
Then, solutions to problems (11)–(13) satisfy the estimate
(36)∫BR[12w,t2+ψ(w,r,wr)]r2dr+∫BRt(w,rt2+w,t2r2)r2drdt′=∫BR[12w12+ψ(w0,r,w0r)]r2dr=c02,foranyt>0.Next, we need the following.Lemma 3 (see [13, Lemma  3.2]).
We consider problems (11)–(13). Let ϑ=w,r, η=w/r. Assume that ψ(ϑ,η)=ψ1(ϑ)+ψ2(η). Assume existence of positive constants α1, α2, β1, and β2 such that
(37)ψ1(ϑ)≥α1ϑ2,ψ2(η)≥α2η2,(38)|∂ψ1∂ϑ|≤β1|ϑ|,|∂ψ2∂η|≤β2|η|.
Finally, assume that(39)w1∈L2,μ(0,R),w0∈Hμ1(0,R),μ∈(0,32).
Then solutions to problems (11)–(13) satisfy the following estimate:
(40)∥w,t∥L∞(0,t;L2,μ(BR))+∥w∥L∞(0,t;Hμ1(BR))+∥w,t∥L2(0,t;Hμ1(BR))≤c(R,μ,t)(∥w1∥L2,μ(BR)+∥w0∥Hμ1(BR))≤c1(R,μ,T),
where μ∈(0,3/2), t≤T.Continuing, we have the following.Lemma 4 (see [13, Lemma  3.3]).
We consider problems (11)–(13). Assume that there exist positive constants α-1, α-2, β-1, c-1, and c-2 such that
(41)ψ1,ϑϑ≥α-1,ψ2,ηη≥α-2,|ψ1,ϑϑ|≤β-1,|∂ϑ3ψ1|≤c-1,|∂η3ψ2|≤c-2.
Assume also that(42)η,t,ϑ,t∈L3(0,T;L3,2/3μ(BR)),w1∈Hμ1(BR),w,tt(0)∈L2,μ(BR),
where
(43)w,tt(0)=1r2[(ψ,ϑ(w0,r,w0r)+μ0w1,r)r2],r-1r[ψ,η(w0,r,w0r)+w1r].
Then solutions to (11)–(13) satisfy the following inequality:
(44)∥w,tt∥L∞(0,t;L2,μ(BR))+∥w,t∥L∞(0,t;Hμ1(BR))+μ0∥w,tt∥L2(0,t;Hμ1(BR))≤c(t)(∥ϑ,t∥L3(0,t;L3,2/3μ(BR))3/2+∥η,t∥L3(0,t;L3,2/3μ(BR))3/2)+c(t)c2,
where μ∈(0,1+(1/2)5), t≤T and
(45)c2=∥w,tt(0)∥L2,μ(BR)+∥w,t(0)∥Hμ1(BR).Remark 5.
In general constantsαi and α-i, i=1,2, are different. Let us consider the example
(46)ψ1(ϑ)=α1ϑ2+a1+ϑ2,
where a is a positive constant. Then we have
(47)ψ1(ϑ)≥α1ϑ2.
Continuing, we derive(48)ψ1,ϑϑ=2α1+a(2ϑ2-1)(1+ϑ2)5/2.
Forϑ2≥1/2, we obtain ψ1,ϑϑ≥2α1. But for ϑ2<1/2, we have
(49)ψ1,ϑϑ≥2α1-a(1+ϑ2)5/2≥2α1-a≡α-1.
It is clear that many examples can be invented.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
(50)(∫0T∫BR|w,rt|3r2μdrdt)1/2+(∫0T∫BR|w,tr|3r2μdrdt)1/2≡I.Hence, by the Hardy inequality (see [14, Chapter 1, Section  2.15]), the above expression is bounded by
(51)I≤c(∫0T∫BR|w,rt|3r2μdrdt)1/2≡I1.To estimateI1, we need the Pego transformation
(52)p(r,t)=∫0rw,t(r′,t)r′2dr′,q(r,t)=μ0w,r(r,t)r2-p(r,t).Hence,(53)w,rtr2=1μ0(q,t+p,t).To calculate the above expression we need problems forp and q. From Lemma 3.4 from [13], we have the following problems for p and q:
(54)p,t-μ0p,rr=ψ,ϑr2-∫0r(ψ,η+μ0w,tr)rdr-2μ0w,tr,p|t=0=∫0rw1(r)r2dr≡p0(r),p,r|r=R=w,t(R,t)R2(55)q,t=-ψ,ϑr2+∫0r(ψ,η+μ0w,tr)rdr,q|t=0=μ0w0,rr2q|t=0=-∫0rw1(r)r2dr≡q0(r).Lemma 6.
Let the assumptions of Lemma3 be satisfied. Let w,r∈Lσ,2-ν(BRT), w,t,∂t2-1/σw∈Lσ(0,T;L2,1(BR)), ∂t2-1/σ means the fractional derivative. Then solutions to problems (54) and (55) satisfy
(56)∥p∥Wσ,-ν2,1(BRT)≤c∥w,r∥Lσ,2-ν(BRT)+c∥w,tr2∥Wσ1,1/2(BR,R0T)+cc1,(57)∥q,t∥Lσ,-ν(BRT)≤c∥w,r∥Lσ,2-ν(BRT)+cc1,
where c1 is introduced in (40), BR,R0={r:0<R0<r<R}, ν<1/σ+3/2-μ, ν<1, σ∈(1,6], and μ∈(0,3/2) is introduced in Lemma 3.Proof.
For solutions to problem (54), we have (see Lemma 1),
(58)∥p∥Wσ,-ν2,1(BRT)≤c(∥w,rr2∥Lσ,-ν(BRT)+∥∫0r(|w|+|w,t|)dr∥Lσ,-ν(BRT)+∥w,tr∥Lσ,-ν(BRT)+∥p0∥Wσ,-ν2-2/σ(BR)+∥w,t(R,t)R2∥Wσ,-ν1-1/σ,1/2-1/2σ(SRT)).
Now, we examine the terms from the r.h.s. of (58).
The first norm on the r.h.s. of (58) equals
(59)(∫BRT|w,r|σr(2-ν)σdrdt)1/σ.
Applying the Hölder inequality, the second integral on the r.h.s. of (58) is bounded by
(60)(∏k=1nAk∫BRT|(∫0rr-2ν1dr)1/2vvv×(∫0r(w2+w,t2)r2ν1dr)1/2|σr-νσdrdt)1/σ≤cc1,
where the last inequality holds in virtue of Lemma3 and under the assumption
(61)ν1+ν<1σ+12,ν1=μ-1,
where μ∈(0,3/2) is introduced in Lemma 3. Hence,
(62)μ+ν<1σ+32.
The third integral on the r.h.s. of (58) we express in the form
(63)(∫BRT|w,t|σr(1-ν)σdrdt)1/σ≡J1.
Assumingν<1, setting μ=1-ν>0, and recalling imbedding (22) and Lemma 3, we obtain for σ≤6 the estimate
(64)J1≤csupt(∫BR|(wrμ),t|2dr)1/2+c(∫BRT|∂r(wrμ),t|2drdt)1/2≤cc1.
Finally, the last term on the r.h.s. of (58) is bounded by
(65)∥w,tr2∥Wσ1-1/σ,1/2-1/2σ(SRT)≤c∥w,tr2∥Wσ1,1/2(BR,R0T).
Using the above estimates in the r.h.s. of (58) yields (56).
Repeating the considerations leading to (56) to problem (55) gives (57). This concludes the proof.Using (56) and (57) in (53) gives
(66)∥w,rt∥Lσ,2-ν(BRT)≤c∥w,r∥Lσ,2-ν(BRT)+c(T)∥w,tr2∥Wσ1,1/2(BR,R0T)+cc1.Applying the equality(67)w,r(t)=∫0tw,rt′dt′+w,r(0)in the first term on the r.h.s. of (66) and using the Gronwall lemma, we get
(68)∥w,rt∥Lσ,2-ν(BRT)≤c(T)∥w,tr2∥Wσ1,1/2(BR,R0T)+c(T)c1.Now, we have to estimate the norm on the r.h.s. of (68). For the purpose we need the following.Lemma 7.
Letw be a solution to (11)–(13). Assume that ψ=ψ1(ϑ)+ψ2(η) and (37), (38) are satisfied. Assume that w1∈Wσ1-1/σ(BR), w0∈Wσ1(BR,R0/2), where σ∈(1,∞), 0<R0<R. Then
(69)∥w,tr2∥Wσ1,1/2(BR,R0T)≤c(T)(c0+∥w0∥Wσ1(BR,R0/2)+∥w1∥Wσ1-1/σ(BR)),
where c0 is introduced in (36).Proof.
Let us introduce a smooth functionζ=ζ(r) such that ζ(r)=0 for r<R0/2 and ζ(r)=1 for r>R0, where R0<R.
First, we obtain a local estimate inBR,R0 for solutions to problems (11)–(13). Multiplying (11) by ζ2w,tr2 and integrating over BR yields
(70)∫BRw,ttζ2w,tr2dr+∫BR(ψϑ+μ0w,rt)(ζ2w,t),rr2dr+∫BR(ψ,η+μ0η,t)ζ2w,trdr=0.
Introducing the notation(71)w~=wζ,
we obtain(72)12ddt∫BRw~,t2r2dr+∫BR(ψ,ϑϑ,t+ψ,ηη,t)ζ2r2dr+2∫BRψ,ϑw,tζζ,rr2dr+μ0∫BR[w,rt(ζ2w,t),r+w~,tr2]r2dr=0.
Continuing, we have(73)ddt(12∫BRw~,t2r2dr+∫BRψζ2r2dr)+μ0∫BR(w,rt2ζ2+w~,t2r2)r2dr=-2μ0∫BRw,rtw,tζζ,rr2dr-2∫BRψ,ϑw,tζζ,rr2dr.
Applying the Hölder and the Young inequalities to the r.h.s. of the above equality and using (38) yields
(74)ddt(12∫BRw~,t2r2dr+∫BRψζ2r2dr)+μ02∫BR(w,rt2ζ2+w~,t2r2)r2dr≤cμ0∫BRw,t2ζ,r2r2dr+c∫BRw,r2ζ,r2r2dr.
Using thatw,rt2ζ2=(w~,rt-w,tζ,r)2 in the above inequality implies
(75)ddt(12∫BRw~,t2r2dr+∫BRψζ2r2dr)+μ0∫BR(w~,rt2+w~,t2r2)r2dr≤cμ0∫BRw,t2ζ,r2r2dr+c∫BRw,r2ζ,r2r2dr.
Integrating the above inequality with respect to time and using (37), we have
(76)12∫BRw~,t2r2dr+∫BR(α1w,r2ζ2+α2w~2r2)r2dr+μ0∫BRt(w,rt2+w~,t2r2)r2drdt′≤cμ0∫BRtw,t2ζ,r2r2drdt′+c∫BRtw,r2ζ,r2r2drdt′+12∫BRw~1r2dr+∫BRψ(w0,r,w0r)r2dr≡c∫BRt(w,t2+w,r2)ζ,r2r2drdt′+cc02.
Sincew~,r2≤2((w,rζ)2+w2ζ,r2), the above inequality yields
(77)12∫BRw~,t2r2dr+∫BR(12α1w~,r2r2+α2w~2)dr+μ0∫BRt(w~,rt2r2+w~,t2)drdt′≤c∫BRt(w,t2+w,r2+w2)ζ,r2r2drdt′+cc02≤cc02,
where (36) was used again.
Now, we introduce the following Pego transformation(78)p~(r,t)=∫0rw~,t(r′,t)r′2dr′,q~(r,t)=μ0w~,rr2-p~(r,t).
Moreover, we have the initial-boundary value problem forp~,
(79)p~,t-μ0p~,rr=∫0rw~,ttr2dr-μ0(w~,tr2),r≡I,p~|t=0=∫0rw~1r2dr,p~(R,t)=∫0Rw~,tr2dr.
The nonhomogeneous Dirichlet boundary condition is not convenient so we introduce the new function(80)p~′=p~-p~(R,t),
which is a solution to the problem
(81)p~′,t-μ0p~′,rr=I-∫0Rw~,tt2r2dr≡I1,p~′|t=0=∫0rw~1r2dr-∫0Rw~1r2dr≡p~′0,p~′|r=R=0.We have to calculate I1. For this purpose, we need the expressions which follow from (11) multiplied by ζ(82)w~,tt=1r2[(ψ,ϑζ+μ0w,rtζ)r2],r-(ψ,ϑ+μ0w,rt)ζ,r-1r(ψ,ηζ+μ0w~,tr),
and from (13) multiplied by ζ(83)(ψ,ϑ+μ0w,rt)ζ|r=R=0.
Using (82) and (83) in I1 yields
(84)I1=∫0r[(ψ,ϑζ+μ0w,rtζ)r2],rdr-∫0r(ψ,ϑ+μ0w,rt)ζ,rr2dr-∫0r(ψ,ηζr+μ0w~,t)dr-μ0(w~,tr2),r-∫0R[(ψ,ϑζ+μ0w,rtζ)r2],rdr+∫0R(ψ,ϑ+μ0w,rt)ζ,rr2dr+∫0R(ψ,ηζr+μ0w~,t)dr=ψ,ϑζr2+μ0w~,rtr2-μ0w,tζ,rr2+∫rR(ψ,ϑ+μ0w,rt)ζ,rr2dr+∫rR(ψ,ηζ+μ0w~,t)dr-μ0(w~,rtr2+2rw~,t)=ψ,ϑζr2-μ0w,tζ,rr2-2μ0rw~,t+∫rR(ψ,ϑζ,rr2+ψ,ηζr+μ0w~,t)dr-μ0∫rRw,t(ζ,rr2),rdr.
For solutions to problem (81), we have
(85)∥p~′∥Wσ2,1(BRT)≤c(∥I1∥Lσ(BRT)+∥p~′0∥Wσ2-2/σ(BR)).
Hence,(86)∥p~∥Wσ2,1(BRT)≤c(∥I1∥Lσ(BRT)+∥p~,t(R,t)∥Lσ(BRT)+∥p~′0∥Wσ2-2/σ(BR)).
From the expression ofI1, we have
(87)∥I1∥Lσ(BRT)≤c(∥w,r∥Lσ(BR,R0/2T)+∥w,t∥Lσ(BR,R0/2T)+∥w∥Lσ(BR,R0/2T)).
In view of (82) and (83), we obtain
(88)p~,t(R,t)=∫0Rw~,ttr2dr=∫0R(ψ,ϑ+μ0w,rt)ζ,rr2dr-∫0R(ψ,ηζr+μ0w~,t)dr.
Hence,(89)|p~,t(R,t)|≤|∫0R(ψ,ϑζ,rr2+ψ,ηζr+μ0w~,t)dr|+|∫0Rμ0w,t(ζ,rr2),rdr|.
Employing (38) yields
(90)∥p~,t(R,t)∥Lσ(BRT)≤c(∥w,r∥Lσ(BR,R0/2T)+∥w,t∥Lσ(BR,R0/2T)+∥w∥Lσ(BR,R0/2T)).
From the form ofp~′0 (see (81)2), we derive
(91)∥p~′0∥Wσ2-2/σ(BR)=∥∫0rw~1r2dr∥Wσ2-2/σ(BR)≤∥w~1r2∥Wσ1-2/σ(BR)≤c∥w~1∥Wσ1-2/σ(BR).
Using (90) in (86) yields
(92)∥p~∥Wσ2,1(BRT)≤c(∥w,r∥Lσ(BR,R0/2T)+∥w,t∥Lσ(BR,R0/2T)+∥w∥Lσ(BR,R0/2T)+∥w1∥Wσ1-2/σ(BR,R0/2T)).
Sincep~,r=w~,tr2 and that
(93)∥p~,r∥Wσ1,1/2(BRT)≤c∥p~∥Wσ2,1(BRT),
we obtain from (92) the inequality
(94)∥w,tr2∥Wσ1,1/2(BR,R0/2T)≤c(T)(∥w,r∥Lσ(BR,R0/2T)+∥w,t∥Lσ(BR,R0/2T)vvvvv+∥w0∥Lσ(BR,R0/2)+∥w1∥Wσ1-2/σ(BR,R0/2)),
where the relation w(t)=∫0tw,t′dt′+w0 was exploited.
Using the interpolation inequality(95)∥w,t∥Lσ(BR,R0/2T)≤ε∥w,rt∥Lσ(BR,R0/2T)+c(1ε)∥w,t∥L2(BR,R0/2T)
in (94), assuming that ε is sufficiently small, and using (77), we obtain
(96)∥w,tr2∥Wσ1,1/2(BR,R0/2T)≤c(T)(∥w,r∥Lσ(BR,R0/2T)+∥w0∥Lσ(BR,R0/2)vvvvvvvvv+∥w1∥Wσ1-2/σ(BR,R0/2)+c0).
Employing(97)w,r(t)=∫0tw,rt′dt′+w,r(0)
in (96) and applying the Gronwall lemma, we get
(98)∥w,tr2∥Wσ1,1/2(BR,R0/2T)≤c(T)(∥w0∥Wσ1(BR,R0/2)+∥w1∥Wσ1-2/σ(BR,R0/2)+c0).
From (98), we obtain (69). This concludes the proof.To derive an estimate for (44), we have to take into account Lemmas 6 and 7. The lemmas imply an estimate for the first two terms on the r.h.s. of (44). For this purpose, we need the following.Lemma 8.
Letw be a solution to problems (11)–(13). Assume that there exist constants α-i, c-i, i=1,2, β-1 such that (41) holds, where ψ=ψ1(ϑ)+ψ2(η), ϑ=w,r, η=w/r. Assume that wtt(0)∈L2,μ1(BR), w0∈W31(BR,R0/2)∩Hμ21(BR), w1∈Hμ11(BR)∩L2,μ2(BR)∩W32/3(BR,R0/2), c0 is defined in (36),
(99)c1=∥w1∥L2,μ2(BR)+∥w0∥Hμ21(BR),c2=∥w,tt(0)∥L2,μ1(BR)+∥w1∥Hμ11(BR),μ1∈(32,1+125),μ2∈(56,32).
Then the following estimate holds:(100)∥w,tt∥L∞(0,t;L2,μ1(BR))+∥w,t∥L∞(0,t;Hμ11(BR))+∥w,tt∥L2(0,t;Hμ11(BR))≤c(t)[+∥w1∥W32/3(BR,R0/2))3/2(12c1+c0+∥w0∥W31(BR,R0/2)vvvvvv+∥w1∥W32/3(BR,R0/2))3/2+c2],t≤T.Proof.
From (44) and (51), we have
(101)∥w,tt∥L∞(0,t;L2,μ1(Br))+∥w,t∥L∞(0,t;Hμ11(BR))+∥w,tt∥L2(0,t;Hμ11(BR))≤c(t)[(∫BRT|w,rt|3r2μ1drdt)1/2+c2],
where μ1∈(0,1+(1/2)5),t≤T,c2=∥w,tt(0)∥L2,μ1(Br)+∥w,t(0)∥Hμ11(BR).
Inequality (68) takes the form
(102)∥w,rt∥Lσ,2-ν(BRt)≤c(t)(∥w,tr2∥Wσ1,1/2(BR,R0t)+c1),
where c1=∥w1∥L2,μ2(BR)+∥w0∥Hμ21(BR), μ2∈(0,3/2).
Finally, (69) yields
(103)∥w,tr2∥Wσ1,1/2(BR,R0t)≤c(t)(c0+∥w0∥Wσ1(BR,R0/2)+∥w1∥Wσ1-1/σ(BR,R0/2)).
To apply (102) and (103) in the r.h.s. of (101), we assume ν=2-(2/3)μ1, σ=3. In view of the assumptions of Lemma 6, we have that ν<1 so 2-(2/3)μ1<1. This implies that μ1>3/2. Hence, μ1∈(3/2,1+(1/2)5). Since ν=2-(2/3)μ1, assumptions of Lemma 6 imply that μ2<1/3+3/2-ν=11/6-2+(2/3)μ1, so μ2∈(5/6,1/2+(1/3)5)∩(0,3/2)=(5/6,3/2). Employing (102) and (103) in (101) yields (100). This concludes the proof.Remark 9.
In view of the expression ofw,tt(0) appeared in the assumptions of Lemma 4 we derive
(104)∥w,tt(0)∥L2,μ1(BR)≤c(∥w0∥Hμ12(BR)+∥w1∥Hμ12(BR)).
Sinceμ1-2∈(-1/2,(1/2)5-1) can be chosen positive, we see that all weights in the norms ∥w0∥Hμ12(BR), ∥w1∥Hμ12(BR) can be chosen as power functions with positive exponents.Assuming that the r.h.s. of (104) is finite assumptions (99) can be replaced by
(105)w1,w0∈Hμ12(BR),μ1∈(32,1+52),μ2≥μ1-1,μ2∈(56,32).Up to now there is no estimate forw,rr. Hence, to get the space regularity, we need the following.Lemma 10.
Let (105) hold. Let
(106)ψ,ϑϑ≥α1,|ψ,ϑ|≤c|ϑ|,|ψ,η|≤c|η|.
Then solutions to problems (11)–(13) satisfy
(107)∫0T∫0Rw,rr2r2μdrdt+∫0Rw,rr2r2μdr≤φ(c0,c1,c3,T),
where 2<μ<(1/3)(5+5), c0 is from (36), c1 from (40) and
(108)c3=∥w0∥Hμ2(BR)+∥w1∥Hμ2(BR).Proof.
Multiplying (11) by w,rrr2μ and integrating over BR, we obtain
(109)∫0Rw,ttw,rrr2μdr=∫0R1r2[(ψ,ϑ+μ0w,rt)r2],rw,rrr2μdr-∫0R(1rψ,η+μ0w,tr)w,rrr2μdr.
Continuing, we have(110)∫0R(ψ,ϑϑw,rr2+μ0w,rrtw,rr)r2μdr=∫0Rw,ttw,rrr2μdrvv-2∫0R(ψ,ϑ+μ0w,rt)w,rrr2μ-1drvv+∫0R(ψ,η+μ0w,t)w,rrr2μ-1dr.
Using (106), we obtain
(111)α1∫0Rw,rr2r2μdr+μ02ddt∫0Rw,rr2r2μdr≤ε∫0Rw,rr2r2μdr+c(1ε)∫0Rw,tt2r2μdrv+c∫0R(w,r2+w,rt2+w,t2+w2r-2)r2μ-2dr.
Integrating the inequality with respect to time, assuming thatε is sufficiently small, using that μ=1+ν, ν∈(0,3/2), and employing (40) with μ=ν and (102), (103) with σ=2, ν=2-(2/3)μ1, μ=(2/3)μ1+1, we obtain (101). Hence, μ∈(1,5/2)∩(2,5/3+5/3)=(2,(1/3)(5+5)). 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].Lemma 11.
Let the assumptions of Lemma3 be satisfied. Then
(112)|w(r,t)|≤R1/2-μ1-2μc1,μ∈(0,12),∥w,t∥L2(0,T;C1/2-μ/2(BR))≤cc1,μ∈(0,12),∥w∥B(0,T;C1/2-μ/2(BR))≤cc1,μ∈(0,12),∥w∥Lβ(0,R;C(β-1)/β(0,T))≤cc1,1<β<22μ+1,μ∈(0,12).


## 4. Existence

We prove the existence of solutions to problems (11)–(13) by the following method of successive approximations:
(113)w,tt(n+1)-μ01r2(w,rt(n+1)r2),r+μ01r2w,t(n+1)=1r2(ψ,ϑ(n)r2),r-1rψ,η(n),w(n+1)|t=0=w0,w,t(n+1)|t=0=w1,w,r(n+1)|r=R=0,n∈ℕ∪{0},
where ϑ(n)=wr(n), η(n)=w(n)/r, and we assume that the zero approximation is defined by
(114)w(0)=w0+tw1.The boundary condition (113)3 implies (13) because
(115)|∂ψ∂ϑ|≤c|ϑ|.First, we examine the linear problem(116)u,t-μ0u,rr-2μ0u,rr+μ0ur2=finBRT,u|t=0=u1inBR,u,r|r=R=0onSR.Lemma 12.
Assume thatf∈L2,ν(BRT), u1∈Hν1(BR), ν∈(2-(1/2)5,2+(1/2)5). Then there exists a solution to problem (116) such that u∈Hν2,1(BRT) and the estimate holds
(117)∥u∥Hν2,1(BRT)≤c(∥f∥L2,ν(BRT)+∥u1∥Hν1(BR)).Proof.
The existence can be proved by the Galerkin method. Therefore, we shall restrict the proof to show only estimate (117).
Multiplying (116)1 by ur2μ and integrating the result over BR yields
(118)12ddt∫BRu2r2μdr+μ0∫BRu,r2r2μdr+2(μ-1)μ0∫BRu,rur2μ-1dr+μ0limr→0(u,rur2μ)+μ0∫BRu2r2μ-2dr=∫BRfur2μdr.
Looking for solutions thatu,ru is finite at r=0, the last but one term on the l.h.s. of (118) vanishes for μ>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
(119)12ddt∫BRu2r2μdr+μ0∫BRu,r2r2μdr+μ0μ(3-2μ)∫BRu2r2μ-2dr+(1-μ)μ0u2r2μ-1|r=0=(1-μ)μ0u2(R,t)R2μ-1+∫BRfur2μdr.
Forμ∈(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
(120)∫0T∫0Ru2r2μ-2dr<∞.
Hence,u~rε near r=0, where ε>-μ+1/2. Then the last term on the l.h.s. of (119) vanishes. For μ>1, the last term on the l.h.s. of (119) vanishes under the assumption that we are looking for solutions such that u(r,t)~rε near r=0, ε>-μ+1/2.
Finally, we estimate the first term on the r.h.s. of (119). Let us recall the set
(121)BR,R0={r∈(0,R):0<R0<r<R}.
Then by imbedding, we get(122)|u(R,t)|2≤ε∫BR,R0u,r2dr+c(1ε)∫BR,R0u2dr≤ε′∫BRu,r2r2μdr+c(1ε′)∫BRu2r2μdr,
where ε′=ε/R02μ.
Using thatε′ is sufficiently small, we obtain
(123)12ddt∫BRu2r2μdr+μ0′∫BRu,r2r2μdr+μ0′μ(3-2μ)∫BRu2r2μ-2dr≤c∫BRu2r2μdr+(∫BRf2r2μdr)1/2(∫BRu2r2μdr)1/2≤c∫BRu2r2μdr+c∫BRf2r2μdr.
Integrating (123) with respect to time, we get the estimate
(124)∫BRu2r2μdr+μ0′∫BRtu,r2r2μdrdt+μ0μ(3-2μ)∫BRtu2r2μ-2drdt≤ect(∫BRtf2r2μdr+∫BRu12r2μdr)≡α1(t).
The above inequality implies the estimate forμ∈[0,3/2]. To increase the interval, we use the Hardy inequality
(125)|12-μ|1/2∫0Ru2r2μ-2dr≤∫0Ru,r2r2μdr.
Combining (124) and (125), we see that (124) gives an estimate for μ satisfying the inequality
(126)α|12-μ|2+μ(3-2μ)>0,α=μ0′μ0.
Solving (126), we see that μ∈((3-α-9-4α)/2(2-α),(3-α+9-4α)/2(2-α)). Since α is very close to 1, we can assume that
(127)μ∈(1-125,1+125).
Hence,[0,3/2]⊂(1-(1/2)5,1+(1/2)5).
Finally, forμ satisfying (127), we have estimate (124).
In the next step, we consider the parabolic problem(128)u,t-μ0u,rr=2μ0u,rr-μ0ur2+f,u|t=0=u1,u,r|r=R=0.
For solutions to problem (128), we have the inequality (see Lemma 1)
(129)∥u∥Hμ′2,1(BRT)≤c(∥u,rr∥L2,μ′(BRT)+∥ur2∥L2,μ′(BRT)vv+∥f∥L2,μ′(BRT)+∥u1∥Hμ′1(BR)∥u,rr∥L2,μ′(BRT)).
The first two norms on the r.h.s. of (129) can be estimated in view of (124). Then we obtain some restriction on μ′. We have
(130)∥u,rr∥L2,μ′(BRT)≤∥u,r∥L2,μ′-1(BRT)≤α1(T),
where the last inequality holds forμ′-1=μ, so μ′∈(2-(1/2)5,2+(1/2)5). Hence, μ′ can be chosen less than 1. Similarly, the inequalities
(131)∥ur2∥L2,μ′(BRT)≤∥u∥L2,μ′-2(BRT)≤α1(T)
hold forμ′-2=μ-1 so μ′-1=μ. Hence, the same restriction on μ′ as before is satisfied. Since the existence follows from the Galerkin method, Lemma 12 is proved.In view of Lemma12, we have the following.Lemma 13.
Assume thatw0∈Hν2(BR), w1∈Hν2(BR),ν∈(2,1+(1/2)5). Assume the Lipschitz continuity
(132)|ψ,ϑ(u)-ψ,ϑ(v)|≤c|u-v|,|ψ,η(u)-ψ,η(v)|≤c|u-v|.
Then there exists a solution to problems (11)–(13) such that w,t∈Hν2,1(BRT) and
(133)∥w,t∥Hν2,1(BRT)≤c(∥w0∥Hν2(BR)+∥w1∥Hν1(BR)).Proof.
Applying Lemma12 to problem (113) yields
(134)∥w,t(n+1)∥Hν2,1(BRT)≤c(∥w,rr(n)∥L2,ν(BRT)+∥w,r(n)∥L2,ν-1(BRT)+∥w(n)∥L2,ν-2(BRT)+∥w1∥Hν1(BR)),
where ν∈(2-(1/2)5,2+(1/2)5) and we used that
(135)|ψ,ϑϑ|≤c,|ψ,ϑ|≤c|ϑ|,|ψ,η|≤c|η|.
In view of the relation(136)w=∫0tw,t′dt′+w0,
we obtain from (134) the inequality
(137)∥w,t(n+1)∥Hν2,1(BRT)≤cT1/2∥w,t(n)∥Hν2,1(BRT)+c∥w0∥Hν2(BR)+c∥w1∥Hν1(BR).
Hence, for sufficiently smallT, we have
(138)∥w,t(n)∥Hν2,1(BRT)≤c(∥w0∥Hν2(BR)+∥w1∥Hν2(BR)),
for anyn∈ℕ.
To show convergence of the sequence{w(n)}n∈ℕ we introduce the differences
(139)W(n)=w(n)-w(n-1),
which are solutions to the problems
(140)W,tt(n+1)-μ0r2(W,rt(n+1)r2),r+μ0r2W,t(n+1)=1r2[(ψ,ϑ(n)-ψ,ϑ(n-1))r2],r-1r(ψ,η(n)-ψ,η(n-1)),W(n+1)|t=0=0,W,t(n+1)|t=0=0,W,rn+1|r=R=0.
To show convergence we multiply (140) by W,t(n+1)r2ν and integrate the result over BR. Then we have
(141)12ddt∫BR|W,t(n+1)|2r2νdr+μ0∫BR|W,tr(n+1)|2r2νdr+μ0ν(3-2ν)∫BR|W,t(n+1)|2r2ν-2dr=∫BR1r2[(ψ,ϑ(n)-ψ,ϑ(n-1))r2],rW,t(n+1)r2νdr-∫BR1r(ψ,η(n)-ψ,η(n-1))W,t(n+1)r2νdr≡I1+I2.
Integrating by parts inI1 yields
(142)I1=-∫BR(ψ,ϑ(n)-ψ,ϑ(n-1))W,tr(n+1)r2νdr-2(ν-1)∫BR(ψ,ϑ(n)-ψ,ϑ(n-1))W,t(n+1)r2ν-1dr≡I11+I12,
where
(143)|I11|≤ε1∫BR|W,rt(n+1)|2r2νdr+c(1ε1)∫BR|ψ,ϑ(n)-ψ,ϑ(n-1)|r2dr,
and the second integral in view of the assumption of the Lipschitz continuity is bounded by(144)c∫BR|W,r(n)|2r2νdr.
Next, we examine(145)|I12|≤c∫BR|ψ,ϑ(n)-ψ,ϑ(n-1)||W,t(n+1)|r2ν-1dr≤ε2∫BR|W,t(n+1)|2r2ν-2dr+c(1ε2)∫BR|W,r(n)|2r2νdr.
Finally, we estimateI2,
(146)|I2|≤c∫BR1r2|W(n)||W,t(n+1)|r2νdr≤ε3∫BR|W,t(n+1)|2r2ν-2dr+c(1ε3)∫BR|W(n)|2r2ν-2dr.
Using the above estimates in (141) and assuming that ε1-ε3 are sufficiently small, we obtain
(147)ddt∫BR|W,t(n+1)|2r2νdr+μ0′∫BR|W,rt(n+1)|2r2νdr+μ0ν(3-2ν)∫BR|W,t(n+1)|2r2ν-2dr≤c(∫BR|W,r(n)|2r2νdr+∫BR|W(n)|2r2ν-2dr),
whereμ0′<μ0 and the same considerations as in (125)–(127) are repeated. Integrating (147) with respect to time and employing the formula
(148)W(n)(t)=∫0tW,t(n)dt,
we derive from (147) the inequality
(149)∫BR|W,t(n+1)|2r2νdr+μ0′∫BRT|W,rt(n+1)|2r2νdrdt+μ0ν(3-2ν)∫BRT|W,t(n+1)|2r2ν-2drdt≤cT(∫BRT|W,rt(n)|2r2νdrdt+∫BRT|W,t(n)|2r2ν-2drdt).
Hence, we have that the sequence{w(n)}n∈ℕ converges and then we show the existence of solutions to problems (11)–(13) on a small interval (0,T). Since we have the estimate on the interval (0,T) with arbitrary T, the above considerations can be repeated step by step.
In view of Lemmas8 and 10 and Remark 9, the extension can be made under the restriction that ν∈(2,1+(1/2)5). This concludes the proof.Proof.
Proof of the Main Theorem. In view of assumptions of Lemmas8 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.



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*Source: 268505-2013-03-12.xml*