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

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.

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 time 𝑡, 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]): 𝜚𝑢_,𝑡𝑡=div𝜎+𝜚𝑓, (1) 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 𝑛⋅𝜎_𝑆=0, (2) where 𝑆=𝜕Ω, 𝑛 is the unit outward vector normal to 𝑆.

Moreover, we add the initial conditions 𝑢_𝑡=0=𝑢₀,𝑢_𝑡𝑡=0=𝑢₁. (3)

We will assume that 𝜎=𝜕𝐹(𝜀)𝜕𝜀+𝜇₀𝜀_𝑡, (4) 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 that Ω is a ball 𝐵_𝑅 with radius 𝑅 centered at the origin of the introduced Cartesian coordinates. We introduce the spherical coordinates 𝑟,𝜑,𝜗 by the relations 𝑥₁𝑥=𝑟cos𝜑sin𝜗,₂𝑥=𝑟sin𝜑sin𝜗,₃=𝑟cos𝜗. (5)

With these coordinates, we connect the orthonormal vectors 𝑒_𝑟=cos𝜑sin𝜗,sin𝜑sin𝜗,cos𝜗,𝑒_𝜗=cos𝜑cos𝜗,sin𝜑cos𝜗,−sin𝜗,𝑒_𝜑=−sin𝜑,cos𝜑,0. (6)

Then, we define 𝑢_𝑟=𝑢⋅𝑒_𝑟, 𝑢_𝜗=𝑢⋅𝑒_𝜗, 𝑢_𝜑=𝑢⋅𝑒_𝜑, 𝜀_𝑟𝑟=𝑒_𝑟⋅𝜀⋅𝑒_𝑟=𝑢_𝑟,𝑟, 𝜀_𝜑𝜑=𝑢_𝑟/𝑟, 𝜀_𝜗𝜗=𝑢_𝑟/𝑟. Since the spherically symmetric case is considered, we have 𝑢_𝜗=𝑢_𝜑=0.

To simplify the notation, we introduce 𝑤=𝑢_𝑟. (7)

Assuming 𝜚=1 and transforming (1) to the spherical coordinates, we obtain 𝑤_,𝑡𝑡=1𝑟²𝜎_𝑟𝑟𝑟²_,𝑟−1𝑟𝜎_𝜗𝜗+𝜎_𝜑𝜑+𝑓_𝑟, (8) where 𝜎_𝑟𝑟=𝜕𝐹𝜕𝜀_𝑟𝑟+𝜇₀𝜀_{𝑟𝑟,𝑡},𝜎_𝜗𝜗=𝜕𝐹𝜕𝜀_𝜗𝜗+𝜇₀𝜀_{𝜗𝜗,𝑡}𝜎_𝜑𝜑=𝜕𝐹𝜕𝜀_𝜑𝜑+𝜇₀𝜀_{𝜑𝜑,𝑡}. (9)

Let us introduce the quantity 𝐹(𝜀)=𝜓𝑤_,𝑟,𝜂, (10) where 𝜂=𝑤/𝑟.

Then, (8) takes the form 𝑤_,𝑡𝑡=1𝑟²𝜕𝜓𝜕𝑤_,𝑟+𝜇₀𝑤_,𝑟𝑡𝑟²_,𝑟−1𝑟𝜕𝜓𝜕𝜂+𝜇₀𝑤_,𝑡𝑟, (11)

and in view of (3), we have the initial conditions 𝑤_𝑡=0=𝑤₀,𝑤_,𝑡𝑡=0=𝑤₁, (12)

and in view of (2) and (9), the boundary condition 𝜕𝜓𝜕𝑤_,𝑟+𝜇₀𝑤_{,𝑟𝑡𝑆𝑅}=0, (13) where 𝑆_𝑅=𝜕𝐵_𝑅.

Global existence of regular solutions to problems (11) and (12) with the Dirichlet boundary condition 𝑤_𝑆𝑅=0 (14)is 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 constants 𝛼₁, 𝛼₂, 𝛽₁, 𝛽₂ such that 𝜕²𝜓₁𝜕𝜗²≥𝛼₁,𝜕²𝜓₂𝜕𝜂²≥𝛼₂,𝜕𝜓₁𝜕𝜗≤𝛽₁|𝜗|,𝜕𝜓₂𝜕𝜂≤𝛽₂𝜂, (15)(3)𝑤₀∈𝐻²_𝜇1(0,𝑅),𝑤₁∈𝐻²_𝜇1(0,𝑅), 𝜇₁∈(2,1+√5/2),(4)𝑤₁∈𝐿_2,𝜇2(0,𝑅),𝑤₀∈𝐻¹_𝜇2(0,𝑅), 𝜇₂∈(0,3/2), 𝜓₁≥𝛼₁𝜗², 𝜓₂≥𝛼₂𝜂², |𝜓_1,𝜗|≤𝛽₁|𝜗|, |𝜓_2,𝜂|≤𝛽₂|𝜂|,(5)𝑤₁∈𝐿_2,𝜇3(0,𝑅), 𝑤₀∈𝐻¹_𝜇3(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 𝑤_,𝑡𝑡∈𝐵0,𝑇;𝐿_2,𝜇1(0,𝑅),𝑤_,𝑡∈𝐵0,𝑇;𝐻¹_𝜇1(0,𝑅),𝑤∈𝐵0,𝑇;𝐻²_𝜇1(0,𝑅),𝑤_,𝑡𝑡∈𝐿₂0,𝑇;𝐻¹_𝜇1(0,𝑅). (16)

(2) Let Assumptions (1) and (4) hold, then 𝑤_,𝑡∈𝐵0,𝑇;𝐿_2,𝜇2(0,𝑅)∩𝐿₂0,𝑇;𝐻¹_𝜇2(0,𝑅),𝑤∈𝐵0,𝑇;𝐻¹_𝜇2(0,𝑅). (17)(3) Let Assumptions (1), (4), and (5) be satisfied. Then 𝑤∈𝐿_∞((0,𝑅)×(0,𝑇)),𝑤∈𝐵0,𝑇;𝐶^{1/2−𝜇3/2}(0,𝑅),𝑤∈𝐿_𝛽0,𝑇;𝐶^{(𝛽−1)/𝛽}(20,𝑇),𝛽∈1,2𝜇₃,𝑤+1_,𝑡∈𝐿₂0,𝑇;𝐶^{1/2−𝜇3/2}(0,𝑅). (18)

Our paper is organized as follows. In Section 1, 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 Section 4 the existence of the global solution for nonsmall data of problems (11)–(13) is proved.

2. Notation and Auxiliary Results

By 𝑐 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 mark 𝑤_,𝑡=𝜕_𝑡𝑤, 𝑤_,𝑟=𝜕_𝑟𝑤, and so on.

By 𝐵(𝐼) we denote the space of bounded functions on the interval 𝐼.

By 𝐻^𝑘_𝜇(0,𝑅), 𝜇∈ℝ, 𝑘∈ℕ∪{0}, we denote a weighted Sobolev space with the finite norm ‖𝑢‖_𝐻^𝑘_𝜇(0,𝑅)=^𝑘_𝛼=0^𝑅₀𝜕^𝛼_𝑟𝑢²𝑟^{2(𝜇−𝑘+|𝛼|)}𝑑𝑟^1/2 (19)

and 𝐿_2,𝜇(0,𝑅)=𝐻⁰_𝜇(0,𝑅).

By 𝐶^𝛼(𝐼), 𝛼∈(0,1) we denote the Hölder space with the finite norm ‖𝑢‖_𝐶^𝛼_(𝐼)=sup_𝜏∈𝐼|𝑢(𝜏)|+sup_{𝜏′,𝜏′′∈𝐼}𝑢𝜏^′−𝑢𝜏^′′𝜏^′−𝜏^′′𝛼. (20)

Next, we recall the Hardy inequality (see [14, Chapter 1, Section  2.15]) 1𝑝^′−𝜇^𝑝∞₀𝑟^{𝑝(𝜇−1)}𝑓^𝑝𝑑𝑟≤^∞₀𝑟^𝑝𝜇𝑓_,𝑟^𝑝𝑑𝑟, (21) where 1/𝑝+1/𝑝^′=1, 𝜇∈ℝ, and𝜇≠1/𝑝^′.

The inequality holds also for functions with compact support. Assuming that supp𝑓⊂[0,𝑅], we introduce 𝐹(𝑥)=∫^∞_𝑥𝑓(𝑦)𝑑𝑦 and repeat the proof from [14, Chapter 1, Section  2.15].

From [15, Chapter 2, Section  3], we have the imbedding ‖𝑢‖_{𝐿𝑞(0,𝑇;𝐿𝑝(0,𝑅))}≤𝑐‖𝑢‖_{𝐿∞(0,𝑇;𝐿2(0,𝑅))}+∇_𝑟𝑢_{𝐿2(0,𝑇;𝐿2(0,𝑅))}, (22) where 1/𝑝+2/𝑞≥1/2.

Finally, we consider the problem 𝑢_,𝑡−𝑢_,𝑟𝑟=𝑓,𝑢_𝑡=0=𝑢₀,𝑢_{,𝑟𝑟=𝑅}=0. (23)

To examine nonstationary problem (23), we need anisotropic weighted Sobolev spaces 𝑉^2,1_𝑝,𝜈((0,𝑅)×(0,𝑇)), 𝑝∈(1,∞), 𝜈∈ℝ, of functions with the finite norm ‖𝑢‖_𝑉^2,1_{𝑝,𝜈((0,𝑅)×(0,𝑇))}=_{𝛼+2𝑎≤2}^𝑇₀^𝑅₀𝜕^𝛼_𝑟𝜕^𝑎_𝑡𝑢^𝑝𝑟^{𝑝(𝜈−2+𝛼+2𝑎)}𝑑𝑟𝑑𝑡^1/𝑝. (24)

Spaces 𝑉²_𝑝,𝜈(0,𝑅) appropriate for elliptic problems were introduced in [16]. Moreover, we assume that 𝐿_𝑝,𝜈(0,𝑅)=𝑉⁰_𝑝,𝜈(0,𝑅).

The following result is valid.

Lemma 1. We assume that 𝑓∈𝐿_𝑝,𝜈((0,𝑅)×(0,𝑇)), 𝑢₀∈𝑉^2−2/𝑝_𝑝,𝜈(0,𝑅). Then there exists a solution to problem (23) such that 𝑢∈𝑉^2,1_𝑝,𝜈((0,𝑅)×(0,𝑇)) and ‖𝑢‖_𝑉^2,1_{𝑝,𝜈((0,𝑅)×(0,𝑇))}≤𝑐𝑓_{𝐿𝑝,𝜈((0,𝑅)×(0,𝑇))+‖𝑢0‖𝑉2−2/𝑝𝑝,𝜈(0,𝑅)}. (25)

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, Lemma 1 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 space 𝐵^𝑙_𝑝,𝜈(𝐵_𝑅), 𝑙∈ℝ₊, 𝑝∈(1,∞), 𝜈∈ℝ, ‖𝑢‖_𝐵^𝑙_{𝑝,𝜈(𝐵𝑅)}=‖𝑢‖_𝐵^{•𝑙−[𝑙]}_{𝑝,𝜈(𝐵𝑅)}+‖𝑢‖_𝑉^[𝑙]_{𝑝,𝜈(𝐵𝑅)}, (26) where [𝑙] is the integer part of 𝑙, ‖𝑢‖_𝐵^•𝛼_{𝑝,𝜈(𝐵𝑅)}=^𝑅₀^𝑅₀𝑢𝑟₁𝑟^𝜈₁−𝑢𝑟₂𝑟^𝜈₂^𝑝𝑟₁−𝑟₂^1+𝑝𝛼𝑑𝑟₁𝑑𝑟₂^1/𝑝, (27) where 𝛼∈(0,1) and ‖𝑢‖_𝑉^𝑘_{𝑝,𝜈(𝐵𝑅)}=_𝛼≤𝑘^𝑅₀𝜕^𝛼_𝑟𝑢^𝑝𝑟^{𝑝(𝜈−𝑘+𝛼)}𝑑𝑟^1/𝑝, (28)

for 𝑘 integer.

𝐵(𝐵_𝑅×(0,𝑇)) is the space of bounded functions.

𝐶^𝛼(𝐵_𝑅), 𝐶^𝛼,𝛽(𝐵_𝑅×[0,𝑇]), 𝛼,𝛽∈(0,1), are the Hölder spaces with the finite norms ‖𝑢‖_𝐶^𝛼_(𝐵𝑅)=‖𝑢‖_{𝐵(𝐵𝑅)}+sup_{𝑟1,𝑟2∈𝐵𝑅}𝑢𝑟₁−𝑢𝑟₂𝑟₁−𝑟₂^𝛼,‖𝑢‖_𝐶^𝛼,𝛽_{(𝐵𝑅×[0,𝑇])}=‖𝑢‖_{𝐵(𝐵𝑅×[0,𝑇])}+sup_{[]𝑡∈0,𝑇}sup_{𝑟1,𝑟2∈𝐵𝑅}𝑢𝑟₁,𝑡−𝑢𝑟₂,𝑡𝑟₁−𝑟₂^𝛼+sup_{𝑟∈𝐵𝑅}sup_{𝑡′,𝑡′′∈[]0,𝑇}𝑢𝑟,𝑡^′−𝑢𝑟,𝑡^′′𝑡^′−𝑡^′′𝛽, (29) where 𝐵_𝑅 is the closure of 𝐵_𝑅.

By 𝐿^𝑙_𝑞(0,𝑇;𝑊^𝑘_𝑝(𝐵_𝑅)), 𝑙,𝑘∈ℕ∪{0}, 𝑝,𝑞∈[1,∞], we denote a space of functions with the finite norm 𝜕^𝑙_𝑡𝑢_{𝐿𝑞(0,𝑇;𝑊𝑘𝑝(𝐵𝑅))}. (30)

We introduce also the Sobolev spaces 𝑊^𝑘_𝑝𝐵_𝑅=𝑢=𝑢(𝑥)∶‖𝑢‖_𝑊^𝑘_{𝑝(𝐵𝑅)}=_𝑙≤𝑘𝜕^𝑙_𝑟𝑢_{𝐿𝑝(𝐵𝑅)}<∞, (31) where 𝑘∈ℤ₊∪{0}, 𝑝∈[1,∞], 𝑊_𝑝^𝑘,𝑘/2𝐵^𝑇_𝑅=𝑢=𝑢(𝑟,𝑡)∶‖𝑢‖_𝑊𝑝^𝑘,𝑘/2_{(𝐵𝑇𝑅)}=_{𝑙+2𝑎≤𝑘}𝜕^𝑙_𝑟𝜕^𝑎_𝑡𝑢_{𝐿𝑝(𝐵𝑇𝑅)}<∞, (32) where 𝑘,𝑘/2∈ℤ₊∪{0}, 𝑝∈[1,∞], 𝐵^𝑇_𝑅=𝐵_𝑅×(0,𝑇), 𝐵_𝑅={𝑟∈ℝ₊∶𝑟<𝑅}, 𝑊^𝑘_𝑝(0,𝑇)=𝑢=𝑢(𝑡)∶‖𝑢‖_𝑊^𝑘_𝑝(0,𝑇)=_𝑙≤𝑘𝜕^𝑙_𝑡𝑢_{𝐿𝑝(0,𝑇)}<∞, (33)

𝑘∈ℤ₊∪{0}, 𝑝∈[0,1]. For 𝑘=0, we have 𝑊⁰_𝑝(𝐵_𝑅)=𝐿_𝑝(𝐵_𝑅), 𝑊_𝑝^0,0(𝐵^𝑇_𝑅)=𝐿_𝑝(𝐵^𝑇_𝑅), 𝑊⁰_𝑝(0,𝑇)=𝐿_𝑝(0,𝑇), 𝐵^𝛼_𝑝(𝐵_𝑅)=𝐵^𝛼_𝑝,𝑝(𝐵_𝑅) is the Besov space introduced in [14, Chapter 4, Section  18] and 𝑊^𝛼_𝑝(𝐵_𝑅) is the Sobolev-Slobodetsky space, where 𝛼∉ℤ₊,𝑝∈[1,∞].

Moreover, 𝑉^𝑘_𝑝,0𝐵_𝑅⊂𝑊^𝑘_𝑝𝐵_𝑅,𝑉^𝑘,𝑘/2_𝑝,0𝐵^𝑇_𝑅⊂𝑊_𝑝^𝑘,𝑘/2𝐵^𝑇_𝑅. (34)

Finally 𝐻^𝑘(𝐵_𝑅)=𝑊^𝑘₂(𝐵_𝑅), 𝐻^𝑘,𝑘/2(𝐵^𝑇_𝑅)=𝑊₂^𝑘,𝑘/2(𝐵^𝑇_𝑅), 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 _𝐵𝑅12𝑤²₁+𝜓𝑤_0,𝑟,𝑤₀𝑟𝑟²𝑑𝑟≡𝑐²₀<∞, (35) where 𝜓 is a positive function differentiable with respect to its arguments introduced by (10).
Then, solutions to problems (11)–(13) satisfy the estimate _𝐵𝑅12𝑤²_,𝑡+𝜓𝑤_,𝑟,𝑤𝑟𝑟²𝑑𝑟+_𝐵𝑡𝑅𝑤²_,𝑟𝑡+𝑤²_,𝑡𝑟²𝑟²𝑑𝑟𝑑𝑡^′=_𝐵𝑅12𝑤²₁+𝜓𝑤_0,𝑟,𝑤₀𝑟𝑟²𝑑𝑟=𝑐²₀,𝑓𝑜𝑟𝑎𝑛𝑦𝑡>0. (36)

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 𝜓₁(𝜗)≥𝛼₁𝜗²,𝜓₂𝜂≥𝛼₂𝜂²,𝜕𝜓₁𝜕𝜗≤𝛽₁|𝜗|,𝜕𝜓₂𝜕𝜂≤𝛽₂𝜂. (37)(38)
Finally, assume that 𝑤₁∈𝐿_2,𝜇(0,𝑅),𝑤₀∈𝐻¹_𝜇()30,𝑅,𝜇∈0,2. (39)
Then solutions to problems (11)–(13) satisfy the following estimate: 𝑤_{,𝑡𝐿∞(0,𝑡;𝐿2,𝜇(𝐵𝑅))}+‖𝑤‖_{𝐿∞(0,𝑡;𝐻}¹_{𝜇(𝐵𝑅))}+𝑤_{,𝑡𝐿2(0,𝑡;𝐻1𝜇(𝐵𝑅))}≤𝑐𝑅,𝜇,𝑡𝑤_{1𝐿2,𝜇(𝐵𝑅)}+𝑤_{0𝐻1𝜇(𝐵𝑅)}≤𝑐₁𝑅,𝜇,𝑇, (40) 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 𝛼₁, 𝛼₂, 𝛽₁, 𝑐₁, and 𝑐₂ such that 𝜓_1,𝜗𝜗≥𝛼₁,𝜓_2,𝜂𝜂≥𝛼₂,𝜓_1,𝜗𝜗≤𝛽₁,𝜕³_𝜗𝜓₁≤𝑐₁,𝜕³_𝜂𝜓₂≤𝑐₂. (41)
Assume also that 𝜂_,𝑡,𝜗_,𝑡∈𝐿₃0,𝑇;𝐿_3,2/3𝜇𝐵_𝑅,𝑤₁∈𝐻¹_𝜇𝐵_𝑅,𝑤_,𝑡𝑡(0)∈𝐿_2,𝜇𝐵_𝑅, (42) where 𝑤_,𝑡𝑡(0)=1𝑟²𝜓_,𝜗𝑤_0,𝑟,𝑤₀𝑟+𝜇₀𝑤_1,𝑟𝑟²_,𝑟−1𝑟𝜓_,𝜂𝑤_0,𝑟,𝑤₀𝑟+𝑤₁𝑟. (43)
Then solutions to (11)–(13) satisfy the following inequality: 𝑤_{,𝑡𝑡𝐿∞(0,𝑡;𝐿2,𝜇(𝐵𝑅))}+𝑤_{,𝑡𝐿∞(0,𝑡;𝐻1𝜇(𝐵𝑅))}+𝜇₀𝑤_{,𝑡𝑡𝐿2(0,𝑡;𝐻1𝜇(𝐵𝑅))}≤𝑐(𝑡)𝜗_,𝑡𝐿^3/2_{30,𝑡;𝐿3,2/3𝜇𝐵𝑅}+𝜂_,𝑡𝐿^3/2_{30,𝑡;𝐿3,2/3𝜇𝐵𝑅}+𝑐(𝑡)𝑐₂, (44) where 𝜇∈(0,1+(1/2)√5), 𝑡≤𝑇 and 𝑐₂=𝑤_,𝑡𝑡(0)_{𝐿2,𝜇(𝐵𝑅)}+𝑤_,𝑡(0)_{𝐻1𝜇(𝐵𝑅).} (45)

Remark 5. In general constants 𝛼_𝑖 and 𝛼_𝑖, 𝑖=1,2, are different. Let us consider the example 𝜓₁(𝜗)=𝛼₁𝜗²+𝑎1+𝜗², (46) where 𝑎 is a positive constant. Then we have 𝜓₁(𝜗)≥𝛼₁𝜗². (47)
Continuing, we derive 𝜓_1,𝜗𝜗=2𝛼₁+𝑎2𝜗²−11+𝜗^25/2. (48)
For 𝜗²≥1/2, we obtain 𝜓_1,𝜗𝜗≥2𝛼₁. But for 𝜗²<1/2, we have 𝜓_1,𝜗𝜗≥2𝛼₁−𝑎1+𝜗^25/2≥2𝛼₁−𝑎≡𝛼₁. (49)
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 ^𝑇_0𝐵𝑅𝑤_,𝑟𝑡³𝑟^2𝜇𝑑𝑟𝑑𝑡^1/2+^𝑇_0𝐵𝑅𝑤_,𝑡𝑟³𝑟^2𝜇𝑑𝑟𝑑𝑡^1/2≡𝐼. (50)

Hence, by the Hardy inequality (see [14, Chapter 1, Section  2.15]), the above expression is bounded by 𝐼≤𝑐^𝑇_0𝐵𝑅𝑤_,𝑟𝑡³𝑟^2𝜇𝑑𝑟𝑑𝑡^1/2≡𝐼₁. (51)

To estimate 𝐼₁, we need the Pego transformation 𝑝(𝑟,𝑡)=^𝑟₀𝑤_,𝑡𝑟^′,𝑡𝑟^′2𝑑𝑟^′,𝑞(𝑟,𝑡)=𝜇₀𝑤_,𝑟(𝑟,𝑡)𝑟²−𝑝(𝑟,𝑡). (52)

Hence, 𝑤_,𝑟𝑡𝑟²=1𝜇₀𝑞_,𝑡+𝑝_,𝑡. (53)

To calculate the above expression we need problems for 𝑝 and 𝑞. From Lemma 3.4 from [13], we have the following problems for 𝑝 and 𝑞: 𝑝_,𝑡−𝜇₀𝑝_,𝑟𝑟=𝜓_,𝜗𝑟²−^𝑟₀𝜓_,𝜂+𝜇₀𝑤_,𝑡𝑟𝑟𝑑𝑟−2𝜇₀𝑤_,𝑡𝑝𝑟,_𝑡=0=^𝑟₀𝑤₁(𝑟)𝑟²𝑑𝑟≡𝑝₀(𝑟),𝑝_{,𝑟𝑟=𝑅}=𝑤_,𝑡(𝑅,𝑡)𝑅² (54)𝑞_,𝑡=−𝜓_,𝜗𝑟²+^𝑟₀𝜓_,𝜂+𝜇₀𝑤_,𝑡𝑟𝑞𝑟𝑑𝑟,_𝑡=0=𝜇₀𝑤_0,𝑟𝑟²−^𝑟₀𝑤₁(𝑟)𝑟²𝑑𝑟≡𝑞₀(𝑟). (55)

Lemma 6. Let the assumptions of Lemma 3 be satisfied. Let 𝑤_,𝑟∈𝐿_{𝜎,2−𝜈}(𝐵^𝑇_𝑅), 𝑤_,𝑡,𝜕_𝑡^2−1/𝜎𝑤∈𝐿_𝜎(0,𝑇;𝐿_2,1(𝐵_𝑅)), 𝜕_𝑡^2−1/𝜎 means the fractional derivative. Then solutions to problems (54) and (55) satisfy 𝑝_{𝑊2,1𝜎,−𝜈(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎,2−𝜈(𝐵𝑇𝑅)}+𝑐𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅)}+𝑐𝑐₁,𝑞_{,𝑡𝐿𝜎,−𝜈(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎,2−𝜈(𝐵𝑇𝑅)}+𝑐𝑐₁, (56)(57) where 𝑐₁ is introduced in (40), 𝐵_𝑅,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), 𝑝_{𝑊2,1𝜎,−𝜈(𝐵𝑇𝑅)}≤𝑐𝑤_,𝑟𝑟²_{𝐿𝜎,−𝜈(𝐵𝑇𝑅)}+^𝑟₀|𝑤|+𝑤_,𝑡𝑑𝑟_{𝐿𝜎,−𝜈(𝐵𝑇𝑅)}+𝑤_,𝑡𝑟_{𝐿𝜎,−𝜈(𝐵𝑇𝑅)}+𝑝_{0𝑊2−2/𝜎𝜎,−𝜈(𝐵𝑅)}+𝑤_,𝑡(𝑅,𝑡)𝑅²_{𝑊1−1/𝜎,1/2−1/2𝜎𝜎,−𝜈(𝑆𝑇𝑅)}. (58)
Now, we examine the terms from the r.h.s. of (58).
The first norm on the r.h.s. of (58) equals _𝐵𝑇𝑅𝑤_,𝑟^𝜎𝑟^{(2−𝜈)𝜎}𝑑𝑟𝑑𝑡^1/𝜎. (59)
Applying the Hölder inequality, the second integral on the r.h.s. of (58) is bounded by _𝐵𝑇𝑅^𝑟₀𝑟^−2𝜈1𝑑𝑟^1/2×^𝑟₀(𝑤²+𝑤²_,𝑡)𝑟^2𝜈1𝑑𝑟^1/2𝜎𝑟^−𝜈𝜎𝑑𝑟𝑑𝑡^1/𝜎≤𝑐𝑐₁, (60)
where the last inequality holds in virtue of Lemma 3 and under the assumption 𝜈₁1+𝜈<𝜎+12,𝜈₁=𝜇−1, (61) where 𝜇∈(0,3/2) is introduced in Lemma 3. Hence, 1𝜇+𝜈<𝜎+32. (62)
The third integral on the r.h.s. of (58) we express in the form _𝐵𝑇𝑅𝑤_,𝑡^𝜎𝑟^{(1−𝜈)𝜎}𝑑𝑟𝑑𝑡^1/𝜎≡𝐽₁. (63)
Assuming 𝜈<1, setting 𝜇=1−𝜈>0, and recalling imbedding (22) and Lemma 3, we obtain for 𝜎≤6 the estimate 𝐽₁≤𝑐sup_𝑡𝐵𝑅(𝑤𝑟^𝜇)_,𝑡²𝑑𝑟^1/2+𝑐_𝐵𝑇𝑅𝜕_𝑟(𝑤𝑟^𝜇)_,𝑡²𝑑𝑟𝑑𝑡^1/2≤𝑐𝑐₁. (64)
Finally, the last term on the r.h.s. of (58) is bounded by 𝑤_,𝑡𝑟²_{𝑊𝜎𝑇𝑅)}^{1−1/𝜎,1/2−1/2𝜎}_(𝑆≤𝑐𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅)}. (65)
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 𝑤_{,𝑟𝑡𝐿𝜎,2−𝜈(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎,2−𝜈(𝐵𝑇𝑅)}+𝑐(𝑇)𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅)}+𝑐𝑐₁. (66)

Applying the equality 𝑤_,𝑟(𝑡)=^𝑡₀𝑤_,𝑟𝑡′𝑑𝑡^′+𝑤_,𝑟(0) (67)

in the first term on the r.h.s. of (66) and using the Gronwall lemma, we get 𝑤_,𝑟𝑡‖_{𝐿𝜎,2−𝜈(𝐵}^𝑇_𝑅)≤𝑐(𝑇)𝑤_,𝑡𝑟²‖_𝑊𝜎^1,1/2_{(𝐵𝑇0𝑅,𝑅)}+𝑐(𝑇)𝑐₁. (68)

Now, we have to estimate the norm on the r.h.s. of (68). For the purpose we need the following.

Lemma 7. Let 𝑤 be a solution to (11)–(13). Assume that 𝜓=𝜓₁(𝜗)+𝜓₂(𝜂) and (37), (38) are satisfied. Assume that 𝑤₁∈𝑊_𝜎^1−1/𝜎(𝐵_𝑅), 𝑤₀∈𝑊¹_𝜎(𝐵_𝑅,𝑅0/2), where 𝜎∈(1,∞), 0<𝑅₀<𝑅. Then 𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅)}≤𝑐(𝑇)𝑐₀+𝑤_{0𝑊1𝜎(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−1/𝜎_(𝐵𝑅), (69) where 𝑐₀ is introduced in (36).

Proof. Let us introduce a smooth function 𝜁=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 in 𝐵_𝑅,𝑅0 for solutions to problems (11)–(13). Multiplying (11) by 𝜁²𝑤_,𝑡𝑟² and integrating over 𝐵_𝑅 yields _𝐵𝑅𝑤_,𝑡𝑡𝜁²𝑤_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝜓_𝜗+𝜇₀𝑤_,𝑟𝑡𝜁²𝑤_,𝑡,𝑟𝑟²+𝑑𝑟_𝐵𝑅𝜓_,𝜂+𝜇₀𝜂_,𝑡𝜁²𝑤_,𝑡𝑟𝑑𝑟=0. (70)
Introducing the notation 𝑤=𝑤𝜁, (71)
we obtain 12𝑑𝑑𝑡_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝜓_,𝜗𝜗_,𝑡+𝜓_,𝜂𝜂_,𝑡𝜁²𝑟²𝑑𝑟+2_𝐵𝑅𝜓_,𝜗𝑤_,𝑡𝜁𝜁_,𝑟𝑟²𝑑𝑟+𝜇_0𝐵𝑅𝑤_,𝑟𝑡𝜁²𝑤_,𝑡,𝑟+𝑤_,𝑡𝑟²𝑟²𝑑𝑟=0. (72)
Continuing, we have 𝑑1𝑑𝑡2_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝜓𝜁²𝑟²𝑑𝑟+𝜇_0𝐵𝑅𝑤²_,𝑟𝑡𝜁²+𝑤²_,𝑡𝑟²𝑟²𝑑𝑟=−2𝜇_0𝐵𝑅𝑤_,𝑟𝑡𝑤_,𝑡𝜁𝜁_,𝑟𝑟²𝑑𝑟−2_𝐵𝑅𝜓_,𝜗𝑤_,𝑡𝜁𝜁_,𝑟𝑟²𝑑𝑟. (73)
Applying the Hölder and the Young inequalities to the r.h.s. of the above equality and using (38) yields 𝑑1𝑑𝑡2_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝜓𝜁²𝑟²+𝜇𝑑𝑟₀2_𝐵𝑅𝑤²_,𝑟𝑡𝜁²+𝑤²_,𝑡𝑟²𝑟²𝑑𝑟≤𝑐𝜇_0𝐵𝑅𝑤²_,𝑡𝜁²_,𝑟𝑟²𝑑𝑟+𝑐_𝐵𝑅𝑤²_,𝑟𝜁²_,𝑟𝑟²𝑑𝑟. (74)
Using that 𝑤²_,𝑟𝑡𝜁²=(𝑤_,𝑟𝑡−𝑤_,𝑡𝜁_,𝑟)² in the above inequality implies 𝑑1𝑑𝑡2_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝜓𝜁²𝑟²𝑑𝑟+𝜇_0𝐵𝑅𝑤²_,𝑟𝑡+𝑤²_,𝑡𝑟²𝑟²𝑑𝑟≤𝑐𝜇_0𝐵𝑅𝑤²_,𝑡𝜁²_,𝑟𝑟²𝑑𝑟+𝑐_𝐵𝑅𝑤²_,𝑟𝜁²_,𝑟𝑟²𝑑𝑟. (75)
Integrating the above inequality with respect to time and using (37), we have 12_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅𝛼₁𝑤²_,𝑟𝜁²+𝛼₂𝑤²𝑟²𝑟²𝑑𝑟+𝜇_{0𝐵𝑡𝑅}𝑤²_,𝑟𝑡+𝑤²_,𝑡𝑟²𝑟²𝑑𝑟𝑑𝑡^′≤𝑐𝜇_{0𝐵𝑡𝑅}𝑤²_,𝑡𝜁²_,𝑟𝑟²𝑑𝑟𝑑𝑡^′+𝑐_𝐵𝑡𝑅𝑤²_,𝑟𝜁²_,𝑟𝑟²𝑑𝑟𝑑𝑡^′+12_𝐵𝑅𝑤₁𝑟²𝑑𝑟+_𝐵𝑅𝜓𝑤_0,𝑟,𝑤₀𝑟𝑟²𝑑𝑟≡𝑐_𝐵𝑡𝑅𝑤²_,𝑡+𝑤²_,𝑟𝜁²_,𝑟𝑟²𝑑𝑟𝑑𝑡^′+𝑐𝑐²₀. (76)
Since 𝑤²_,𝑟≤2((𝑤_,𝑟𝜁)²+𝑤²𝜁²_,𝑟), the above inequality yields 12_𝐵𝑅𝑤²_,𝑡𝑟²𝑑𝑟+_𝐵𝑅12𝛼₁𝑤²_,𝑟𝑟²+𝛼₂𝑤²𝑑𝑟+𝜇_{0𝐵𝑡𝑅}𝑤²_,𝑟𝑡𝑟²+𝑤²_,𝑡𝑑𝑟𝑑𝑡^′≤𝑐_𝐵𝑡𝑅𝑤²_,𝑡+𝑤²_,𝑟+𝑤²𝜁²_,𝑟𝑟²𝑑𝑟𝑑𝑡^′+𝑐𝑐²₀≤𝑐𝑐²₀, (77) where (36) was used again.
Now, we introduce the following Pego transformation 𝑝(𝑟,𝑡)=^𝑟₀𝑤_,𝑡𝑟^′,𝑡𝑟^′2𝑑𝑟^′,𝑞(𝑟,𝑡)=𝜇₀𝑤_,𝑟𝑟²−𝑝(𝑟,𝑡). (78)
Moreover, we have the initial-boundary value problem for 𝑝, 𝑝_,𝑡−𝜇₀𝑝_,𝑟𝑟=^𝑟₀𝑤_,𝑡𝑡𝑟²𝑑𝑟−𝜇₀𝑤_,𝑡𝑟²_,𝑟𝑝≡𝐼,_𝑡=0=^𝑟₀𝑤₁𝑟²𝑑𝑟,𝑝(𝑅,𝑡)=^𝑅₀𝑤_,𝑡𝑟²𝑑𝑟. (79)
The nonhomogeneous Dirichlet boundary condition is not convenient so we introduce the new function 𝑝^′=𝑝−𝑝(𝑅,𝑡), (80) which is a solution to the problem 𝑝^′_,𝑡−𝜇₀𝑝^′_,𝑟𝑟=𝐼−^𝑅₀𝑤²_,𝑡𝑡𝑟²𝑑𝑟≡𝐼₁,𝑝^′_𝑡=0=^𝑟₀𝑤₁𝑟²𝑑𝑟−^𝑅₀𝑤₁𝑟²𝑑𝑟≡𝑝^′₀,𝑝^′_𝑟=𝑅=0. (81)We have to calculate 𝐼₁. For this purpose, we need the expressions which follow from (11) multiplied by 𝜁𝑤_,𝑡𝑡=1𝑟²𝜓_,𝜗𝜁+𝜇₀𝑤_,𝑟𝑡𝜁𝑟²_,𝑟−𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_,𝑟−1𝑟𝜓_,𝜂𝜁+𝜇₀𝑤_,𝑡𝑟, (82)
and from (13) multiplied by 𝜁𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_𝑟=𝑅=0. (83)
Using (82) and (83) in 𝐼₁ yields 𝐼₁=^𝑟₀𝜓_,𝜗𝜁+𝜇₀𝑤_,𝑟𝑡𝜁𝑟²_,𝑟−𝑑𝑟_𝑟0𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_,𝑟𝑟²−𝑑𝑟_𝑟0𝜓_,𝜂𝜁𝑟+𝜇₀𝑤_,𝑡𝑑𝑟−𝜇₀𝑤_,𝑡𝑟²_,𝑟−^𝑅₀𝜓_,𝜗𝜁+𝜇₀𝑤_,𝑟𝑡𝜁𝑟²_,𝑟+𝑑𝑟_𝑅0𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_,𝑟𝑟²+𝑑𝑟_𝑅0𝜓_,𝜂𝜁𝑟+𝜇₀𝑤_,𝑡𝑑𝑟=𝜓_,𝜗𝜁𝑟²+𝜇₀𝑤_,𝑟𝑡𝑟²−𝜇₀𝑤_,𝑡𝜁_,𝑟𝑟²+^𝑅_𝑟𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_,𝑟𝑟²𝑑𝑟+^𝑅_𝑟𝜓_,𝜂𝜁+𝜇₀𝑤_,𝑡𝑑𝑟−𝜇₀𝑤_,𝑟𝑡𝑟²+2𝑟𝑤_,𝑡=𝜓_,𝜗𝜁𝑟²−𝜇₀𝑤_,𝑡𝜁_,𝑟𝑟²−2𝜇₀𝑟𝑤_,𝑡+^𝑅_𝑟𝜓_,𝜗𝜁_,𝑟𝑟²+𝜓_,𝜂𝜁𝑟+𝜇₀𝑤_,𝑡𝑑𝑟−𝜇₀^𝑅_𝑟𝑤_,𝑡𝜁_,𝑟𝑟²_,𝑟𝑑𝑟. (84)
For solutions to problem (81), we have 𝑝^′_{𝑊𝜎2,1(𝐵𝑇𝑅)}≤𝑐𝐼_{1𝐿𝜎(𝐵𝑇𝑅)}+𝑝^′_{0𝑊𝜎2−2/𝜎(𝐵𝑅)}. (85)
Hence, 𝑝_{𝑊𝜎2,1(𝐵𝑇𝑅)}≤𝑐𝐼_{1𝐿𝜎(𝐵𝑇𝑅)}+𝑝_,𝑡(𝑅,𝑡)_{𝐿𝜎(𝐵𝑇𝑅)}+𝑝^′_{0𝑊𝜎2−2/𝜎(𝐵𝑅)}. (86)
From the expression of 𝐼₁, we have 𝐼_{1𝐿𝜎(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{,𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+‖𝑤‖_{𝐿𝜎(𝐵}^𝑇_{0𝑅,𝑅/2)}. (87)
In view of (82) and (83), we obtain 𝑝_,𝑡(𝑅,𝑡)=^𝑅₀𝑤_,𝑡𝑡𝑟²=𝑑𝑟_𝑅0𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝜁_,𝑟𝑟²𝑑𝑟−^𝑅₀𝜓_,𝜂𝜁𝑟+𝜇₀𝑤_,𝑡𝑑𝑟. (88)
Hence, 𝑝_,𝑡(≤𝑅,𝑡)_𝑅0𝜓_,𝜗𝜁_,𝑟𝑟²+𝜓_,𝜂𝜁𝑟+𝜇₀𝑤_,𝑡+𝑑𝑟_𝑅0𝜇₀𝑤_,𝑡𝜁_,𝑟𝑟²_,𝑟𝑑𝑟. (89)
Employing (38) yields 𝑝_,𝑡(𝑅,𝑡)_{𝐿𝜎(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{,𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+‖𝑤‖_{𝐿𝜎(𝐵}^𝑇_{0𝑅,𝑅/2)}. (90)
From the form of 𝑝^′₀ (see (81)₂), we derive 𝑝^′_{0𝑊𝜎2−2/𝜎(𝐵𝑅)}=^𝑟₀𝑤₁𝑟²𝑑𝑟_{𝑊𝜎2−2/𝜎(𝐵𝑅)}≤𝑤₁𝑟²_{𝑊𝜎1−2/𝜎(𝐵𝑅)}≤𝑐𝑤_1𝑊𝜎^1−2/𝜎_(𝐵𝑅). (91)
Using (90) in (86) yields 𝑝_{𝑊𝜎2,1(𝐵𝑇𝑅)}≤𝑐𝑤_{,𝑟𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{,𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+‖𝑤‖_{𝐿𝜎(𝐵}^𝑇_{0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−2/𝜎_{(𝐵𝑇0𝑅,𝑅/2)}. (92)
Since 𝑝_,𝑟=𝑤_,𝑡𝑟² and that 𝑝_{,𝑟𝑊𝜎1,1/2(𝐵𝑇𝑅)}≤𝑐𝑝_{𝑊𝜎2,1(𝐵𝑇𝑅)}, (93)
we obtain from (92) the inequality 𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅/2)}≤𝑐(𝑇)𝑤_{,𝑟𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{,𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{0𝐿𝜎(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−2/𝜎_{(𝐵0𝑅,𝑅/2)}, (94) where the relation 𝑤(𝑡)=∫^𝑡₀𝑤_,𝑡′𝑑𝑡^′+𝑤₀ was exploited.
Using the interpolation inequality 𝑤_{,𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}≤𝜀𝑤_{,𝑟𝑡𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}1+𝑐𝜀𝑤_{,𝑡𝐿2(𝐵𝑇0𝑅,𝑅/2)} (95)
in (94), assuming that 𝜀 is sufficiently small, and using (77), we obtain 𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅/2)}≤𝑐(𝑇)𝑤_{,𝑟𝐿𝜎(𝐵𝑇0𝑅,𝑅/2)}+𝑤_{0𝐿𝜎(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−2/𝜎_{(𝐵0𝑅,𝑅/2)}+𝑐₀. (96)
Employing 𝑤_,𝑟(𝑡)=^𝑡₀𝑤_,𝑟𝑡′𝑑𝑡^′+𝑤_,𝑟(0) (97)
in (96) and applying the Gronwall lemma, we get 𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑇0𝑅,𝑅/2)}≤𝑐(𝑇)𝑤_{0𝑊1𝜎(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−2/𝜎_{(𝐵0𝑅,𝑅/2)}+𝑐₀. (98)
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. Let 𝑤 be a solution to problems (11)–(13). Assume that there exist constants 𝛼_𝑖, 𝑐_𝑖, 𝑖=1,2, 𝛽₁ such that (41) holds, where 𝜓=𝜓₁(𝜗)+𝜓₂(𝜂), 𝜗=𝑤_,𝑟, 𝜂=𝑤/𝑟. Assume that 𝑤_𝑡𝑡(0)∈𝐿_2,𝜇1(𝐵_𝑅), 𝑤₀∈𝑊¹₃(𝐵_𝑅,𝑅0/2)∩𝐻¹_𝜇2(𝐵_𝑅), 𝑤₁∈𝐻¹_𝜇1(𝐵_𝑅)∩𝐿_2,𝜇2(𝐵_𝑅)∩𝑊₃^2/3(𝐵_𝑅,𝑅0/2), 𝑐₀ is defined in (36), 𝑐₁=𝑤_{1𝐿22,𝜇(𝐵𝑅)}+𝑤_{0𝐻1𝜇2(𝐵𝑅)},𝑐₂=𝑤_,𝑡𝑡(0)_{𝐿12,𝜇(𝐵𝑅)}+𝑤_{1𝐻1𝜇1(𝐵𝑅)},𝜇₁∈321,1+2√5,𝜇₂∈56,32. (99)
Then the following estimate holds: 𝑤_{,𝑡𝑡𝐿∞(0,𝑡;𝐿12,𝜇(𝐵𝑅))}+𝑤_{,𝑡𝐿∞(0,𝑡;𝐻1𝜇1(𝐵𝑅))}+𝑤_{,𝑡𝑡𝐿2(0,𝑡;𝐻1𝜇1(𝐵𝑅))}≤𝑐(𝑡)𝑐₁+𝑐₀+𝑤_{0𝑊13(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊3^2/3_{(𝐵𝑅,𝑅0/2)}^3/2+𝑐₂,𝑡≤𝑇. (100)

Proof. From (44) and (51), we have 𝑤_{,𝑡𝑡𝐿∞(0,𝑡;𝐿12,𝜇(𝐵𝑟))}+𝑤_{,𝑡𝐿∞(0,𝑡;𝐻1𝜇1(𝐵𝑅))}+𝑤_{,𝑡𝑡𝐿2(0,𝑡;𝐻1𝜇1(𝐵𝑅))}≤𝑐(𝑡)_𝐵𝑇𝑅𝑤_,𝑟𝑡³𝑟^2𝜇1𝑑𝑟𝑑𝑡^1/2+𝑐₂, (101) where 𝜇₁∈(0,1+(1/2)√5),𝑡≤𝑇,𝑐₂=‖𝑤_,𝑡𝑡(0)‖_{𝐿12,𝜇(𝐵𝑟)}+‖𝑤_,𝑡(0)‖_{𝐻1𝜇1(𝐵𝑅).}
Inequality (68) takes the form 𝑤_{,𝑟𝑡𝐿𝜎,2−𝜈(𝐵𝑡𝑅)}≤𝑐(𝑡)𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑡0𝑅,𝑅)}+𝑐₁, (102) where 𝑐₁=‖𝑤₁‖_{𝐿22,𝜇(𝐵𝑅)}+‖𝑤₀‖_𝐻¹_{𝜇2(𝐵𝑅)}, 𝜇₂∈(0,3/2).
Finally, (69) yields 𝑤_,𝑡𝑟²_{𝑊𝜎1,1/2(𝐵𝑡0𝑅,𝑅)}≤𝑐(𝑡)𝑐₀+𝑤_{0𝑊1𝜎(𝐵0𝑅,𝑅/2)}+𝑤_1𝑊𝜎^1−1/𝜎_{(𝐵0𝑅,𝑅/2)}. (103)
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 of 𝑤_,𝑡𝑡(0) appeared in the assumptions of Lemma 4 we derive 𝑤_,𝑡𝑡(0)_{𝐿12,𝜇(𝐵𝑅)}≤𝑐𝑤_{0𝐻2𝜇1(𝐵𝑅)}+𝑤_{1𝐻2𝜇1(𝐵𝑅)}. (104)
Since 𝜇₁−2∈(−1/2,(1/2)√5−1) can be chosen positive, we see that all weights in the norms ‖𝑤₀‖_𝐻²_{𝜇1(𝐵𝑅)}, ‖𝑤₁‖_𝐻²_{𝜇1(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 𝑤₁,𝑤₀∈𝐻²_𝜇1𝐵_𝑅,𝜇₁∈32√,1+52,𝜇₂≥𝜇₁−1,𝜇₂∈56,32. (105)

Up to now there is no estimate for 𝑤_,𝑟𝑟. Hence, to get the space regularity, we need the following.

Lemma 10. Let (105) hold. Let 𝜓_,𝜗𝜗≥𝛼₁,𝜓_,𝜗≤𝑐|𝜗|,𝜓_,𝜂≤𝑐𝜂. (106)
Then solutions to problems (11)–(13) satisfy ^𝑇₀^𝑅₀𝑤²_,𝑟𝑟𝑟^2𝜇𝑑𝑟𝑑𝑡+^𝑅₀𝑤²_,𝑟𝑟𝑟^2𝜇𝑑𝑟≤𝜑𝑐₀,𝑐₁,𝑐₃,𝑇, (107) where 2<𝜇<(1/3)(5+√5), 𝑐₀ is from (36), 𝑐₁ from (40) and 𝑐₃=𝑤_{0𝐻2𝜇(𝐵𝑅)}+𝑤_{1𝐻2𝜇(𝐵𝑅)}. (108)

Proof. Multiplying (11) by 𝑤_,𝑟𝑟𝑟^2𝜇 and integrating over 𝐵_𝑅, we obtain ^𝑅₀𝑤_,𝑡𝑡𝑤_,𝑟𝑟𝑟^2𝜇𝑑𝑟=^𝑅₀1𝑟²𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝑟²_,𝑟𝑤_,𝑟𝑟𝑟^2𝜇−𝑑𝑟_𝑅01𝑟𝜓_,𝜂+𝜇₀𝑤_,𝑡𝑟𝑤_,𝑟𝑟𝑟^2𝜇𝑑𝑟. (109)
Continuing, we have ^𝑅₀𝜓_,𝜗𝜗𝑤²_,𝑟𝑟+𝜇₀𝑤_{,𝑟𝑟𝑡}𝑤_,𝑟𝑟𝑟^2𝜇=𝑑𝑟_𝑅0𝑤_,𝑡𝑡𝑤_,𝑟𝑟𝑟^2𝜇𝑑𝑟−2^𝑅₀𝜓_,𝜗+𝜇₀𝑤_,𝑟𝑡𝑤_,𝑟𝑟𝑟^2𝜇−1+𝑑𝑟_𝑅0𝜓_,𝜂+𝜇₀𝑤_,𝑡𝑤_,𝑟𝑟𝑟^2𝜇−1𝑑𝑟. (110)
Using (106), we obtain 𝛼₁^𝑅₀𝑤²_,𝑟𝑟𝑟^2𝜇𝜇𝑑𝑟+⁰2𝑑𝑑𝑡^𝑅₀𝑤²_,𝑟𝑟𝑟^2𝜇𝑑𝑟≤𝜀^𝑅₀𝑤²_,𝑟𝑟𝑟^2𝜇1𝑑𝑟+𝑐𝜀^𝑅₀𝑤²_,𝑡𝑡𝑟^2𝜇𝑑𝑟+𝑐^𝑅₀𝑤²_,𝑟+𝑤²_,𝑟𝑡+𝑤²_,𝑡+𝑤²𝑟⁻²𝑟^2𝜇−2𝑑𝑟. (111)
Integrating the inequality with respect to time, assuming that 𝜀 is sufficiently small, using that 𝜇=1+𝜈, 𝜈∈(0,3/2), and employing (40) with