Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model

Yaxin Liu; Huijuan Song; Zejia Wang

doi:10.1515/anona-2025-0128

Article Open Access

Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model

Yaxin Liu , Huijuan Song and Zejia Wang

Published/Copyright: November 21, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we consider a free boundary problem modeling the growth of a double-layered tumor, which contains quiescent cells and proliferating cells. Compared with that composed of necrotic cells and proliferating cells, the remarkable feature of this model is that quiescent cells actively consume nutrients, introducing an additional nutrient decay term, but at a rate different from that of proliferating cells. The parameter μ in model represents the “aggressiveness” of the tumor. A threshold μ * is determined such that the radially symmetric stationary solution is linearly stable for μ < μ * and unstable for μ > μ * under non-radially symmetric perturbations. Moreover, it is shown that there exist a sequence { μ n } ( ≥ μ * ) and a positive integer n * * ≥ 2 , such that a branch of symmetry-breaking stationary solutions bifurcates from the radially symmetric stationary solution for μ = μ n ( even n ≥ n * * ) .

Keywords: free boundary problem; double-layered tumor; linear stability; bifurcation; stationary solution

MSC 2020: 35B40; 35B32; 35R35

1 Introduction

It is recognized that due to the nonuniform distribution of nutrient materials, when a tumor grows to a detectable size, it usually contains different populations of cells, such as proliferating cells, necrotic cells, and “in-between” quiescent cells, and the different populations are segregated by interface boundaries, where necrotic cells occupy the central region, living cells inhabit the surrounding area, the outer part of which is occupied by proliferating cells, and the inner part of which is occupied by quiescent cells [2,3]. The purpose of this article is to study such tumor model, neglecting the necrotic cells for mathematical simplicity.

Denote by Ω ( t ) ⊂ R 3 the domain occupied by the tumor at time t and Q ( t ) ⊂ Ω ( t ) the domain occupied by quiescent cells. Let σ be the nutrient concentration within the tumor. Then σ satisfies

(1.1) c σ t = Δ σ − λ 1 σ I Q ( t ) − λ 2 σ I Ω ( t ) \ Q ( t ) in Ω ( t ) , t > 0 ,

where c is the ratio of the nutrient diffusion timescale to the tumor doubling timescale, and typical values for the numerator and the denominator are of the order of minutes and days; thus, 0 < c ≪ 1 . The positive constants λ 1 and λ 2 are the nutrient consumption rates by quiescent cells and proliferating cells, respectively. Taking into account that proliferating cells consume nutrients at a rate no less than quiescent cells, we assume that λ 2 ≥ λ 1 . I E stands for the characteristic function of a set E ⊂ R 3 : I E ( x ) = 1 for x ∈ E ; I E ( x ) = 0 for x ∉ E . Let σ ¯ be a positive constant representing the nutrient concentration outside the tumor, and let σ Q > 0 be a critical value such that in the region where σ ≤ σ Q , all tumor cells are quiescent, and in the region where σ > σ Q , all tumor cells are proliferating. Thus,

(1.2) σ = σ Q on ∂ Q ( t ) , t > 0 ,

(1.3) σ = σ ¯ on ∂ Ω ( t ) , t > 0 .

The pressure p stems from the transport of cells that proliferate or die. Let V be the velocity field of the tumor cell movement. As in [19], we assume that the mitosis rate of proliferating cells is proportional to the nutrient level σ − σ Q with a proportionality constant μ > 0 , and cells in the proliferating layer and the quiescent region undergo apoptosis at rates ν 1 > 0 and ν 2 > 0 , respectively. Then the conservation of mass gives that

div V = μ ( σ − σ Q ) I Ω ( t ) \ Q ( t ) − ν 1 I Ω ( t ) \ Q ( t ) − ν 2 I Q ( t ) .

If the tissue has the structure of a porous medium for which Darcy’s law V = − ∇ p holds, then we obtain, for p the equation

(1.4) − Δ p = μ ( σ − σ ˜ ) I Ω ( t ) \ Q ( t ) − μ ν I Q ( t ) in Ω ( t ) , t > 0 ,

where σ ˜ = σ Q + ν 1 ⁄ μ and ν = ν 2 ⁄ μ . In addition, the cell-to-cell adhesiveness leads to the boundary condition

(1.5) p = κ on ∂ Ω ( t ) , t > 0 ,

where κ is the mean curvature. Furthermore, supposing that the velocity field is continuous up to the boundary yields that

(1.6) V n = − ∂ n p on ∂ Ω ( t ) , t > 0 ,

where V n is the outward normal velocity of the free boundary ∂ Ω ( t ) and n is the outward normal.

We also point out that as in [5,16,18], (1.1) and (1.4) imply that

(1.7) [ ∂ n σ ] = 0 , [ p ] = 0 , [ ∂ n p ] = 0 on ∂ Q ( t ) , t > 0 .

Here, the notation [ p ] denotes the jump of p across ∂ Q ( t ) , that is, [ p ] = ϒ p + − ϒ p − for p + = p ∣ Ω ( t ) \ Q ( t ) and p − = p ∣ Q ( t ) , where ϒ is the trace operator on ∂ Q ( t ) . [ ∂ n p ] and [ ∂ n σ ] are similarly defined.

Finally, we prescribe the initial conditions

(1.8) σ ( x , 0 ) = σ 0 ( x ) in Ω 0 ,

(1.9) Ω ( 0 ) = Ω 0 .

If the tumor contains only proliferating cells, Friedman and Reitich [9] showed that there exists a unique radially symmetric stationary solution provided that 0 < σ ˜ < σ ¯ , which is globally asymptotically stable for any μ > 0 under radially symmetric perturbations. Fontelos and Friedman [7] obtained the existence of a sequence of symmetry-breaking branches of stationary solutions at an increasing sequence μ = μ n ( n = 2 , 4 , 6 , … ). Friedman and Hu [11,12] further investigated the linear stability and the asymptotic stability of the radially symmetric stationary solution under the non-radially symmetric perturbations. They found a critical value of the proliferation rate μ , denoted by μ ∗ , such that the radially symmetric stationary solution is asymptotically stable for μ < μ ∗ and unstable for μ > μ ∗ , in particular, μ ∗ ≤ μ 2 . While if the tumor is with a necrotic core, where necrotic cells do not consume the nutrient so that the nutrient concentration is constant in the necrotic core, Cui [3] proved that a unique necrotic-proliferating radially symmetric stationary solution exists if and only if σ ¯ > σ * for some σ * > σ ˜ , and when c = 0 , all radially symmetric transient solutions converge to this stationary one when the time tends to infinity. Hao et al. [13] performed the bifurcation analysis with the aid of numerical calculations, which was recently improved by Chen and Xing [5]. They took the surface tension coefficient γ as the bifurcation parameter, and rigorously derived a positive sequence { γ n } such that γ 2 , γ 3 , γ 4 , … are all bifurcation points. For the case with c = 0 , Cui [4] solved the asymptotic stability problem of the radially symmetric stationary solution with respect to the non-radially symmetric perturbations, and a critical value of the surface tension coefficient was obtained.

The study of the necrotic problem is more complicated than the non-necrotic problem, see [3–5,13] and the references therein. The main reason lies in that the non-necrotic problem has only one free boundary ∂ Ω ( t ) , whose evolution is governed by (1.6), but the necrotic tumor model has two free boundaries: in addition to the outer free boundary ∂ Ω ( t ) , the interface between the proliferating shell and the necrotic core is an obstacle-type free surface, namely, its evolution is determined by some stationary-type equation. The model (1.1)–(1.9) is obviously with two free boundaries of Stefan-type and obstacle-type. However, in contrast to the necrotic problem, there are two remarkable differences. One is that quiescent cells consume nutrients, which introduces the additional nutrient decay term − λ 1 σ I Q ( t ) in (1.1). The other is that since cells often require less nutrients and energy when they are in a quiescent state, the different rates at which quiescent and proliferating cells consume nutrients are assumed. These make the analysis of the dependence of the inner free interface ∂ Q ( t ) on the outer free boundary ∂ Ω ( t ) , as well as the eigenvalues of the linearized problems more intricate and challenging, which are crucial steps to obtain the linear stability and bifurcation results, see Lemmas 3.1–3.3.

For the radially symmetric problem of (1.1)–(1.9) with c = 0 , it was proved in [15] that when

(1.10) σ Q − σ ˜ + ν ≥ 0 and σ ¯ > σ * ,

there exists a unique quiescent-proliferating stationary solution ( σ s ( r ) , p s ( r ) , η s , R s ) with r = ∣ x ∣ , which is globally asymptotically stable for any μ > 0 , where σ * ∈ ( σ ˜ , ∞ ) is a critical value of the external nutrient concentration. The first condition in (1.10) is equal to ν 2 ≥ ν 1 , which means that the apoptosis rate of quiescent cells is not less than that of proliferating cells. While the second condition means that the stationary tumor furnished with a large quantity of nutrients can possibly be with the quiescent region. For more detailed biological explanations, we refer the reader to [3,15,19]. In this study, we shall always suppose that (1.10) holds and first study the linear stability of this unique radially symmetric stationary solution under non-radially symmetric perturbations. Precisely speaking, assume that c = 0 and the initial condition is perturbed as follows:

(1.11) ∂ Ω ( 0 ) : r = R s + ε ρ 0 ( θ , ϕ ) .

Substituting

(1.12) ∂ Ω ( t ) : r = R s + ε ρ ( θ , ϕ , t ) + O ( ε 2 ) , ∂ Q ( t ) : r = η s + ε ξ ( θ , ϕ , t ) + O ( ε 2 ) , σ ( r , θ , ϕ , t ) = σ ˆ 1 ( r ) + ε w ( r , θ , ϕ , t ) + O ( ε 2 ) in Ω ( t ) \ Q ( t ) , σ ˆ 2 ( r ) + ε w ( r , θ , ϕ , t ) + O ( ε 2 ) in Q ( t ) , p ( r , θ , ϕ , t ) = p ˆ 1 ( r ) + ε q ( r , θ , ϕ , t ) + O ( ε 2 ) in Ω ( t ) \ Q ( t ) , p ˆ 2 ( r ) + ε q ( r , θ , ϕ , t ) + O ( ε 2 ) in Q ( t )

into (1.1)–(1.7) and collecting the ε -order terms, we obtain the linearized system at the radially symmetric stationary solution ( σ s ( r ) , p s ( r ) , η s , R s ). Here,

(1.13) σ ˆ 1 ( r ) = 2 π λ 2 σ Q η s 2 k 1 ( λ 2 η s ) + λ 1 i 1 ( λ 1 η s ) λ 2 i 0 ( λ 1 η s ) k 0 ( λ 2 η s ) i 0 ( λ 2 r ) + i 1 ( λ 2 η s ) − λ 1 i 1 ( λ 1 η s ) λ 2 i 0 ( λ 1 η s ) i 0 ( λ 2 η s ) k 0 ( λ 2 r ) , σ ˆ 2 ( r ) = σ Q i 0 ( λ 1 r ) i 0 ( λ 1 η s ) , p ˆ 1 ( r ) = μ λ 2 ( σ s ( R s ) − σ s ( r ) ) + 1 6 μ σ ˜ ( r 2 − R s 2 ) + 1 3 μ σ ˜ η s 3 1 r − 1 R s + 1 3 μ ν η s 3 1 R s − 1 r + μ λ 2 σ s ′ ( η s ) η s 2 1 R s − 1 r + 1 R s , p ˆ 2 ( r ) = μ λ 2 ( σ s ( R s ) − σ s ( η s ) ) + 1 6 μ σ ˜ ( η s 2 − R s 2 ) + 1 3 μ σ ˜ η s 3 1 η s − 1 R s + 1 3 μ ν η s 3 1 R s − 1 η s + μ λ 2 σ s ′ ( η s ) η s 2 1 R s − 1 η s + 1 6 μ ν ( r 2 − η s 2 ) + 1 R s .

The linear stability result is then stated as follows.

Theorem 1.1

Assume that ρ 0 ∈ L 2 ( Σ ) . Then, there exists μ * > 0 such that if μ < μ * , the radially symmetric stationary solution ( σ s ( r ) , p s ( r ) , η s , R s ) of the problem (1.1)–(1.9) with c = 0 is linearly stable in the sense that for any positive integer k, there exist positive constants δ , C , and t 0 such that

ρ ( θ , ϕ , t ) − ∑ m = − 1 m = 1 ρ 1 , m ( 0 ) Y 1 , m ( θ , ϕ ) H k + 1 ⁄ 2 ( Σ ) ≤ C e − δ t , t > t 0 .

While if μ > μ * , then the radially symmetric stationary solution ( σ s ( r ) , p s ( r ) , η s , R s ) is linearly unstable. Here and below, Σ is the unit sphere.

Next, we discuss the existence of symmetry-breaking stationary solutions to (1.1)–(1.9), and derive the bifurcation result.

Theorem 1.2

There exists a positive integer n * * ≥ 2 such that for every even integer n ≥ n * * , μ n defined by (3.55) is a bifurcation point of the steady state of the system (1.1)–(1.9) with free boundary

∂ Ω ε : r = R s + ε Y n , 0 ( θ ) + O ( ε 2 ) , ∂ Q ε : r = η s + ε b n Y n , 0 ( θ ) + O ( ε 2 ) ,

where b n is given by (3.28).

Remark 1.3

In contrast to the result obtained by Liu and Zhuang [15] that the stability holds for any μ > 0 with respect to radially symmetric perturbations, the linear stability holds only for μ < μ * under non-radially symmetric perturbations.

Remark 1.4

In fact, μ * = min { μ 2 , μ 3 , μ 4 , … } . Note that bifurcation solutions with outer boundary r = R s + ε Y n , 0 ( θ ) + O ( ε 2 ) may be regarded as protrusions, which are associated with the invasion of tumors into their surrounding stroma. Thus, the existence of bifurcation solutions at μ = μ n also indicates some instability of the radially symmetric stationary solution at μ = μ n .

The rest of the article is organized as follows: in Section 2, we present some preliminaries; in Section 3, we discuss the linear stability of the radially symmetric stationary solution under non-radially symmetric perturbations, and the bifurcation analysis is made in Section 4.

2 Preliminaries

In this section, we first collect some properties of the spherical harmonics and the modified spherical Bessel functions. Then, we give the radially symmetric stationary solution to (1.1)–(1.9). Finally, we present an auxiliary lemma and state the Crandall-Rabinowitz theorem.

In R 3 , the family of spherical harmonic functions { Y n , m } forms a complete orthonormal basis for L 2 ( Σ ) , and

(2.1) Δ ω Y n , m = − n ( n + 1 ) Y n , m ,

where

Δ ω = 1 sin θ ∂ ∂ θ sin θ ∂ ∂ θ + 1 sin 2 θ ∂ 2 ∂ ϕ 2

is the Laplace operator on Σ . Then, the Laplace operator in R 3 can be written as

(2.2) Δ = ∂ 2 ∂ r 2 + 2 r ∂ ∂ r + 1 r 2 Δ ω .

The modified spherical Bessel functions given by

i n ( s ) = π 2 s I n + 1 ⁄ 2 ( s ) , k n ( s ) = π 2 s K n + 1 ⁄ 2 ( s ) , s > 0 , n = 0 , 1 , 2 , …

form a fundamental solution set of the differential equation (see [8])

(2.3) y ″ ( s ) + 2 s y ′ ( s ) − 1 + n ( n + 1 ) s 2 y ( s ) = 0 ,

where I n + 1 ⁄ 2 ( s ) and K n + 1 ⁄ 2 ( s ) are the modified Bessel functions. For s > 0 , i n ( s ) and k n ( s ) satisfy

(2.4) i n − 1 ( s ) − i n + 1 ( s ) = 2 n + 1 s i n ( s ) , k n − 1 ( s ) − k n + 1 ( s ) = − 2 n + 1 s k n ( s ) , n ≥ 1 ,

(2.5) i n ′ ( s ) = i n − 1 ( s ) − n + 1 s i n ( s ) , k n ′ ( s ) = − k n − 1 ( s ) − n + 1 s k n ( s ) , n ≥ 1 ,

(2.6) i n ′ ( s ) = i n + 1 ( s ) + n s i n ( s ) , k n ′ ( s ) = − k n + 1 ( s ) + n s k n ( s ) , n ≥ 0 ,

(2.7) i n ′ ( s ) > 0 , k n ′ ( s ) < 0 , n ≥ 0 ,

(2.8) i n ( s ) ∼ 1 2 ( 2 n + 1 ) s e s 2 n + 1 n + 1 2 as n → ∞ ,

(2.9) k n ( s ) ∼ π 2 ( 2 n + 1 ) s e s 2 n + 1 − n − 1 2 as n → ∞ ,

(2.10) i n ( s ) k n + 1 ( s ) + i n + 1 ( s ) k n ( s ) = π 2 1 s 2 , n ≥ 0 .

Denote the unique radially symmetric stationary solution to the system (1.1)–(1.9) by ( σ s ( r ) , p s ( r ) , η s , R s ) , where r = ∣ x ∣ . Then, it was obtained in [15] that σ s ( r ) = σ ( r ; R s ) , η s = η ( R s ) , where

(2.11) σ ( r ; R ) = σ Q i 0 ( λ 1 r ) i 0 ( λ 1 η ) , 0 < r ≤ η , 2 π λ 2 σ Q η 2 k 1 ( λ 2 η ) + λ 1 i 1 ( λ 1 η ) λ 2 i 0 ( λ 1 η ) k 0 ( λ 2 η ) i 0 ( λ 2 r ) + i 1 ( λ 2 η ) − λ 1 i 1 ( λ 1 η ) λ 2 i 0 ( λ 1 η ) i 0 ( λ 2 η ) k 0 ( λ 2 r ) , η < r < R ,

η = η ( R ) ∈ ( 0 , R ) is uniquely determined by the equation

(2.12) k 1 ( λ 2 η ) + λ 1 i 1 ( λ 1 η ) λ 2 i 0 ( λ 1 η ) k 0 ( λ 2 η ) i 0 ( λ 2 R ) + i 1 ( λ 2 η ) − λ 1 i 1 ( λ 1 η ) λ 2 i 0 ( λ 1 η ) i 0 ( λ 2 η ) k 0 ( λ 2 R ) = π σ ¯ 2 λ 2 σ Q η 2 .

Since σ ¯ > σ Q , the maximum principle shows that σ s ′ ( η s ) > 0 and σ s ′ ( R s ) > 0 . In view of (1.4), (1.5), and (1.7), p s satisfies

(2.13) − p s ″ ( r ) − 2 r p s ′ ( r ) = − μ ν , 0 < r < η s ,

(2.14) − p s ″ ( r ) − 2 r p s ′ ( r ) = μ ( σ s ( r ) − σ ˜ ) , η s < r < R s ,

(2.15) p s ′ ( 0 ) = 0 , p s ( η s − ) = p s ( η s + ) , p s ′ ( η s − ) = p s ′ ( η s + ) , p s ( R s ) = 1 R s .

Solving (2.13)–(2.15), we obtain

(2.16) p s ( r ) = μ λ 2 ( σ s ( R s ) − σ s ( η s ) ) + 1 6 μ σ ˜ ( η s 2 − R s 2 ) + 1 3 μ σ ˜ η s 3 1 η s − 1 R s + 1 3 μ ν η s 3 1 R s − 1 η s + μ λ 2 σ s ′ ( η s ) η s 2 1 R s − 1 η s + 1 6 μ ν ( r 2 − η s 2 ) + 1 R s , 0 < r ≤ η s , μ λ 2 ( σ s ( R s ) − σ s ( r ) ) + 1 6 μ σ ˜ ( r 2 − R s 2 ) + 1 3 μ σ ˜ η s 3 1 r − 1 R s + 1 3 μ ν η s 3 1 R s − 1 r + μ λ 2 σ s ′ ( η s ) η s 2 1 R s − 1 r + 1 R s , η s < r < R s .

Finally, using (2.16) and the condition p s ′ ( R s ) = 0 implied by (1.6), we find that

1 λ 2 1 R s σ s ′ ( R s ) − 1 λ 2 η s 2 R s 3 σ s ′ ( η s ) + σ ˜ − ν 3 η s 3 R s 3 − σ ˜ 3 = 0 ,

namely, R s is the unique positive root of

(2.17) 1 λ 2 1 R 3 R 2 ∂ σ ∂ r ( R ; R ) − η 2 ( R ) ∂ σ ∂ r ( η ( R ) ; R ) + σ ˜ − ν 3 η 3 ( R ) R 3 − σ ˜ 3 = 0 .

The next lemma will be used in Sections 3 and 4.

Lemma 2.1

(Lemma 8.2 in [10]) Let s be a nonnegative integer, and let

f ( θ , ϕ ) = ∑ n = 0 ∞ ∑ m = − n n f n , m Y n , m ( θ , ϕ ) .

Then, there exist positive constants c 1 and c 2 independent of f such that

c 1 ‖ f ‖ H s + 1 ⁄ 2 ( Σ ) 2 ≤ ∑ n = 0 ∞ ∑ m = − n n ( 1 + n 2 s + 1 ) ∣ f n , m ∣ 2 ≤ c 2 ‖ f ‖ H s + 1 ⁄ 2 ( Σ ) 2 .

We conclude this section by presenting an important theorem in the bifurcation analysis.

Theorem 2.2

(Crandall-Rabinowitz theorem [6]) Let X and Y be real Banach spaces and F ( x , μ ) be a C p map, p ≥ 3 , of a neighborhood ( 0 , μ 0 ) in X × R into Y. Suppose

F ( 0 , μ ) = 0 for all μ in a neighborhood of μ 0 ;
Ker F x ( 0 , μ 0 ) is a one-dimensional space, spanned by x 0 ;
Im F x ( 0 , μ 0 ) = Y 1 has codimension 1;
F μ x ( 0 , μ 0 ) x 0 ∉ Y 1 .

Then, ( 0 , μ 0 ) is a bifurcation point of the equation F ( x , μ ) = 0 in the following sense: in a neighborhood of ( 0 , μ 0 ) , the set of solutions of F ( x , μ ) = 0 consists of two C p − 2 smooth curves Γ 1 and Γ 2 , which intersect only at the point ( 0 , μ 0 ) ; Γ 1 is the curve ( 0 , μ ) and Γ 2 can be parameterized as follows:

Γ 2 : ( x ( ε ) , μ ( ε ) ) , ∣ ε ∣ small , ( x ( 0 ) , μ ( 0 ) ) = ( 0 , μ 0 ) , x ′ ( 0 ) = x 0 .

3 Linear stability

In this section, we consider the linear stability of the unique radially symmetric stationary solution ( σ s ( r ) , p s ( r ) , η s , R s ) to (1.1)–(1.9) under non-radially symmetric perturbations in the quasi-steady state case c = 0 .

Substituting (1.12) into (1.1)–(1.7), using

V n ∣ ∂ Ω ( t ) = ε ρ t + O ( ε 2 ) , κ ∣ ∂ Ω ( t ) = 1 R s − ε R s 2 ρ + 1 2 Δ ω ρ + O ( ε 2 ) ,

and observing from (1.13), (2.11), and (2.16) that

(3.1) σ s ( r ) = σ ˆ 2 ( r ) , 0 < r < η s , σ ˆ 1 ( r ) , η s < r < R s , p s ( r ) = p ˆ 2 ( r ) , 0 < r < η s , p ˆ 1 ( r ) , η s < r < R s ,

by collecting the ε -order terms, we obtain the linearized system

(3.2) Δ w ( r , θ , ϕ , t ) = λ 1 w ( r , θ , ϕ , t ) in B η s , t > 0 ,

(3.3) Δ w ( r , θ , ϕ , t ) = λ 2 w ( r , θ , ϕ , t ) in B R s \ B ¯ η s , t > 0 ,

(3.4) w = − σ s ′ ( η s ) ξ ( θ , ϕ , t ) on ∂ B η s , t > 0 ,

(3.5) [ ∂ n ω ] = − ( λ 2 − λ 1 ) σ Q ξ ( θ , ϕ , t ) on ∂ B η s , t > 0 ,

(3.6) w = − σ s ′ ( R s ) ρ ( θ , ϕ , t ) on ∂ B R s , t > 0 ,

and

(3.7) − Δ q ( r , θ , ϕ , t ) = 0 in B η s , t > 0 ,

(3.8) − Δ q ( r , θ , ϕ , t ) = μ w ( r , θ , ϕ , t ) in B R s \ B ¯ η s , t > 0 ,

(3.9) [ q ] = 0 , [ ∂ n q ] = μ ( σ Q − σ ˜ + ν ) ξ ( θ , ϕ , t ) on ∂ B η s , t > 0 ,

(3.10) q = − 1 R s 2 ρ + 1 2 Δ ω ρ on ∂ B R s , t > 0 ,

(3.11) d ρ d t = − p s ″ ( R s ) ρ ( θ , ϕ , t ) − ∂ q ∂ r ( R s , θ , ϕ , t ) for t > 0 .

We next solve the linearized problem (3.2)–(3.11). For this purpose, assuming that

ρ ( θ , ϕ , t ) = ∑ n = 0 ∞ ∑ m = − n n ρ n , m ( t ) Y n , m ( θ , ϕ ) , ξ ( θ , ϕ , t ) = ∑ n = 0 ∞ ∑ m = − n n ξ n , m ( t ) Y n , m ( θ , ϕ ) , w ( r , θ , ϕ , t ) = ∑ n = 0 ∞ ∑ m = − n n w n , m ( r , t ) Y n , m ( θ , ϕ ) , q ( r , θ , ϕ , t ) = ∑ n = 0 ∞ ∑ m = − n n q n , m ( r , t ) Y n , m ( θ , ϕ ) ,

by (2.1) and (2.2), it follows from (3.2) to (3.11) that

(3.12) ∂ 2 w n , m ∂ r 2 ( r , t ) + 2 r ∂ w n , m ∂ r ( r , t ) − n ( n + 1 ) r 2 w n , m ( r , t ) = λ 1 w n , m ( r , t ) , 0 < r < η s , t > 0 ,

(3.13) ∂ 2 w n , m ∂ r 2 ( r , t ) + 2 r ∂ w n , m ∂ r ( r , t ) − n ( n + 1 ) r 2 w n , m ( r , t ) = λ 2 w n , m ( r , t ) , η s < r < R s , t > 0 ,

(3.14) w n , m ( η s , t ) = − σ s ′ ( η s ) ξ n , m ( t ) , t > 0 ,

(3.15) ∂ w n , m ∂ r ( η s + , t ) − ∂ w n , m ∂ r ( η s − , t ) = − ( λ 2 − λ 1 ) σ Q ξ n , m ( t ) , t > 0 ,

(3.16) w n , m ( R s , t ) = − σ s ′ ( R s ) ρ n , m ( t ) , t > 0 ,

and

(3.17) ∂ 2 q n , m ∂ r 2 ( r , t ) + 2 r ∂ q n , m ∂ r ( r , t ) − n ( n + 1 ) r 2 q n , m ( r , t ) = 0 , 0 < r < η s , t > 0 ,

(3.18) ∂ 2 q n , m ∂ r 2 ( r , t ) + 2 r ∂ q n , m ∂ r ( r , t ) − n ( n + 1 ) r 2 q n , m ( r , t ) = − μ w n , m ( r , t ) , η s < r < R s , t > 0 ,

(3.19) q n , m ( η s + , t ) = q n , m ( η s − , t ) , t > 0 ,

(3.20) ∂ q n , m ∂ r ( η s + , t ) − ∂ q n , m ∂ r ( η s − , t ) = μ ( σ Q − σ ˜ + ν ) ξ n , m ( t ) , t > 0 ,

(3.21) q n , m ( R s , t ) = − 1 R s 2 1 − n ( n + 1 ) 2 ρ n , m ( t ) , t > 0 ,

(3.22) d ρ n , m ( t ) d t = − p s ″ ( R s ) ρ n , m ( t ) − ∂ q n , m ∂ r ( R s , t ) , t > 0 .

Using the separation of variables, if we further suppose that

(3.23) w n , m ( r , t ) = − σ s ′ ( R s ) E n ( r ) ρ n , m ( t ) , 0 < r < η s , t > 0 , − σ s ′ ( R s ) Q n ( r ) ρ n , m ( t ) , η s < r < R s , t > 0 , q n , m ( r , t ) = P n ( r ) ρ n , m ( t ) , 0 < r < R s , t > 0

and denote

b n = σ s ′ ( R s ) σ s ′ ( η s ) E n ( η s ) ,

then one obtains from (3.12) to (3.21) that

(3.24) E n ″ ( r ) + 2 r E n ′ ( r ) − n ( n + 1 ) r 2 E n ( r ) = λ 1 E n ( r ) , 0 < r < η s ,

(3.25) E n ( η s ) = σ s ′ ( η s ) σ s ′ ( R s ) b n , Q n ′ ( η s ) − E n ′ ( η s ) = λ 2 − λ 1 σ s ′ ( R s ) σ Q b n ,

(3.26) Q n ″ ( r ) + 2 r Q n ′ ( r ) − n ( n + 1 ) r 2 Q n ( r ) = λ 2 Q n ( r ) , η s < r < R s ,

(3.27) Q n ( η s ) = σ s ′ ( η s ) σ s ′ ( R s ) b n , Q n ( R s ) = 1 ,

ξ n , m ( t ) = b n ρ n , m ( t ) , t > 0 ,

and

P n ″ ( r ) + 2 r P n ′ ( r ) − n ( n + 1 ) r 2 P n ( r ) = 0 , 0 < r < η s , P n ″ ( r ) + 2 r P n ′ ( r ) − n ( n + 1 ) r 2 P n ( r ) = μ σ s ′ ( R s ) Q n ( r ) , η s < r < R s , P n ( η s + ) = P n ( η s − ) , P n ′ ( η s + ) − P n ′ ( η s − ) = μ ( σ Q − σ ˜ + ν ) b n , P n ( R s ) = − 1 R s 2 1 − n ( n + 1 ) 2 .

Based on (2.3), (2.6), and (2.10), a direct calculation immediately shows that

(3.28) b n = π 2 1 λ 2 η s 2 1 i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) σ s ′ ( R s ) σ s ′ ( η s ) ( λ 2 − λ 1 ) σ Q σ s ′ ( η s ) + λ 1 i n + 1 ( λ 1 η s ) i n ( λ 1 η s ) + λ 2 i n ( λ 2 R s ) k n + 1 ( λ 2 η s ) + i n + 1 ( λ 2 η s ) k n ( λ 2 R s ) i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) ,

(3.29) E n ( r ) = b n σ s ′ ( η s ) σ s ′ ( R s ) i n ( λ 1 r ) i n ( λ 1 η s ) , 0 < r < η s ,

(3.30) Q n ( r ) = k n ( λ 2 η s ) − b n σ s ′ ( η s ) σ s ′ ( R s ) k n ( λ 2 R s ) i n ( λ 2 r ) i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) − i n ( λ 2 η s ) − b n σ s ′ ( η s ) σ s ′ ( R s ) i n ( λ 2 R s ) k n ( λ 2 r ) i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) , η s < r < R s ,

(3.31) P n ( r ) = − μ λ 2 n η s σ s ′ ( η s ) b n − μ λ 2 σ s ′ ( R s ) Q n ′ ( η s ) + μ ( σ Q − σ ˜ + ν ) b n η s n + 2 2 n + 1 1 η s 2 n + 1 − 1 R s 2 n + 1 r n + μ λ 2 σ s ′ ( η s ) b n r n η s n − μ λ 2 σ s ′ ( R s ) r n R s n − r n R s n + 2 1 − n ( n + 1 ) 2 , 0 < r ≤ η s , − μ λ 2 n η s σ s ′ ( η s ) b n − μ λ 2 σ s ′ ( R s ) Q n ′ ( η s ) + μ ( σ Q − σ ˜ + ν ) b n η s n + 2 2 n + 1 1 r 2 n + 1 − 1 R s 2 n + 1 r n + μ λ 2 σ s ′ ( R s ) Q n ( r ) − r n R s n − r n R s n + 2 1 − n ( n + 1 ) 2 , η s < r < R s .

Note that (2.14) together with the facts that σ s ( R s ) = σ ¯ and p s ′ ( R s ) = 0 implies that

(3.32) p s ″ ( R s ) = − μ ( σ ¯ − σ ˜ ) ,

and by (3.23) and (3.31),

∂ q n , m ∂ r ( R s , t ) = μ λ 2 n η s σ s ′ ( η s ) b n − μ λ 2 σ s ′ ( R s ) Q n ′ ( η s ) + μ ( σ Q − σ ˜ + ν ) b n η s n + 2 R s n + 2 + μ λ 2 σ s ′ ( R s ) Q n ′ ( R s ) − n R s + n R s 3 n ( n + 1 ) 2 − 1 ρ n , m ( t ) .

We thus derive from (3.22) that

d ρ n , m ( t ) d t = − ( A n − μ B n ) ρ n , m ( t )

with

(3.33) A n = n R s 3 n ( n + 1 ) 2 − 1 ,

(3.34) B n = σ ¯ − σ ˜ − 1 λ 2 n η s σ s ′ ( η s ) b n − 1 λ 2 σ s ′ ( R s ) Q n ′ ( η s ) + ( σ Q − σ ˜ + ν ) b n η s n + 2 R s n + 2 − 1 λ 2 σ s ′ ( R s ) Q n ′ ( R s ) − n R s .

Hence,

(3.35) ρ n , m ( t ) = ρ n , m ( 0 ) e − ( A n − μ B n ) t .

Clearly, A 0 = A 1 = 0 and A n > 0 for n ≥ 2 .

To proceed further, we need the following lemmas.

Lemma 3.1

0 < b n + 1 < b n for every n ≥ 0 , where b n is defined by (3.28).

Proof

First, since λ 2 ≥ λ 1 , it is seen from (3.28) that b n > 0 . Applying the maximum principle to (3.26)–(3.27), we thus see that Q n ( r ) > 0 for η s ≤ r ≤ R s . Next, we claim that

(3.36) Q n + 1 ( r ) < Q n ( r ) , η s ≤ r < R s .

As a matter of fact, setting

F n ( r ) = Q n ( r ) − Q n + 1 ( r ) , η s ≤ r ≤ R s , G n ( r ) = E n ( r ) − E n + 1 ( r ) , 0 ≤ r ≤ η s ,

it follows from (3.26) to (3.27) and (3.24) to (3.25) that

− F n ″ ( r ) − 2 r F n ′ ( r ) + λ 2 + n ( n + 1 ) r 2 F n ( r ) = 2 ( n + 1 ) r 2 Q n + 1 ( r ) > 0 , η s < r < R s , F n ( η s ) = Q n ( η s ) − Q n + 1 ( η s ) , F n ( R s ) = 0

and

(3.37) − G n ″ ( r ) − 2 r G n ′ ( r ) + λ 1 + n ( n + 1 ) r 2 G n ( r ) = 2 ( n + 1 ) r 2 E n + 1 ( r ) , 0 < r < η s , G n ( 0 ) ≥ 0 , G n ( η s ) = F n ( η s ) , G n ′ ( η s ) = F n ′ ( η s ) − λ 2 − λ 1 σ s ′ ( η s ) σ Q F n ( η s ) .

If F n ( η s ) ≤ 0 , then by the maximum principle, F n ′ ( η s ) > 0 . Thus, G n ′ ( η s ) > 0 . This, combined with G n ( 0 ) ≥ 0 and G n ( η s ) ≤ 0 , implies that there exists r n ∈ ( 0 , η s ) such that G n ( r n ) = min 0 ≤ r ≤ η s G n ( r ) < 0 , from which it follows that G n ″ ( r n ) ≥ 0 and G n ′ ( r n ) = 0 . Consequently,

− G n ″ ( r n ) − 2 r n G n ′ ( r n ) + λ 1 + n ( n + 1 ) r n 2 G n ( r n ) < 0 ,

which contradicts to (3.37), because E n + 1 ( r n ) > 0 . Hence, F n ( η s ) > 0 . Using the maximum principle again, we therefore obtain F n ( r ) > 0 for η s < r < R s ; the assertion (3.36) is obtained. In particular, Q n + 1 ( η s ) < Q n ( η s ) , so that b n + 1 < b n . The proof is complete.□

Lemma 3.2

The sequence { B n } , defined by (3.34), possesses the following properties:

B n < B n + 1 for every n ≥ 0 ;
B 1 = 0 ;
lim n → ∞ B n = σ ¯ − σ ˜ .

Proof

(i) Using (3.26), we observe that

( r n + 2 Q n ′ ( r ) − n r n + 1 Q n ( r ) ) ′ = r n + 2 Q n ″ ( r ) + 2 r n + 1 Q n ′ ( r ) − n ( n + 1 ) r n Q n ( r ) = r n + 2 Q n ″ ( r ) + 2 r Q n ′ ( r ) − n ( n + 1 ) r 2 Q n ( r ) = λ 2 r n + 2 Q n ( r ) .

By the second boundary condition in (3.27), there holds that

∫ η s R s ( r n + 2 Q n ′ ( r ) − n r n + 1 Q n ( r ) ) ′ d r = R s n + 2 Q n ′ ( R s ) − n R s n + 1 Q n ( R s ) − ( η s n + 2 Q n ′ ( η s ) − n η s n + 1 Q n ( η s ) ) = R s n + 2 Q n ′ ( R s ) − n R s n + 1 − ( η s n + 2 Q n ′ ( η s ) − n η s n + 1 Q n ( η s ) ) .

Thus,

(3.38) Q n ′ ( R s ) − n R s = Q n ′ ( η s ) − n η s Q n ( η s ) η s n + 2 R s n + 2 + λ 2 ∫ η s R s r n + 2 R s n + 2 Q n ( r ) d r .

Applying (3.38) and the first boundary condition in (3.27), one can then rewrite (3.34) as

B n = σ ¯ − σ ˜ − ( σ Q − σ ˜ + ν ) η s n + 2 R s n + 2 b n − σ s ′ ( R s ) ∫ η s R s r n + 2 R s n + 2 Q n ( r ) d r ,

which together with Lemma 3.1, (3.36), and the facts that σ Q − σ ˜ + ν ≥ 0 , η s < R s , and σ s ′ ( R s ) > 0 proves the assertion (i).

(ii) We first show that

(3.39) b 1 = 1 ,

namely,

(3.40) π 2 1 λ 2 η s 2 σ s ′ ( R s ) = ( λ 2 − λ 1 ) σ Q + λ 1 σ s ′ ( η s ) i 2 ( λ 1 η s ) i 1 ( λ 1 η s ) [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] + λ 2 σ s ′ ( η s ) [ i 1 ( λ 2 R s ) k 2 ( λ 2 η s ) + i 2 ( λ 2 η s ) k 1 ( λ 2 R s ) ]

by (3.28). Indeed, in view of (2.11),

(3.41) σ s ′ ( η s ) = λ 1 σ Q i 1 ( λ 1 η s ) i 0 ( λ 1 η s ) ,

(3.42) σ s ′ ( R s ) = 2 π λ 2 3 ⁄ 2 σ Q η s 2 k 1 ( λ 2 η s ) + λ 1 i 1 ( λ 1 η s ) λ 2 i 0 ( λ 1 η s ) k 0 ( λ 2 η s ) i 1 ( λ 2 R s ) − i 1 ( λ 2 η s ) − λ 1 i 1 ( λ 1 η s ) λ 2 i 0 ( λ 1 η s ) i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) .

Since (3.41) implies that

(3.43) λ 1 λ 2 i 1 ( λ 1 η s ) i 0 ( λ 1 η s ) = σ s ′ ( η s ) σ Q λ 2 ,

(3.42) can be rewritten as

(3.44) σ s ′ ( R s ) = 2 π λ 2 3 ⁄ 2 σ Q η s 2 { i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) + σ s ′ ( η s ) σ Q λ 2 [ i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ] ,

from which it immediately follows that

(3.45) π 2 1 λ 2 η s 2 σ s ′ ( R s ) = λ 2 σ Q [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] + λ 2 σ s ′ ( η s ) [ i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ] .

Note that (3.41) also implies that

(3.46) σ Q = σ s ′ ( η s ) λ 1 i 0 ( λ 1 η s ) i 1 ( λ 1 η s ) .

Substituting this and (3.45) into (3.40), we obtain

(3.47) λ 2 [ i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ] = − λ 1 i 0 ( λ 1 η s ) i 1 ( λ 1 η s ) + λ 1 i 2 ( λ 1 η s ) i 1 ( λ 1 η s ) [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] + λ 2 [ i 1 ( λ 2 R s ) k 2 ( λ 2 η s ) + i 2 ( λ 2 η s ) k 1 ( λ 2 R s ) ] .

Applying the facts that

i 2 ( s ) = i 0 ( s ) − 3 s i 1 ( s ) , k 2 ( s ) = k 0 ( s ) + 3 s k 1 ( s ) , s > 0

implied by (2.4), one further finds that

− λ 1 i 0 ( λ 1 η s ) i 1 ( λ 1 η s ) + λ 1 i 2 ( λ 1 η s ) i 1 ( λ 1 η s ) [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] + λ 2 [ i 1 ( λ 2 R s ) k 2 ( λ 2 η s ) + i 2 ( λ 2 η s ) k 1 ( λ 2 R s ) ] = − 3 η s [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] + λ 2 i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + 3 λ 2 η s k 1 ( λ 2 η s ) + λ 2 i 0 ( λ 2 η s ) − 3 λ 2 η s i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) = λ 2 i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + λ 2 i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ,

which proves (3.47). Thus, (3.39) is obtained.

Now we prove that B 1 = 0 . Using (2.5), the second boundary condition in (3.25), (3.29), and (3.46), we have

(3.48) Q 1 ′ ( η s ) = E 1 ′ ( η s ) + λ 2 − λ 1 σ s ′ ( R s ) σ Q b 1 = b 1 λ 1 σ s ′ ( η s ) σ s ′ ( R s ) i 0 ( λ 1 η s ) − 2 λ 1 η s i 1 ( λ 1 η s ) i 1 ( λ 1 η s ) + λ 2 σ Q b 1 σ s ′ ( R s ) − λ 1 b 1 σ s ′ ( R s ) σ s ′ ( η s ) λ 1 i 0 ( λ 1 η s ) i 1 ( λ 1 η s ) = b 1 σ s ′ ( R s ) λ 2 σ Q − 2 η s σ s ′ ( η s ) .

Besides, by (2.17),

(3.49) σ ˜ = 3 λ 2 1 R s σ s ′ ( R s ) + σ ˜ − ν − 3 λ 2 1 η s σ s ′ ( η s ) η s 3 R s 3 .

Substituting (3.48) and (3.49) into (3.34) with n = 1 , we thus arrive at

B 1 = σ ¯ + ( 1 − b 1 ) 3 λ 2 1 η s σ s ′ ( η s ) − ( σ ˜ − ν ) η s 3 R s 3 − 2 λ 2 1 R s σ s ′ ( R s ) − 1 λ 2 σ s ′ ( R s ) Q 1 ′ ( R s ) = 1 λ 2 λ 2 σ s ( R s ) − 2 R s σ s ′ ( R s ) − 1 λ 2 σ s ′ ( R s ) Q 1 ′ ( R s ) = 1 λ 2 ( σ s ″ ( R s ) − σ s ′ ( R s ) Q 1 ′ ( R s ) ) ,

where we have used (3.39) and the fact that σ ¯ = σ ( R s ) . Hence, it suffices to verify that

(3.50) σ s ″ ( R s ) = σ s ′ ( R s ) Q 1 ′ ( R s ) .

By (2.11) and (3.43),

σ s ″ ( R s ) = 2 π λ 2 2 σ Q η s 2 [ k 1 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) − i 1 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) ] + 2 π λ 2 3 2 η s 2 σ s ′ ( η s ) [ k 0 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) + i 0 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) ] .

On the other hand, it follows from (3.30), (3.39), and (3.44) that

σ s ′ ( R s ) Q 1 ′ ( R s ) = λ 2 [ σ s ′ ( R s ) k 1 ( λ 2 η s ) − σ s ′ ( η s ) k 1 ( λ 2 R s ) ] i 1 ′ ( λ 2 R s ) i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) − λ 2 [ σ s ′ ( R s ) i 1 ( λ 2 η s ) − σ s ′ ( η s ) i 1 ( λ 2 R s ) ] k 1 ′ ( λ 2 R s ) i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) = 2 π λ 2 2 σ Q η s 2 [ k 1 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) − i 1 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) ] + 2 π λ 2 3 2 η s 2 σ s ′ ( η s ) i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) k 1 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) − 2 π λ 2 3 2 η s 2 σ s ′ ( η s ) i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) i 1 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) − λ 2 σ s ′ ( η s ) k 1 ( λ 2 R s ) i 1 ′ ( λ 2 R s ) − i 1 ( λ 2 R s ) k 1 ′ ( λ 2 R s ) i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) .

Consequently, in order to obtain (3.50), we only need to prove

2 π λ 2 3 2 η s 2 [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] k 0 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) + 2 π λ 2 3 2 η s 2 [ i 1 ( λ 2 R s ) k 1 ( λ 2 η s ) − i 1 ( λ 2 η s ) k 1 ( λ 2 R s ) ] i 0 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) = 2 π λ 2 3 2 η s 2 [ i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ] k 1 ( λ 2 η s ) i 1 ′ ( λ 2 R s ) − 2 π λ 2 3 2 η s 2 [ i 1 ( λ 2 R s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 R s ) ] i 1 ( λ 2 η s ) k 1 ′ ( λ 2 R s ) − λ 2 [ k 1 ( λ 2 R s ) i 1 ′ ( λ 2 R s ) − i 1 ( λ 2 R s ) k 1 ′ ( λ 2 R s ) ] ,

namely,

− 2 π λ 2 3 2 η s 2 [ i 1 ( λ 2 η s ) k 0 ( λ 2 η s ) + i 0 ( λ 2 η s ) k 1 ( λ 2 η s ) ] k 1 ( λ 2 R s ) i 1 ′ ( λ 2 R s ) + 2 π λ 2 3 2 η s 2 [ i 0 ( λ 2 η s ) k 1 ( λ 2 η s ) + i 1 ( λ 2 η s ) k 0 ( λ 2 η s ) ] i 1 ( λ 2 R s ) k 1 ′ ( λ 2 R s ) = − λ 2 k 1 ( λ 2 R s ) i 1 ′ ( λ 2 R s ) + λ 2 i 1 ( λ 2 R s ) k 1 ′ ( λ 2 R s ) ,

which holds true by (2.10). The proof of the assertion (ii) is complete.

(iii) A direct calculation based on (3.30), (2.6), and the second boundary condition in (3.27) gives that

(3.51) Q n ′ ( R s ) − n R s = λ 2 k n ( λ 2 η s ) − b n σ s ′ ( η s ) σ s ′ ( R s ) k n ( λ 2 R s ) i n + 1 ( λ 2 R s ) i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) + λ 2 i n ( λ 2 η s ) − b n σ s ′ ( η s ) σ s ′ ( R s ) i n ( λ 2 R s ) k n + 1 ( λ 2 R s ) i n ( λ 2 R s ) k n ( λ 2 η s ) − i n ( λ 2 η s ) k n ( λ 2 R s ) = λ 2 1 − b n σ s ′ ( η s ) σ s ′ ( R s ) k n ( λ 2 R s ) k n ( λ 2 η s ) i n + 1 ( λ 2 R s ) i n ( λ 2 R s ) + i n ( λ 2 η s ) i n ( λ 2 R s ) − b n σ s ′ ( η s ) σ s ′ ( R s ) k n + 1 ( λ 2 R s ) k n ( λ 2 η s ) 1 − i n ( λ 2 η s ) k n ( λ 2 R s ) i n ( λ 2 R s ) k n ( λ 2 η s ) .

Applying Lemma 3.1, (2.7)–(2.9), and the fact that 0 < η s < R s , we immediately derive from (3.51) that

(3.52) lim n → ∞ Q n ′ ( R s ) − n R s = 0 .

Besides, since Q n ( r ) > 0 for η s ≤ r ≤ R s , from (3.26) we see that

R s 2 Q n ′ ( R s ) − η s 2 Q n ′ ( η s ) = ∫ η s R s ( r 2 Q n ′ ( r ) ) ′ d r = ∫ η s R s λ 2 + n ( n + 1 ) r 2 r 2 Q n ( r ) d r > 0 .

Thus, there holds

(3.53) 0 < Q n ′ ( η s ) η s n + 2 R s n + 2 ≤ Q n ′ ( R s ) η s n R s n = Q n ′ ( R s ) − n R s + n R s η s n R s n ,

which, combined with (3.52) and η s < R s , implies that

(3.54) lim n → ∞ Q n ′ ( η s ) η s n + 2 R s n + 2 = 0 .

Using Lemma 3.1, (3.52), (3.54), and η s < R s , by (3.34), we therefore conclude that lim n → ∞ B n = σ ¯ − σ ˜ and complete the proof of the lemma.□

The assertions (i) and (ii) of Lemma 3.2 imply that B n > 0 for each n ≥ 2 . Denote

(3.55) μ n = A n B n , n ≥ 2 ,

for which we have the next result.

Lemma 3.3

There exists an integer n * ≥ 2 such that μ n is monotonically increasing for n ≥ n * ; moreover, lim n → ∞ μ n = + ∞ .

Proof

By (3.33) and the assertion (iii) of Lemma 3.2, there holds that

(3.56) lim n → ∞ μ n n 3 = 1 2 R s 3 ( σ ¯ − σ ˜ ) ,

which proves the lemma.□

With the lemmas described earlier, we now establish estimates for ρ n , m ( t ) given by (3.35). Applying the assertions (i) and (ii) of Lemma 3.2, we obtain estimates for ρ 0,0 ( t ) and ρ 1 , m ( t ) .

Lemma 3.4

For any μ > 0 ,

(3.57) ρ 0,0 ( t ) = ρ 0,0 ( 0 ) e μ B 0 t w i t h B 0 < 0 , t > 0 ,

(3.58) ρ 1 , m ( t ) = ρ 1 , m ( 0 ) , t > 0 .

Remark 3.5

For n = 0 , r = R s + ε ρ 0,0 ( t ) Y 0,0 , which indicates that the perturbation is radially symmetric. (3.57) is just another indication that the stability discussed in [15] is valid for all μ > 0 .

Since Lemma 3.4 holds for all μ > 0 , we define μ 0 = μ 1 = + ∞ . Set

μ * = min { μ 0 , μ 1 , μ 2 , μ 3 , μ 4 , … } ,

more precisely, by Lemma 3.3, μ * = min { μ 2 , μ 3 , μ 4 , … , μ n * } . We now prove Theorem 1.1.

Proof of Theorem 1.1

Fixing μ < μ * , by Lemma 3.2 and (3.56), there exists δ > 0 , independent of n , such that for every n ≥ 0 and n ≠ 1 ,

A n − μ B n ≥ δ ( n 3 + 1 ) .

Then, it follows from (3.35) that for n ≥ 0 and n ≠ 1 ,

(3.59) ∣ ρ n , m ( t ) ∣ ≤ ∣ ρ n , m ( 0 ) ∣ e − δ ( n 3 + 1 ) t , t > 0 .

Thus, for any positive integer k , using Lemma 2.1, (3.58), and (3.59), we obtain

(3.60) ρ ( θ , ϕ , t ) − ∑ m = − 1 m = 1 ρ 1 , m ( 0 ) Y 1 , m ( θ , ϕ ) H k + 1 ⁄ 2 ( Σ ) 2 = ∑ n = 0 , n ≠ 1 ∞ ∑ m = − n m = n ρ n , m ( t ) Y n , m ( θ , ϕ ) H k + 1 ⁄ 2 ( Σ ) 2 ≤ C ∑ n = 0 , n ≠ 1 ∞ ∑ m = − n m = n ( 1 + n 2 k + 1 ) ∣ ρ n , m ( t ) ∣ 2 ≤ C ∑ n = 0 , n ≠ 1 ∞ ∑ m = − n m = n ( 1 + n 2 k + 1 ) ∣ ρ n , m ( 0 ) ∣ 2 e − 2 δ ( n 3 + 1 ) t ≤ C e − 2 δ t , t > t 0

for some C > 0 and t 0 > 0 .

We next turn to the case μ > μ * . According to the definition of μ * , there exists an integer n * such that 2 ≤ n * ≤ n * and μ * = μ n * . Then, we obtain from (3.35) that

∣ ρ n * , m ( t ) ∣ = ∣ ρ n * , m ( 0 ) ∣ e B n * ( μ − μ * ) t → + ∞

as t → + ∞ , where we have used the fact that B n * > 0 . The proof is complete.□

4 Bifurcation

In this section, we make the bifurcation analysis for (1.1)–(1.9). Concretely speaking, we first obtain and solve the linearization of the steady state of (1.1)–(1.9) at the radially symmetric solution ( σ s , p s , η s , R s ) . In order to obtain Fréchet derivatives, we next derive rigorous mathematical estimates for this linearization. Finally, we give the proof of Theorem 1.2 by applying the Crandall-Rabinowitz theorem.

Consider a family of domains with perturbed boundaries

∂ Q ε : r = η s + ε T ( θ , ϕ ) , ∂ Ω ε : r = R s + ε S ( θ , ϕ ) ,

where both S and T are to be determined. Let ( σ , p ) be the solution to

(4.1) Δ σ = λ 1 σ I Q ε + λ 2 σ I Ω ε \ Q ε in Ω ε ,

(4.2) − Δ p = μ ( σ − σ ˜ ) I Ω ε \ Q ε − μ ν I Q ε in Ω ε ,

(4.3) σ = σ Q , [ ∂ n σ ] = 0 on ∂ Q ε ,

(4.4) σ = σ ¯ on ∂ Ω ε ,

(4.5) [ p ] = 0 , [ ∂ n p ] = 0 on ∂ Q ε ,

(4.6) p = κ on ∂ Ω ε .

Then, the well-posedness of (4.1)–(4.6) follows from the local flattening and theories of diffraction problem for systems, see [14]. If we define

(4.7) F ( R ˜ , μ ) = ∂ p ∂ n ∂ Ω ε ,

where R ˜ = ε S , then ( σ , p , η s + ε T , R s + ε S ) is a stationary solution to the system (1.1)–(1.9) if and only if F ( R ˜ , μ ) = 0 .

Write

(4.8) σ ( r , θ , ϕ ) = σ ˆ 1 ( r ) + ε σ 1 ( r , θ , ϕ ) + O ( ε 2 ) in Ω ε \ Q ε , σ ˆ 2 ( r ) + ε σ 1 ( r , θ , ϕ ) + O ( ε 2 ) in Q ε ,

(4.9) p ( r , θ , ϕ ) = p ˆ 1 ( r ) + ε p 1 ( r , θ , ϕ ) + O ( ε 2 ) in Ω ε \ Q ε , p ˆ 2 ( r ) + ε p 1 ( r , θ , ϕ ) + O ( ε 2 ) in Q ε ,

where σ ˆ 1 ( r ) , σ ˆ 2 ( r ) , p ˆ 1 ( r ) , and p ˆ 2 ( r ) are given as in (1.13). Substituting (4.8) and (4.9) into (4.1)–(4.6), using (3.1) and collecting the ε -order terms, we then obtain the linearized problem for ( σ 1 , p 1 ) :

(4.10) Δ σ 1 = λ 1 σ 1 in B η s ,

(4.11) Δ σ 1 = λ 2 σ 1 in B R s \ B ¯ η s ,

(4.12) σ 1 = − σ s ′ ( η s ) T , [ ∂ n σ 1 ] = − ( λ 2 − λ 1 ) σ Q T on ∂ B η s ,

(4.13) σ 1 = − σ s ′ ( R s ) S on ∂ B R s

and

(4.14) − Δ p 1 = 0 in B η s ,

(4.15) − Δ p 1 = μ σ 1 in B R s \ B ¯ η s ,

(4.16) [ p 1 ] = 0 , [ ∂ n p 1 ] = μ ( σ Q − σ ˜ + ν ) T on ∂ B η s ,

(4.17) p 1 = − 1 R s 2 S + 1 2 Δ ω S on ∂ B R s .

Assume that

(4.18) S ( θ , ϕ ) = ∑ n = 0 ∞ ∑ m = − n n a n , m Y n , m ( θ , ϕ ) .

By calculations similar to those in Section 3, we explicitly solve the problem (4.10)–(4.17) as follows:

(4.19) σ 1 ( r , θ , ϕ ) = − σ s ′ ( R s ) ∑ n = 0 ∞ ∑ m = − n n a n , m Q n ( r ) Y n , m ( θ , ϕ ) in B R s \ B ¯ η s , − σ s ′ ( R s ) ∑ n = 0 ∞ ∑ m = − n n a n , m E n ( r ) Y n , m ( θ , ϕ ) in B η s ,

(4.20) T ( θ , ϕ ) = ∑ n = 0 ∞ ∑ m = − n n a n , m b n Y n , m ( θ , ϕ ) ,

and

(4.21) p 1 ( r , θ , ϕ ) = ∑ n = 0 ∞ ∑ m = − n n a n , m P n ( r ) Y n , m ( θ , ϕ ) ,

with b n , E n ( r ) , Q n ( r ) , and P n ( r ) given by (3.28)–(3.31).

We next show that the formal expansions (4.8) and (4.9) are actually rigorous. Noting that ( σ , p ) is defined on Ω ε , while ( σ s , p s ) is defined on whole R 3 and ( σ 1 , p 1 ) is defined on B R s , we introduce the Hanzawa transformation to transform all these functions to the same domain Ω ε , which is a diffeomorphism defined by

(4.22) ( r , θ , ϕ ) = H ε ( r ′ , θ ′ , ϕ ′ ) = ( r ′ + χ ( η s − r ′ ) ε T ( θ ′ , ϕ ′ ) + χ ( R s − r ′ ) ε S ( θ ′ , ϕ ′ ) , θ ′ , ϕ ′ ) ,

where

χ ∈ C ∞ , χ ( z ) = 1 , if ∣ z ∣ < δ 0 ⁄ 4 , 0 , if ∣ z ∣ ≥ 3 δ 0 ⁄ 4 , d k χ d z k ≤ C δ 0 k ,

and δ 0 is a small positive constant such that δ 0 < min { 4 η s ⁄ 3,2 ( R s − η s ) ⁄ 3 } . Then, H ε maps B R s ( B η s ) onto Ω ε ( Q ε ) while keeping the ball { r ′ < η s − 3 δ 0 ⁄ 4 } fixed, and the inverse Hanzawa transformation H ε − 1 maps Ω ε ( Q ε ) onto B R s ( B η s ). Set

σ ˜ 1 ( r , θ , ϕ ) = σ 1 ( H ε − 1 ( r , θ , ϕ ) ) , p ˜ 1 ( r , θ , ϕ ) = p 1 ( H ε − 1 ( r , θ , ϕ ) ) in Ω ε .

Then, we have the following lemma.

Lemma 4.1

Assume that S ∈ C 4 + α ( Σ ) is given by (4.18) with 1 ⁄ 2 < α < 1 and ‖ S ‖ C 4 + α ( Σ ) ≤ 1 . Then, the estimates

(4.23) ‖ σ − σ ˆ 2 − ε σ ˜ 1 ‖ C 2 + α ( Q ¯ ε ) + ‖ σ − σ ˆ 1 − ε σ ˜ 1 ‖ C 2 + α ( Ω ε \ Q ε ¯ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) ,

(4.24) ‖ p − p ˆ 2 − ε p ˜ 1 ‖ C 2 + α ( Q ¯ ε ) + ‖ p − p ˆ 1 − ε p ˜ 1 ‖ C 2 + α ( Ω ε \ Q ε ¯ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ )

are valid uniformly for small ∣ ε ∣ with C independent of ε and S.

Proof

First, since T admits the representation (4.20), by using Lemmas 2.1 and 3.1, we find

(4.25) ‖ T ‖ C 2 + α ( Σ ) ≤ C ‖ T ‖ C 3 ( Σ ) ≤ C ‖ T ‖ H 9 ⁄ 2 ( Σ ) ≤ C ‖ S ‖ H 9 ⁄ 2 ( Σ ) ≤ C ‖ S ‖ C 4 + α ( Σ ) .

Next, for (4.10)–(4.13), the Schauder estimates (see [17]), combined with (4.25), say that

(4.26) ‖ σ 1 ‖ C 2 + α ( B ¯ η s ) ≤ C ‖ T ‖ C 2 + α ( Σ ) ≤ C ‖ S ‖ C 4 + α ( Σ ) , ‖ σ 1 ‖ C 2 + α ( B R s \ B η s ¯ ) ≤ C ( ‖ T ‖ C 2 + α ( Σ ) + ‖ S ‖ C 2 + α ( Σ ) ) ≤ C ‖ S ‖ C 4 + α ( Σ ) .

Moreover, we derive for σ ˜ 1 the problem

(4.27) − Δ σ ˜ 1 + λ 1 σ ˜ 1 = f ˜ 1 in Q ε ,

(4.28) − Δ σ ˜ 1 + λ 2 σ ˜ 1 = f ˜ 2 in Ω ε \ Q ε ,

(4.29) σ ˜ 1 = − σ s ′ ( η s ) T on ∂ Q ε ,

(4.30) σ ˜ 1 = − σ s ′ ( R s ) S on ∂ Ω ε ,

where f ˜ 1 ⁄ ε and f ˜ 2 ⁄ ε involve at most second-order derivatives of T , S , and σ 1 , and

(4.31) ‖ f ˜ 1 ‖ C α ( Q ¯ ε ) ≤ C ∣ ε ∣ ‖ S ‖ C 4 + α ( Σ ) ,

(4.32) ‖ f ˜ 2 ‖ C α ( Ω ε \ Q ε ¯ ) ≤ C ∣ ε ∣ ‖ S ‖ C 4 + α ( Σ ) .

Now, set

ψ = σ − σ ˆ 1 − ε σ ˜ 1 in Ω ε \ Q ε , σ − σ ˆ 2 − ε σ ˜ 1 in Q ε .

Then, by (3.1), (4.1), (4.3), (4.4), and (4.27)–(4.30), we obtain

− Δ ψ + λ 1 ψ = − ε f ˜ 1 in Q ε , − Δ ψ + λ 2 ψ = − ε f ˜ 2 in Ω ε \ Q ε , ψ − = σ Q − σ ˆ 2 ( η s + ε T ) + ε σ s ′ ( η s ) T on ∂ Q ε , ψ + = σ Q − σ ˆ 1 ( η s + ε T ) + ε σ s ′ ( η s ) T on ∂ Q ε , ψ = σ ¯ − σ ˆ 1 ( R s + ε S ) + ε σ s ′ ( R s ) S on ∂ Ω ε ,

and by (4.25),

‖ σ Q − σ ˆ 2 ( η s + ε T ) + ε σ s ′ ( η s ) T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) , ‖ σ Q − σ ˆ 1 ( η s + ε T ) + ε σ s ′ ( η s ) T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) , ‖ σ ¯ − σ ˆ 1 ( R s + ε S ) + ε σ s ′ ( R s ) S ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 2 + α ( Σ ) ,

which together with (4.31) and (4.32) implies (4.23).

Finally, we turn to the proof of (4.24). The Schauder estimates for solutions of elliptic equations and those near the boundary for solutions of elliptic systems (see [1]), combined with (4.25) and (4.26), show that

(4.33) ‖ p 1 ‖ C 2 + α ( B ¯ η s ) + ‖ p 1 ‖ C 2 + α ( B R s \ B η s ¯ ) ≤ C ‖ S ‖ C 4 + α ( Σ ) .

Similarly, denote

φ = p − p ˆ 1 − ε p ˜ 1 in Ω ε \ Q ε , p − p ˆ 2 − ε p ˜ 1 in Q ε ,

for which there holds

− Δ φ = − ε f ˜ 3 in Q ε , − Δ φ = μ ( σ − σ ˆ 1 − ε σ ˜ 1 ) − ε f ˜ 4 in Ω ε \ Q ε , [ φ ] = − p ˆ 1 ( η s + ε T ) + p ˆ 2 ( η s + ε T ) on ∂ Q ε , [ ∂ n φ ] = − ∂ n p ˆ 1 + ∂ n p ˆ 2 − ε μ ( σ Q − σ ˜ + ν ) T − ε g ˜ 1 on ∂ Q ε , φ = κ − p ˆ 1 ( R s + ε S ) + ε 1 R s 2 S + 1 2 Δ ω S on ∂ Ω ε ,

and

‖ f ˜ 3 ‖ C α ( Q ¯ ε ) ≤ C ∣ ε ∣ ‖ S ‖ C 4 + α ( Σ ) , ‖ f ˜ 4 ‖ C α ( Ω ε \ Q ε ¯ ) ≤ C ∣ ε ∣ ‖ S ‖ C 4 + α ( Σ ) , ‖ g ˜ 1 ‖ C 1 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) , ‖ − p ˆ 1 ( η s + ε T ) + p ˆ 2 ( η s + ε T ) ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) , ∥ − ( ∂ n p ˆ 1 − ∂ n p ˆ 2 ) ∣ ∂ Q ε − ε μ ( σ Q − σ ˜ + ν ) T ∥ C 1 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ T ‖ C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) , κ ∣ ∂ Ω ε − p ˆ 1 ( R s + ε S ) + ε 1 R s 2 S + 1 2 Δ ω S C 2 + α ( Σ ) ≤ C ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) .

Combining with (4.23), we therefore obtain (4.24) and complete the proof of the lemma.□

In the sequel, we take μ as the bifurcation parameter and obtain the existence of symmetry-breaking stationary solutions to (1.1)–(1.9) by using the Crandall-Rabinowitz theorem.

By using (4.24) and (4.33), we compute

(4.34) F ( R ˜ , μ ) = ε p s ″ ( R s ) S + ∂ p 1 ∂ r ( R s , θ , ϕ ) + O ( ∣ ε ∣ 2 ‖ S ‖ C 4 + α ( Σ ) ) .

As in [16], we introduce the Banach spaces

X l + α = { R ˜ ∈ C l + α ( Σ ) , R ˜ is π − periodic in θ , 2 π − periodic in ϕ } ,

X 2 l + α = closure of the linear space spanned by { Y n , 0 ( θ ) , n = 0 , 2 , 4 , … } in X l + α ,

and take X = X 2 4 + α , Y = X 2 1 + α with 1 ⁄ 2 < α < 1 . Observing that Y n , 0 ( π − θ ) = Y n , 0 ( θ ) if and only if n is even, X 2 l + α coincides with the subspace of the C l + α ( Σ ) -closure of the smooth functions consisting of those functions u that are independent of ϕ and satisfy u ( θ ) = u ( π − θ ) . Thus, F maps X into Y . The relation (4.34) shows that the mapping ( R ˜ , μ ) → F ( R ˜ , μ ) from X 2 l + 3 + α to X 2 l + α is bounded if l = 1 , and the same argument shows that the same is true for any l ≥ 1 . Besides, note that the above argument shows that the differentiability is eventually reduced to the regularity of the corresponding PDEs, and an explicit formula is not needed if we are only interested in differentiability; therefore, a similar argument shows that this mapping is Fréchet differentiable in ( R ˜ , μ ) ; furthermore, ∂ F ( R ˜ , μ ) ⁄ ∂ R ˜ (or ∂ F ( R ˜ , μ ) ⁄ ∂ μ ) is obtained by solving a linearized problem about ( R ˜ , μ ) with respect to R ˜ ( or μ ) . By using the Schauder estimates, one can then obtain differentiability of F ( R ˜ , μ ) to any order.

In view of (4.34), the Fréchet derivative of F ( R ˜ , μ ) with respect to R ˜ at ( 0 , μ ) is given by

(4.35) [ F R ˜ ( 0 , μ ) ] S = p s ″ ( R s ) S + ∂ p 1 ∂ r ( R s , θ , ϕ ) .

By (4.21) and (3.31),

∂ p 1 ∂ r ( R s , θ , ϕ ) = ∑ n = 0 ∞ ∑ m = − n n a n , m μ λ 2 n η s σ s ′ ( η s ) b n − μ λ 2 σ s ′ ( R s ) Q n ′ ( η s ) + μ ( σ Q − σ ˜ + ν ) b n η s n + 2 R s n + 2 + μ λ 2 σ s ′ ( R s ) Q n ′ ( R s ) − n R s + n R s 3 n ( n + 1 ) 2 − 1 Y n , m .

Substituting this and (3.32) into (4.35) yields that

[ F R ˜ ( 0 , μ ) ] S = ∑ n = 0 ∞ ∑ m = − n n a n , m ( A n − μ B n ) Y n , m ,

where A n and B n are defined by (3.33) and (3.34), respectively. In particular,

[ F R ˜ ( 0 , μ ) ] Y n , m = ( A n − μ B n ) Y n , m .

Proof of Theorem 1.2

Let

n * * = min { n : n ≥ n * , μ n > max { μ 2 , μ 3 , … , μ n * − 1 } } ,

which is well defined by Lemma 3.3. Noting that B 0 < 0 by the assertions (i) and (ii) of Lemma 3.2, according to the definition of n * * and using Lemma 3.3 again, we have that for even n ≥ n * * , Ker [ F R ˜ ( 0 , μ n ) ] = span { Y n , 0 } , Y 1 = Im [ F R ˜ ( 0 , μ n ) ] has codimension 1, and [ F μ R ˜ ( 0 , μ n ) ] Y n , 0 = − B n Y n , 0 ∉ Y 1 . Applying the Crandall-Rabinowitz theorem, we thus obtain the result and complete the proof of the theorem.□

Acknowledgments

The authors are grateful for the reviewers’ valuable comments that improved the manuscript.

Funding information: This work was partly supported by the National Natural Science Foundation of China (Nos. 12261047, 12161045, and 12561035), the Jiangxi Provincial Natural Science Foundation (Nos. 20243BCE51015, 20224BCD41001, and 20232BAB201010), the Science and Technology Project Founded by the Education Department of Jiangxi Province (No. GJJ2200319), and the Graduate Innovation Fund of Jiangxi Normal University (No. YJS2023072).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript, consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. Y.L. was responsible for the investigation, methodology, and writing the original draft. H.S. and Z.W. contributed to supervision, methodology, reviewing, and editing the manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data were used for the research described in the manuscript.

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Received: 2025-05-28

Revised: 2025-09-13

Accepted: 2025-10-28

Published Online: 2025-11-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0128

Keywords for this article

free boundary problem; double-layered tumor; linear stability; bifurcation; stationary solution

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