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Linearized stability for a multi-dimensional free boundary problem modeling two-phase tumor growth

This paper is concerned with a multi-dimensional free boundary problem modeling the growth of a tumor with two species of cells: proliferating cells and quiescent cells. This free boundary problem has a unique radial stationary solution. By using the Fourier expansion of functions on unit sphere via spherical harmonics, we establish some decay estimates for the solution of the linearized system of this tumor model at the radial stationary solution, so that proving that the radial stationary solution is linearly asymptotically stable when neglecting translations.


Introduction
Since Greenspan first used free boundary problems of partial differential equations to model the growth of solid tumors in 1972 (cf. [27,28]), many different tumor models in terms of free boundary problems of partial differential equations have been proposed by different groups of researchers, cf., the reviewing articles [1,17,19,20,21,29] and references cited therein. Rigorous mathematical analysis of such models has made great progress during the past twenty more years, and many interesting results have been obtained, cf., [3,4,5,6,7,8,9,10,11,13,22,23,24,25,26,34,35] and references cited therein. From these mentioned references it can be seen that dynamics of such tumor models are usually very rich, and some recently developed mathematical tools have played important role in their rigorous analysis. Major concern in this topic is the asymptotic behavior of solutions as time goes to infinity.
Models describing the growth of tumors possessing homogeneous structures or consisting of only one species of cells usually can be reduced into differential equations in Banach spaces possessing parabolic structures. It turns out that asymptotic behavior of their solutions can be well treated by using the abstract theory for parabolic differential equations in Banach spaces, cf. [6,9,10,34,35] and references therein (see also [22,23,24,25,26] and references therein for the analysis by using some classical methods). Unlike this, for tumors with inhomogeneous structures, their growth models are more complex and the corresponding rigorous analysis is much more difficult. Here we particularly mention the models reviewed in [19,20,21], which describes the growth of tumors consisting of more than one species of cells. For those models, some interesting results have been obtained in the spherically symmetric case (cf. [3,4,5,7,11,13]). For the general non-symmetric case, however, only local well-posedness is established (cf. [2]), and large-time behavior of the solution is totally unclear up to the present.
In this paper we study a tumor model describing the growth of spherically non-symmetric tumors consisting of two species of cells: proliferating cells and quiescent cells. Mathematical formulation of this model is the following multi-dimensional free boundary problem: ∆c = F (c) for x ∈ Ω(t), t > 0, (1.1) c = 1 for x ∈ ∂Ω(t), t > 0, (1.2) ∂p ∂t + ∇ · ( vp) = [K B (c) − K Q (c)]p + K P (c)q for x ∈ Ω(t), t > 0, (1.3) ∂q ∂t + ∇ · ( vq) = K Q (c)p − [K P (c) + K D (c)]q for x ∈ Ω(t), t > 0, (1.4) p + q = 1 for x ∈ Ω(t), t > 0, (1.5) v = −∇̟ for x ∈ Ω(t), t > 0, (1.6) ̟ = γκ for x ∈ ∂Ω(t), t > 0, (1.7) V n = v · n for x ∈ ∂Ω(t), t > 0. (1.8) Here Ω(t) is the domain occupied by the tumor at time t, c = c(x, t), p = p(x, t) and q = q(x, t) are the concentration of nutrient, the density of proliferating cells and the density of quiescent cells, respectively, v = v(x, t) is the velocity of tumor cell movement, ̟ = ̟(x, t) is the pressure distribution in the tumor, κ is the mean curvature of the tumor surface whose sign is designated by the convention that κ ≥ 0 at points where ∂Ω(t) is convex, n is the unit outward normal vector of ∂Ω(t), and V n is the normal velocity of the tumor surface. Besides, F (c) is the consumption rate of nutrient by tumor cells, K B (c) is the birth rate of tumor cells, K P (c) and K Q (c) are transferring rate of tumor cells from quiescent state to proliferating state and from proliferating state to quiescent state, respectively, and K D (c) is the death rate of quiescent cells. Typically we have (cf. [31]) F (c) = λc, (1.9) where λ, k B , k D , k P and k Q are positive constants. Finally, γ is a positive constant and is referred as surface tension coefficient. For illustration of biological implications of each equation in the above model, we refer the reader to see [7,19,20,21,31,32] and references therein. By summing up (1.3), (1.4) and using (1.5), we get ∇ · v = K B (c)p − K D (c)q for x ∈ Ω(t), t > 0. (1.11) Besides, we note that due to (1.5), the unknown variables p and q are not independent. In what follows we shall keep p only. It follows that the system (1.1)-(1.8) reduces into the following one: ∆c = F (c) for x ∈ Ω(t), t > 0, (1.12) c = 1 for x ∈ ∂Ω(t), t > 0, (1.13) ∂p ∂t + v · ∇p = f (c, p) for x ∈ Ω(t), t > 0, (1.14) ∇ · v = g(c, p) for x ∈ Ω(t), t > 0, (1.15) v = −∇̟ for x ∈ Ω(t), t > 0, (1.16) ̟ = γκ for x ∈ ∂Ω(t), t > 0, (1.17) V n = −∂ n ̟ for x ∈ ∂Ω(t), t > 0, (1.18) where We note that the equations (1.15)-(1.17) can be regarded as an elliptic boundary value problem for the unknown ̟. Thus, since κ is a (quasi-linear) second-order elliptic operator for the unknown function ρ describing the free boundary ∂Ω(t), by solving (1.15)-(1.17) to get ̟ as a functional of c, p, ρ and next substituting it into (1.18), we see that the equation (1.18) can be reduced into a (quasi-linear) third-order parabolic pseudo-differential equation for ρ (containing other unknown functions), cf. [15,16]. On the other hand, the equation (1.14) is clearly a quasilinear hyperbolic equation for the unknown function p (containing other unknown functions). This determines that the above model can neither be treated as purely parabolic type equations as in [9,10,34,35], nor can it be dealt with as a purely hyperbolic equation, which is the main point where the difficulty of the above problem lies.
As we have mentioned before, local existence and uniqueness of a classical solution of the initial value problem for the above system has been well established by Chen and Friedman in [2] in more general setting. In [11] and [3] it was proved that the above system has a unique radial stationary solution under the following general conditions on the given functions F , K B , K D , K P , and K Q : F, K B , K D , K P and K Q are C ∞ -functions; (1.22) 0) = 0 and K D (1) = 0; K P and K Q satisfy the same conditions as K B and K D , respectively; (1.24) Moreover, in [5] it was proved that this unique radial stationary solution is asymptotically stable under radial perturbations (see [7] for an extension of this result). Naturally, we want to know if this unique radial stationary solution is also asymptotically stable under non-radial perturbations? Up to now we are unable to give a satisfactory answer to such a difficult question. The purpose of this paper is to prove the following weaker result: Theorem 1.1 Let K B , K D , K P and K Q be given by (1.10) with coefficients satisfying the following conditions: There exists a constant γ * > 0 such that for γ > γ * , the unique radial stationary solution of the system (1.12)-(1.18) is linearly asymptotically stable modula translations, i.e., the trivial solution of the linearized system of (1.12)-(1.18) at its unique radial stationary solution is asymptotically stable modula translations in suitable function spaces.
Here the phrase "modula translations" is used to refer to the following property of the system (1.12)-(1.18): Since this system is invariant under translations in the coordinate space, its stationary solutions are not isolated, and by translating a given stationary solution we get a n-parameter family of stationary solutions. Thus stationary solutions of the above system form a n-dimensional manifold -the so-called center manifold. It follows that the trivial solution of the linearized system is also not an isolated stationary solution, but instead, all stationary solutions of the linearized system make up a n-dimensional linear space. Hence, to study asymptotic stability of the trivial solution for the linearized system, we must make analysis in certain quotient spaces. See Theorem 8.1 in Section 8 for more explicit statement of the above result.
We remark that in (1.25), the condition k B > k D is essential and it cannot be removed. Indeed, if this condition is removed then the system (1.12)-(1.18) does not have a stationary solution and the tumor will finally disappear, cf. [12]. Unlike this, the other conditions in (1.25) are imposed just for technical reasons; see Lemma 3.2 in Section 3 and Lemmas 6.2-6.4 in Section 6. We conjecture that the rest conditions can be removed without affecting the validity of the above result.
Throughout this paper we shall make discussion for the general n-dimension version of the system (1.12)-(1.18) with n ≥ 2. This will enable us to use some abstract theory of differential equations and spherical harmonic functions and avoid using concrete expressions of Bessel functions and 3-dimensional spherical harmonics. We note that all discussions made in the literature [2], [3], [5] and [11] can be easily extended to the general n-dimension case, so that, in particular, the conditions (1.22)-(1.24) ensure that the system (1.12)-(1.18) has also a unique radial stationary solution in the general n-dimension case.
The structure of the rest part is as follows: In the next section we compute the linearization of the system (1.10)-(1.16) around the radial stationary solution and use the spherical harmonic expansion to make reduction to the linearized system. In section 3 we collect a few preliminary lemmas. In Sections 4-7 we step by step establish decay estimates for each mono-mode system obtained from the spherical harmonic expansion of the linearized system. In the last section we compose up all the mono-mode estimates to get desired result.

Linearization and reduction
We denote by (c s (r), p s (r), v s (r), σ s (r), Ω s ) (Ω s = {x ∈ R n : r = |x| < R s ) the unique radial stationary solution of the system (1.10)-(1.16), namely, the solution of the following system of equations: Later on we shall use the following notations: , p s (r)), g * p (r) = g p (c s (r), p s (r)).
The system (2.11)-(2.17) can be reduced into a 2-system of linear evolution equations containing only the unknowns ϕ and η. To see this we denote by K , K 0 and G respectively the following operators: For η ∈ C 2 (S n−1 ), let u = K (η) ∈ C 2 * (Ω s ) and v = K 0 (η) ∈ C 2 * (Ω s ), where C 2 * (Ω s ) represents the second-order Zygmund space in Ω s , be respectively solutions of the following elliptic boundary value problems: For h ∈ C(Ω s ), let w = G (η) ∈ C 2 * (Ω s ) be the solution of the following elliptic boundary value problem: Then from (2.11) and (2.12) we have and from (2.15) and (2.16) we have Substituting these expressions into (2.13) and (2.17), we see that the system (2.11)-(2.17) reduces into the following 2-system: where ω represents a variable in the sphere S n−1 , be a spherical harmonics of degree k (cf. [33]), i.e., Y k (ω) is a nontrivial solution of the following equation: . Consider a solution of (2.19) of the form where r = |x| and ω = x |x| . Using the identity ∆ϕ = ∂ 2 ϕ ∂r 2 + n−1 r ∂ϕ ∂r + 1 r 2 ∆ ω ϕ we easily see that where u k is the solution of the following problem: (2.20) Moreover, for any f ∈ C[0, R s ] we have It follows that Hence we get , and for φ = φ(r), and adding it into (2.22), we see that the system (2.22)-(2.23) reduces into the following equivalent one: and Using these facts, one may easily check that This implies that in the case k = 1, (2.24)-(2.25) has the following stationary solution: Or equivalently, in the case k = 1 the system (2.22)-(2.23) has the following stationary solutions: where c is an arbitrary real constant. This means that the system (2.19) has infinite many stationary solutions, and all its stationary solutions form a n-dimensional linear space (cf. (8.2) in Section 8).

Some preliminary lemmas
In this section we collect some preliminary lemmas.
Lemma 3.1 The following inequalities hold for all 0 ≤ r ≤ 1: Proof: From (1.16) and (1. 17) we see that  [3] we know that there exists a constant c 0 > 0 such that as r → 0 + , Hence, we only need to prove that θ > 2, or equivalently, f * Hence θ > 2 if and only if This is equivalent to It is easy to check that the conditions in (1.25) ensure that the above inequality holds. Hence the desired assertion follows. ✷ Lemma 3.3 For the solution u k of the problem (2.20), we have the following assertions: is monotone non-decreasing in k, i.e., u k (r) ≥ u l (r) for 0 ≤ r ≤ R s and k > l.
Proof: The problem (2.20) can be regarded as the spherically symmetric form of the n+2kdimensional elliptic boundary value problem From this fact the assertion (1) immediately follow. Next, from the first equation This proves (3.5). Since u k (R s ) = 1, by integrating (3.5) over (r, R s ) we get (3.4). This proves the assertion (2). The assertion (3) follows from the fact that u ′ k (r) ≥ 0 and the maximum principle for second-order elliptic equations. The assertion (4) follows from direct computation.
Indeed, a direct computation shows that the functionũ 1 satisfies the same equation we see thatũ ′ 1 (0) = 0. Hence, by uniqueness of the solution of the elliptic boundary value problem we getũ 1 (r) = u 1 (r) for 0 ≤ r ≤ R s . Finally, it is easy to check that the functioñ Using the fact that u ′ 0 (r) ≥ 0 we can also easily see that u 0 (r) satisfies the inequality Since u ′ 0 (0) =ũ ′ 0 (0) and u 0 (R s ) =ũ 0 (R s ), by the maximum principle we see that the desired assertion follows. This completes the proof of Lemma 3.3. ✷

Decay estimates for some positive semigroups
In this preliminary section we establish decay estimates for some positive semigroups in For any function ϕ defined in [0, R s ] such that the right-hand side of the following equality makes sense, We shall regard L 0 both as an unbounded closed linear operator in R s ]} and as an unbounded closed linear operator in It is clear that λ 0 (a) ≤ λ 1 (a) ≤ λ * (a). In what follows the notation λ denotes a complex number, and h denotes a complex-valued function defined in [0, R s ].

Lemma 4.1
We have the following assertions: (1) If Reλ > λ 0 (a) then for any h ∈ C(0, R s ] the equation Moreover, for any 0 < r 0 < R s there exists a corresponding constant C λ (r 0 ) > 0 such that If furthermore λ is real and h(r) ≥ 0 for 0 < r ≤ R s then also ϕ λ (r) ≥ 0 for 0 < r ≤ R s .
(2) If Reλ > λ 1 (a) then for any h ∈ C[0, R s ] the unique solution of (4.1) ensured by the above assertion belongs to C[0, R s ] ∩ C 1 (0, R s ), and in addition to (4.2) we have also that Moreover, there exists a constant C λ > 0 such that max 0≤r≤Rs |ϕ λ (r)| ≤ C λ max 0≤r≤Rs |h(r)|.  Proof: We first assume that Reλ > λ 0 (a). Choose a number r 0 ∈ (0, R s ) and set It is easy to see that W λ ∈ C 1 (0, R s ), and where , and C is a nonzero constant (depending on the choice of r 0 ). Note that Re α 1 > 0. Clearly, the equation (4.1) can be rewritten as follows: Letting c = ϕ(r 0 ) and integrating both sides of this equation from r 0 to an arbitrary point 0 < r < R s , we see that the general solution of the equation (4.1) is given by is a finite number (by (4.7) and (2.10)), we see that lim Hence the solution which is bounded near r = R s is unique, and this unique bounded solution is given by which is continuous in (0, R s ], continuously differentiable in (0, R s ), and satisfies (4.2) (as we have seen above). The estimate (4.3) easily follows from (4.9) because the function r → 1 |W λ (r)| Rs r |W λ (η)| |v s (η)| dη is continuous in (0, R s ]. Moreover, if λ is real then W λ (r) > 0 for 0 < r < R s . Using this fact and the expression (4.9) we easily see that if h ≥ 0 then also ϕ λ ≥ 0. This proves the assertion (1). Next we assume that Reλ > λ 1 (a). Then in addition to (4.7) we have also that , and C is a nonzero constant (depending on the choice of r 0 ). Note that Re α 0 > 0. Using this fact we can easily deduce that for any constant c the function ϕ given by (4.8) satisfies Hence, all solutions given by (4.8) are continuous at r = 0 and satisfy (4.4). In particular, the solution ϕ λ given by (4.9) belongs to C[0, R s ] ∩ C 1 (0, R s ) and satisfy both (4.2) and (4.4). The estimate (4.5) easily follows from (4.9) because in the present situation the function r This proves the assertion (2).
Proof: We fulfill the proof through three steps.
Step 1: We first prove that the equation (4.11) has a unique solution in the class C(0, R s ] ∩ C 1 (0, R s ). To this end, we explicitly write out the equation (4.11) as follows: Let W λ (r) be as before but with a(r) replaced with a k (r). By rewriting the above equation in the form we easily see that, as far as solutions which are bounded near r = R s are concerned, the differential-integral equation (4.22) is equivalent to the following integral equation: (4.20) It follows from a standard contraction mapping argument (similar to that used in the proof of Theorem 5.3 (1) of [3]) that there exists a sufficiently small δ > 0 such that (4.19) has a unique bounded solution in the interval (R s − δ, R s ), such that ϕ ∈ C(R s − δ, R s ] ∩ C 1 (R s − δ, R s ), and Since v s (r) = 0 for 0 < r < R s , the equation (4.19) is a regular linear differential-integral equation at any point in (0, R s ), so that by standard ODE theory we can uniquely extend the solution to the whole interval (0, R s ] such that ϕ ∈ C(0, R s ] ∩ C 1 (0, R s ). This fulfills the task of the first step.
We note that if λ is real, λ > µ k and h ≥ 0, then also ϕ ≥ 0. Indeed, since λ is real, we have W λ (r) > 0 for 0 < r < R s . If h(R s ) > 0 then by (4.21) we see that ϕ(R s ) > 0. Let r 0 be the smallest number such that ϕ(r) > 0 for r 0 < r ≤ R s . Then by (4.20) we must have r 0 = 0. The assertion for the case h(R s ) = 0 follows from a limit argument.
Step 2: We next prove that the solution ensured by the above step satisfies To prove this assertion we note that from (4.20) we have It follows that for any 0 < r < r ′ ≤ R s we have

From (4.4) and (4.5) we have
where 0 < C 1 < C 2 , Reα 0 > 0 and Reα 1 > 0. Using these estimates and (3.2) we have Hence there exists a constant δ > 0 independent of k such that if 0 < r ′ − r ≤ δ then which implies that Hence, by dividing the interval [0, R s ] into finite number (depending on k because W λ depends on k) of subintervals and using an iteration argument, we see that there exists a constant C k,λ > 0 depending on k such that This fulfills the task of the second step.
Step 3: From the assertion obtained in the above step, it follows that the function belongs to C[0, R s ], and max 0≤r≤Rs |h 1 (r)| ≤ C k,λ max 0≤r≤Rs |h(r)|. Hence, by rewritting the equation and applying the assertion (2) of Lemma 4.1, we get the desired assertion. This completes the proof of Lemma 4.3. ✷ Corollary 4.5 The operatorL + k generates a positive C 0 -semigroup e tL + k in C[0, R s ] satisfying the following estimate: For any µ > µ k there exists corresponding constant C k,µ > 0 such that max (4.24) and B k is the integral part ofL + k . By Corollary 4.2 we see thatL 0 k generates a positive C 0semigroup in C[0, R s ]. Since clearly B k is a positive bounded linear operator in C[0, R s ], by a standard perturbation theorem (cf. Corollary 1.11 in Chapter VI of [14]) for C 0 -semigroups we see thatL + k also generates a positive C 0 -semigroup in C[0, R s ]. By Lemma 4.4 we see that the spectral bound ofL + k is not greater than µ k : s(L + k ) ≤ µ k . Hence by a similar argument as in the proof of Corollary 4.2 we obtain the estimate (4.23). ✷ Let J be the following continuous linear functional in L 1 [0, R s ]: Proof: Since the semigroup e tL + k is positive, we see that ψ 0 ≥ 0 implies ψ ≥ 0. To prove (4.25) we note that ψ is a solution of the following equation: ∂ t ψ =L + k ψ for 0 < r < R s , t > 0. (4.27) Using this fact we compute It is clear that the first two terms on the right-hand side of the above equality are negative for 0 < r < R s . In what follows we prove that if the conditions in (1.25) are satisfied then We first note that, since g * p (r) = K M (c s (r)) is monotone increasing in r, we have Note that the conditions in (1.25) imply that the function c → monotone increasing for c > 0. Hence c s (r)) .
From these estimates we see that (4.29) follows. Having proved (4.29), we see that andã k (r) ≤ −κ 0 for 0 ≤ r ≤ R s . Using this result and (4.28), we see that (4.25) follows. To prove (4.26) we multiply the equation in (4.27) with sgnψ, which yields the following relation: ∂ t |ψ| ≤L + k |ψ| for 0 < r < R s , t > 0. Using this fact and a similar argument as above we obtain (4.26). This completes the proof of Lemma 4.6. ✷ 5 Decay estimates for the equation ∂ t ϕ =L k (ϕ) for large k In this and the next sections we establish decay estimates for the solution of the following initial value problem: In what follows we consider the case that k is sufficiently large; the rest cases will be treated in the next subsection. We denote Since f * p (r) < 0 for all 0 ≤ r ≤ R s , we see that ν 0 < 0.
Lemma 5.1 For any µ > ν 0 there exist corresponding positive integer k µ and positive constant C = C µ such that for any k ≥ k µ and ϕ 0 ∈ C[0, R s ], the solution of the initial value problem (5.1) satisfies the following estimate: Proof: Let L 0 be the following unbounded closed linear operator in By Corollary 4.2 we see that L 0 generates a positive C 0 -semigroup e tL 0 in C[0, R s ] satisfying the following estimate: For any µ > ν 0 , Now for each integer k ≥ 2 we denote by K k the following bounded linear operator in C[0, R s ]: For any ϕ ∈ C[0, R s ], It is easy to see that there exists a positive constant C independent of k such that for any k ≥ 3, SinceL k = L 0 +K k (k = 2, 3, · · · ), by using a standard perturbation theorem for C 0 -semigroups we deduce from the above estimate and (5.3) that for any µ > ν 0 , The desired assertion immediately follows from this result. This proves Lemma 5.1. ✷ For every α ≥ 1 we denote .
Note that ≤0 for 0 ≤ r ≤ R s .
Lemma 5.2 For any α ≥ 1 and µ > µ * α there exist corresponding positive integer k α,µ and positive constant C = C α,µ such that for any k ≥ k µ and ϕ 0 ∈ L α ([0, R s ]; r n−1 dr), the solution of the initial value problem (5.1) satisfies the following estimate: Proof: We first establish a L α -estimate for e tL 0 . Given ϕ 0 ∈ C[0, R s ] we let ϕ(r, t) = e tL 0 ϕ 0 (r). Then ϕ is a solution of the following initial value problem:  This means that, by using the abbreviation L α for L α ([0, R s ]; r n−1 dr), we have Next, it is not hard to prove that there exists a constant C α > 0 independent of k such that for any k ≥ 3 and φ ∈ L α ([0, R s ]; r n−1 dr), SinceL k = L 0 + K k (k = 2, 3, · · · ), from (5.7), (5.8) and a standard perturbation theorem for C 0 -semigroups we see that the desired assertion follows. ✷ 6 Decay estimates for the equation ∂ t ϕ =L k (ϕ) for small k A similar estimate for small k is much more involved. In what follows we consider this case. For every nonnegative integer k, we denote byL k the following differential-integral operator: where a k (r) is as before (see (4.16)). By using the relations we can easily get the following relation: It follows that if we let ϕ(r, t) = r −(n+k−1) p ′ s (r)ψ(r, t) and ϕ 0 (r) = r −(n+k−1) p ′ s (r)ψ 0 (r), then ϕ is a solution of the problem (5.1) if and only if ψ is a solution of the following problem:    ∂ t ψ =L k ψ for 0 < r < R s , t > 0, Note that by denoting Lemma 6.1 Let ψ(r, t) be the solution of the problem (6.3), κ(r, t) = e tL + k e k (r), and ψ(r, t) = e tL + k ψ 0 (r). Let Ψ(t) = J(ψ), K(t) = J(κ) andΨ(t) = J(ψ). Then the following relation holds: The proof is similar to that of Lemma 8.2 of [3], so that is omitted. ✷ Lemma 6.2 Let K ∈ C 1 [0, ∞) and assume that for some real constant σ. Then for anyΨ ∈ C[0, ∞), the unique solution Ψ of the Volterra integral equation (6.4) satisfies the following estimate: Proof: See Lemma 8.3 of [3]. ✷ Lemma 6.3 Assume that the conditions in (1.25) are satisfied. There exists a constant µ * < 0 independent of k such that for every nonnegative integer k and any µ > µ * there exists corresponding constant C = C k,µ > 0 such that for the solution of the problem (6.3) the following estimate holds: max 0≤r≤Rs |ψ(r, t)| ≤ Ce µt max 0≤r≤Rs |ψ 0 (r)| for t ≥ 0. (6.5) Proof: Let the notation be as in Lemma 6.1. Since e k ≥ 0, by Lemmas 4.5 and 4.6 we see that K(t) ≥ 0 and d dt e κ 0 t K(t) ≤ 0 for t ≥ 0. By Lemma 6.2, it follows that the following estimate holds: Moreover, applying Lemma 4.5 to e tL + k e k (r) and e tL + k ψ 0 (r) we see that for any nonnegative integer k and any µ > µ k there exists corresponding constant C = C k,µ > 0 such that max 0≤r≤Rs |e tL + k e k (r)| ≤ Ce µt for t ≥ 0, (6.7) max 0≤r≤Rs |e tL + k ψ 0 (r)| ≤ Ce µt max 0≤r≤Rs |ψ 0 (r)| for t ≥ 0. (6.8) The latter implies that |Ψ(t)| ≤ Ce µt max 0≤r≤Rs |ψ 0 (r)| for t ≥ 0. (6.9) Since µ k ≤ µ * 0 for all nonnegative integer k, the above estimate holds for any µ > µ * 0 . Substituting (6.9) into (6.6) we easily see that for any nonnegative integer k and any µ > µ * ≡ max{µ * 0 , −κ 0 } there exists corresponding constant C = C k,µ > 0 such that |Ψ(t)| ≤ Ce µt max 0≤r≤Rs |ψ 0 (r)| for t ≥ 0. (6.10) Now, noticing that From this relation and the estimates (6.7), (6.8) and (6.10), we immediately obtain (6.5). This proves Lemma 6.3. ✷ Lemma 6.4 Assume that the conditions in (1.25) are satisfied and let κ 0 be as in Lemma 4.6. For every nonnegative integer k there exists a corresponding constant C = C k > 0 such that for the solution of the problem (6.3) the following estimate holds: (6.11) Proof: Let the notation be as in Lemma 6.1. Applying Lemma 4.5 toψ(r, t) = e tL + k ψ 0 (r) we see that This implies that From this estimate and Lemmas 6.1, 6.2 we get Now, we rewrite the equation ∂ t ψ =L k ψ as follows: ∂ t ψ(r, t) =L + k ψ(r, t) − Ψ(t)e k (r).
Using this relation and a similar argument as in the proof of Lemma 4.6 we get From (6.12) and (6.13) we easily see that (6.11) follows. ✷ We are now ready to study the problem (5.1) for small k.
Step 1: We first prove that Indeed, by the relation (6.2) it follows that if ϕ is a solution of the problem (5.1) then ψ(r, t) = r n+k−1 ϕ(r, t)/p ′ s (r) is a solution of the problem (6.3) with initial data ψ 0 (r) = r n+k−1 ϕ 0 (r)/p ′ s (r). Due to this fact, the above estimate is an immediate consequence of Lemma 6.4.