1. Introduction. In this paper we shall consider the Kuramoto–Siva- shinsky equation in two space dimensions. It can be written as

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POLONICI MATHEMATICI LXXIII.3 (2000)

On the Kuramoto–Sivashinsky equation in a disk

by Vladimir Varlamov (Bogot´a)

Abstract. We consider the first initial-boundary value problem for the 2-D Kura- moto–Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion.

1. Introduction. In this paper we shall consider the Kuramoto–Siva- shinsky equation in two space dimensions. It can be written as

(1.1.) ∂

t

u + ν∆

²

u + ∆u = β|∇u|

²

,

where u = u(x

1

, x

2

, t), ∆ is the Laplace operator in x

1

, x

2

, ∇u = grad u, ν = const > 0, and β = const ∈ R.

The equation (1.1) arises in the theory of long waves in thin films [4], [33], of long waves at an interface between two viscous liquids [13], in sys- tems of the reaction-diffusion type [15], [16], and in the description of the nonlinear evolution of a linearly unstable flame front [29], [30]. The linear terms in (1) describe the interaction of long-wavelength pumping and short-wavelength dissipation, and the nonlinear term characterizes energy redistribution between various modes.

The Kuramoto–Sivashinsky equation and related model equations have been studied extensively in the eighties (mostly in the spatially one-dimensional case), both in the context of inertial manifolds and in numerical sim- ulations of dynamical behavior (see [2], [7], [8], [23], [24], and the references there). Michelson [18] investigated special solutions u(x, t) = −c

²

t + v(x) of the spatially one-dimensional equation (1.1). Setting y = v

^′

(x) he reduced

2000 Mathematics Subject Classification: 35K55, 35K65.

Key words and phrases: Kuramoto–Sivashinsky equation, disk, first initial-boundary value problem, long-time asymptotics.

[227]

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it to the ordinary differential equation

(1.2) y

^′′′

+ y

^′

= c

²

− y

²

which he studied numerically. The equation (1.2) was examined analytically in [26], [34], and in the latter paper from the point of view of singular perturbations. In [19], [20] Michelson showed that a slight modification of (1.1),

(1.3) ∂

t

u + ν∆

²

u + ∆u + |∇u|

²

= c

²

,

possesses stationary solutions. In the context of combustion theory these solutions represent Bunsen flames on infinite linear or circular burners. In [20] Michelson examined the linear stability of the radially symmetric solutions of (1.3) in a disk with the boundary conditions u|

^∂Ω

= ∆u|

^∂Ω

= 0.

We shall also use these conditions below in our analysis of the long-time behavior of solutions of the spatially two-dimensional equation (1.1) in a circular domain.

As regards spatially periodic solutions of the Kuramoto–Sivashinsky equation and their stability, we must point out that Nicolaenko, Scheurer, and Temam [24] showed that the existence of a global absorbing ball implied the existence of a global attractor and gave an upper estimate of its Haus- dorff dimension. Under the assumption that the initial data is odd, they proved the existence of a bounded global absorbing set in L

2

(0, l ) for the derivative Kuramoto–Sivashinsky equation. Collet, Eckmann, Epstein, and Stubbe [6] and independently Goodman [12] got rid of this antisymmetry requirement. Berloff and Howard [3] considered the generalized derivative Kuramoto–Sivashinsky equation

∂

_t

u + ∂

_x⁴

u + σ∂

_x³

u + ∂

_x²

u + 2u∂

_x

u = 0

and constructed a periodic wave train solution by means of the singular manifold method and partial fraction decomposition.

In the two-dimensional case an important problem was to show the existence of a bounded absorbing set in L

₂

(Ω). Sell and Taboada [28] gave the first answer to this question by means of proving the existence of a bounded local absorbing set in H

¹_per

([0, 2π] × [0, 2πε]) for ε small enough.

They adapted the method used by Raugel and Sell [27] for the Navier–Stokes equation in a three-dimensional thin domain. In his interesting study [21]

Molinet improved the results of [28] and examined the local stability of the solutions of the reduced Kuramoto–Sivashinsky equation with spatially periodic boundary conditions in a thin rectangular domain. He gave a sufficient condition on the width l

2

of the domain depending on the length l

1

, so that there exists a bounded local absorbing set in L

2,per

, and estimated this set.

As well as Sell and Taboada, Molinet used the derivative form of (1.1), that

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is, he set ∇u = (v

¹

, v

2

) and reduced (1.1) to the system

∂

t

v

1

+ ν∆

²

v

1

+ ∆v

1

+ v

1

∂

x

v

1

+ v

2

∂

x

v

2

= 0,

∂

t

v

2

+ ν∆

²

v

2

+ ∆v

2

+ v

1

∂

y

v

1

+ v

2

∂

y

v

2

= 0,

∂

_y

v

₁

= ∂

_x

v

₂

, which is convenient for obtaining some estimates.

In our present investigation of the Kuramoto–Sivashinsky equation we shall not use this reduction and will study (1.1) in its original form. We shall consider the first initial-boundary value problem for (1.1) in a unit disk with “small” initial conditions, homogeneous boundary conditions, and periodicity conditions in the angle. We shall prove the existence of a global in time strong solution by means of constructing it in the form of a series in a small parameter present in the initial conditions. The uniqueness will be proved via showing that the difference of two solutions from the required function space equals zero. We shall also obtain a uniform in space long-time asymptotic representation of the solution in question. The method applied includes the use of eigenfunction expansions and the theory of perturbations.

In order to explain its origins we have to give a bit of history.

One of the powerful methods of studying Cauchy problems for nonlinear evolution equations is the inverse scattering transform (IST) (see [1]). Nev- ertheless, solving initial-boundary value problems by this method remained an open question until the breakthrough made by Fokas [9] and Fokas and Its [10], [11]. However, IST does not work for a wide class of dissipative equations which are not completely integrable. Another approach was used by Naumkin and Shishmar¨ev [22] who considered nonlocal dissipative equations of the first order in time. Having applied the Fourier transform and the theory of perturbations, they solved a number of Cauchy problems with small initial data and calculated the major terms of the long-time asymptotic expansions of their solutions. In [35]–[39] this method was further developed and adapted for solving Cauchy problems, spatially periodic problems, and spatially 1-D initial-boundary value problems for nonlinear dissipative equations of the second and third order in time. A radially symmetric mixed problem in a circle was considered in [40].

In the present paper we shall show how this approach can be applied for

solving a spatially two-dimensional initial-boundary value problem in a disk

via the use of eigenfunction expansions. As a result of examining a general

spatially 2-D case, we shall not observe the effect of the “loss of smoothness”,

as in [40]. The increase of the regularity of the initial data via imposing more

periodicity conditions can still influence the smoothness of the solution in

question. After constructing the solution, we shall obtain its uniform in space

long-time asymptotic expansion in the stable case ν > 1/λ

²₀₁

, where λ

01

is

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the first positive zero of the Bessel function J

0

(z), and examine its growth in time for 0 < ν ≤ 1/λ

²01

.

2. Statement of the problem, notations, and technical lemmas.

We consider the first initial-boundary value problem for the Kuramoto–

Sivashinsky equation in the unit disk with small initial data and homogeneous boundary conditions. Using polar coordinates (r, θ) we can write it as follows:

(2.1)

∂

t

u + ν∆

²

u + ∆u = β|∇u|

²

, (r, θ) ∈ Ω, t > 0, u(r, θ, 0) = ε

²

ϕ(r, θ), (r, θ) ∈ Ω,

u|

∂Ω

= ∆u|

∂Ω

= 0,

|u(0, θ, t)| < +∞,

periodicity conditions in θ with period 2π for u and its derivatives included in the equation,

where ∆ =

¹_r

∂

r

(r∂

r

) +

_r¹2

∂

_θ²

, Ω = {(r, θ) : |r| < 1, θ ∈ [−π, π]}; ε, ν = const > 0, β = const ∈ R, and ϕ(r, θ) is a given real-valued function.

Our main tool in examining (2.1) will be the expansions in the series of the eigenfunctions of the Laplace operator in the disk. For a function f (r, θ) ∈ L

^2,r

(Ω) (L

2

(Ω) with a weight r) the corresponding expansion is (see [31])

(2.2) f (r, θ) = X

∞ n=1

X

∞ m=−∞

f b

mn

χ

mn

(r, θ) = X

m,n

f b

mn

χ

mn

(r, θ),

where χ

mn

(r, θ) are the eigenfunctions of the Laplace operator in the disk, i.e., nontrivial solutions of the problem

∆χ = −Λχ, (r, θ) ∈ Ω,

χ|

^∂Ω

= 0, χ(r, θ) = χ(r, θ + 2π), |χ(0, θ)| < ∞.

These eigenfunctions and the corresponding eigenvalues are given by the formulas

χ

mn

(r, θ) = J

m

(λ

mn

r)e

^imθ

, Λ

mn

= λ

²_mn

, m ∈ Z, n ∈ N, where J

m

(z) are the Bessel functions of index m, λ

mn

are its positive zeros numbered in increasing order, and n = 1, 2, . . . is the number of the zero.

The system of functions {χ

^mn

(r, θ)}

^{m∈Z, n∈N}

is orthogonal and complete in the space L

2,r

(Ω) (see [31]). Denoting the scalar product in L

2,r

(Ω) by (·, ·)

^r,0

and the corresponding norm by k · k

^r,0

we can write

(χ

mn

, χ

kl

)

r,0

= δ

mk

δ

nl

kχ

^mn

k

²r,0

,

where δ

ij

is the Kronecker symbol.

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We also have the Parseval identity in L

2,r

(Ω), kfk

²r,0

= X

m,n

| b f

mn

|

²

kχ

^mn

k

²r,0

.

The coefficients of the expansion (2.2) are expressed by the formulas f b

_mn

= (f, χ

mn

)

r,0

kχ

^mn

k

²r,0

= 1

kχ

^mn

k

²r,0 1

\

0

rJ

_m

(λ

_mn

r) dr

π\

−π

f (r, θ)e

^−imθ

dθ.

It will be convenient to use iterated integrals in what follows and to expand first in θ and then in r (the absolute convergence of the integrals will permit us to do that).

We shall need the weighted space L

2,r

(0, 1) (L

2

(0, 1) with a weight r) and shall denote the corresponding scalar product by (·, ·)

^r

and the norm by k · k

r

. Then

kχ

^mn

k

²r,0

= 2πkJ

^m

(λ

mn

r)k

²r

.

For a fixed integer m the system of functions {J

m

(λ

_mn

r)}

^∞n=1

is orthogonal and complete in L

2,r

(0, 1) (see [31], [41]). Expansions of the type

f (r) = X

∞ n=1

f b

m,n

J

m

(λ

mn

r), f b

m,n

= (f, J

m

(λ

mn

r))

r

kJ

m

(λ

_mn

r)k

²r

,

called Fourier–Bessel series are often used for solving radially symmetric problems in a disk (see [32]). However, if m is not fixed, the system {J

^m

(λ

mn

r)}

^{m∈Z, n∈N}

is not orthogonal in L

2,r

(0, 1).

Note that [32, p. 219]

kJ

m

(λ

_mn

r)k

²r

=

1

\

0

rJ

_m²

(λ

_mn

r) dr = J

_m+1²

(λ

_mn

)/2 and for sufficiently large positive λ,

(2.3) C

1

/λ ≤ kJ

^m

(λr)k

²r

≤ C

²

/λ.

We shall also need some properties of the zeros of the Bessel functions J

_m

(z), m ≥ 0. For bounded m large positive zeros of J

m

(z) have the following uniform asymptotics (called McMahon’s expansion; see [14, p. 153], [25, p. 247]):

λ

mn

= µ + O

1 µ

mn

, µ

mn

=

m + 2n − 1 2

π

2 , n → ∞.

For large m and n the major term of this formula still holds [41, p. 514]:

(2.4) λ

_mn

∼ (m + 2n) π

2 .

In what follows we shall need the weighted Sobolev spaces H

_r^s

(Ω), s ∈ R,

which differ from the usual Sobolev spaces H

^s

(Ω) in that instead of L

2

(Ω)

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we use the weighted space L

2,r

(Ω). We introduce the norm in H

_r^s

(Ω) by the formula (see [17])

kfk

²r,s

= X

m,n

λ

^2s_mn

| b f

mn

|

²

kχ

^mn

k

²r,0

.

Here λ

mn

> 0 for all m ∈ Z and n ∈ N. Evidently, H

r⁰

(Ω) ≡ L

^2,r

(Ω).

Our elementary space operator in (2.1) is ∆, and thanks to the orthog- onality of the functions χ

mn

(r, θ) in the space L

2,r

(Ω) we have

k∆f k

²r,0

= (∆f, ∆f )

r,0

= X

m,n

(−λ

²mn

) b f

mn

χ

mn

, X

k,l

(−λ

²kl

) b f

kl

χ

kl

r,0

= X

m,n,k,l

λ

²_mn

λ

²_kl

| b f

mn

| · | b f

kl

|(χ

^mn

, χ

kl

)

r,0

= X

m,n

λ

⁴_mn

| b f

mn

|

²

kχ

^mn

k

²r,0

.

We need to introduce the Banach space C

^k

([0, ∞), H

r^s

(Ω)) equipped with the norm

kuk

C^k

= X

k j=0

sup

t∈[0,∞)

k∂

t^j

u(t)k

^s,r

.

Now we prove two lemmas that will enable us to estimate the magnitude of the coefficients b f

mn

in the eigenfunction expansion (2.2). The first lemma is the extension of the proposition given in [41, p. 595] to the case when the integrand depends on a parameter, that is, f = f (x, α) = f

α

(x), x ∈ [0, 1], α ∈ [a, b], −∞ < a, b < ∞. We denote by V

0¹

(f

α

(x)) the total variation of the function f

α

(x) in x ∈ [0, 1].

Consider the integral I

m

(λ, α) =

1

\

0

xf (x, α)J

m

(λx) dx, m ≥ 0, λ > 0, α ∈ [a, b].

Lemma 1. If for each fixed α ∈ [a, b] the function √

xf (x, α) has a bounded total variation in x on [0, 1], V

₀¹

( √

xf

α

(x)) = V

α

; lim

x→0

√

xf (x, α)

= F

_α

, and V

_α

, F

_α

∈ L

1

(a, b), then for m ≥ 0, λ > 0, α ∈ [a, b],

|I

^m

(λ, α)| ≤ C

^α

/λ

^3/2

,

where C

α

is independent of m and λ and C

α

∈ L

¹

(a, b).

P r o o f. From the asymptotic formula as x → ∞ for the Bessel functions we see that for any z ∈ (0, ∞),

z

\

0

√ xJ

m

(x) dx

≤ c < ∞,

where c is independent of m and z.

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Set √

xf

α

(x) = ψ

α

(x). We can represent ψ

α

(x) as ψ

α

(x) = ψ

α,1

(x) − ψ

^α,2

(x),

where ψ

α,1

(x) = V

₀^x

(ψ

α

(x)) is the variation of ψ

α

(x) in [0, x], x ∈ [0, 1], and ψ

α,2

(x) = V

₀^x

(ψ

α

(x)) − ψ

^α

(x). The functions ψ

α,1

(x) and ψ

α,2

(x) are non- decreasing in x for each fixed α ∈ [a, b]. Then

ψ

_α,1

(0) = 0, ψ

_α,1

(1) = V

₀¹

(ψ

_α

(x)) = V

_α

∈ L

1

(a, b), ψ

α,2

(0) = − ψ

^α

(0) = −F

^α

∈ L

¹

(a, b),

ψ

_α,2

(1) = V

₀¹

(ψ

_α

(x)) − ψ

α

(1) = V

_α

− ψ

α

(1).

We also have |ψ

α

(1) − ψ

α

(0)| ≤ V

0¹

(ψ

_α

(x)), which implies that

|ψ

α

(1)| ≤ |ψ

α

(0)| + V

0¹

(ψ

_α

(x)) = |F

α

| + V

α

. Applying the second mean value theorem for integrals we obtain

1

\

0

ψ

α,1

(x) √

xJ

m

(λx) dx

≤

ψ

^α,1

(0)

ξ

\

0

√ xJ

_m

(λx) dx + ψ

_α,1

(1)

1\

ξ

√ xJ

_m

(λx) dx

≤ cV

^α

λ

^−3/2

,

1

\

0

ψ

α,2

(x) √

xJ

m

(λx) dx

≤

ψ

^α,2

(0)

η

\

0

√ xJ

_m

(λx) dx + ψ

_α,2

(1)

1\

η

√ xJ

_m

(λx) dx

≤ c[2|F

^α

| + V

^α

]λ

^−3/2

.

Hence follows the necessary estimate which completes the proof.

The next statement gives a tool for increasing the decay of I

_m

(λ, α) in λ.

Lemma 2. If f (x, α) has partial derivatives in x in [0, 1] through the third order and f (0, α) = ∂

_x

f (0, α) = ∂

_x²

f (0, α) = 0 (in case m = 0 only

∂

x

f (0, α) = 0), f (1, α) = ∂

x

f (1, α) = ∂

_x²

f (1, α) = 0, lim

x→0+

√

x∂

_x³

f (x, α)

= e F

α

∈ L

¹

(a, b), and for each fixed α the function √

x∂

_x³

f (x, α) has a bounded total variation in x in [0, 1] which is absolutely integrable in α ∈ (a, b), i.e. V

₀¹

( √

x∂

_x³

f

α

(x)) = e V

α

∈ L

¹

(a, b), then for m ≥ 0, λ > 0,

|I

^m

(λ, α)| ≤ C

^α

(m + 1)

³

/λ

^9/2

,

where C

α

is independent of m and λ and C

α

∈ L

¹

(a, b).

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P r o o f. By the well known formula (see [32, p. 205]), we have d

dx (x

^m+1

J

m+1

(x)) = x

^m+1

J

m

(x), m ≥ 0.

Below we shall use the notation f

α

(x) = f (x, α), and a prime will denote the derivative in x. Changing the variable of integration ξ = λx and integrating by parts we deduce that

I

m

(λ, α) = 1 λ

²

1

\

0

ξf

α

(ξ/λ)J

m

(ξ) dξ = ξ

λ

²

f

α

(ξ/λ)J

m+1

(ξ)|

^λ0

− 1 λ

²

λ\

0

1 λ f

_α^′

(ξ/λ) − m

ξ f

α

(ξ/λ)

ξJ

m+1

(ξ) dξ.

Here the first term on the right is zero because of the condition f

α

(1) = 0.

Integrating two more times by parts and using the boundary conditions f

_α^′

(1) = f

_α^′′

(1) = 0 we obtain

I

m

(λ, α) = − 1 λ

²

λ

\

0

ξ

1 λ

³

f

_α^′′′

(ξ/λ) − 2m

ξ

³

f

α

(ξ/λ) + 2m

λξ

²

f

_α^′

(ξ/λ)

− m

λ

²

ξ f

_α^′′

(ξ/λ)

+

1 λ

²

f

_α^′′

(ξ/λ) + m

ξ

²

f (ξ/λ) − m

λξ f

_α^′

(ξ/λ)

− (m + 1)

1 λ

²

f

_α^′′

(ξ/λ) + m

ξ

²

f

_α

(ξ/λ) − m

λξ f

_α^′

(ξ/λ)

− m + 3 ξ

ξ

1 λ

²

f

_α^′′

(ξ/λ) + m

ξ

²

f

_α

(ξ/λ) − m

λξ f

_α^′

(ξ/λ)

− (m + 1)

1 λ f

_α^′

(ξ/λ) − m

ξ f

α

(ξ/λ)

J

m+3

(ξ) dξ

= − 1 λ

³

1

\

0

xf

_α^′′′

(x) − 3(m + 1)f

α^′′

(x) − m

f

_α^′

(x) − f

_α

(x) x

+ (m + 1)(m + 3) x

f

_α^′

(x) − m x f

α

(x)

J

m+3

(λx) dx.

Now we have to justify the formal calculations. Since V

₀¹

( √

xf

_α^′′′

(x)) = V e

α

∈ L

¹

(a, b), lim

x→0+

√

xf

_α^′′′

(x) = e F

α

∈ L

¹

(a, b), there exists a constant M

α

∈ L

¹

(a, b) such that | √

xf

_α^′′′

(x)| ≤ M

^α

for x ∈ [0, 1]. Therefore f

α^′′′

(x)

is absolutely integrable in x in (0, 1). Expanding f

α

(x), f

_α^′

(x), and f

_α^′′

(x)

around x

0

= 0 in the integrand and using the boundary conditions f

α

(0) =

f

_α^′

(0) = f

_α^′′

(0) = 0 we can write that for x ∈ (0, 1],

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f

α

(x) = f

_α^′′′

(ϑ

1

x)

3! x

³

, 0 < ϑ

1

< 1, f

_α^′

(x) = f

_α^′′′

(ϑ

2

x)

2! x

²

, 0 < ϑ

2

< 1, f

_α^′′

(x) = f

_α^′′′

(ϑ

3

x)x, 0 < ϑ

3

< 1.

Substituting these expansions into the integrand we get I

m

(λ, α) = − 1

λ

³

1

\

0

x

f

_α^′′′

(x) − 3(m + 1)f

α^′′′

(ϑ

3

x)

− m

f

_α^′′′

(ϑ

2

x)

2! − f

_α^′′′

(ϑ

1

x) 3!

+ (m + 1)(m + 3)

f

_α^′′′

(ϑ

2

x)

2! − m f

_α^′′′

(ϑ

1

x) 3!

J

m+3

(λx) dx.

All the transformations performed above are valid for m ≥ 0, but in the special case m = 0 it is possible to reduce the number of the boundary conditions at x

0

= 0. Indeed, we have

I

0

(λ, α) = − 1 λ

³

1

\

0

xf

_α^′′′

(x) + 3

f

_α^′

(x)

x − f

α^′′

(x)

J

3

(λx) dx.

By Taylor’s theorem and the condition f

_α^′

(0) = 0 we have for x ∈ (0, 1], f

_α^′

(x)

x − f

α^′′

(x) =

f

_α^′′′

(ϑ

2

x)

2! − f

_α^′′′

(ϑ

3

x) 3!

x, 0 < ϑ

_2,3

< 1.

Therefore, applying Lemma 1 we obtain the required estimate of I

₀

(λ, α).

We shall need eigenfunction expansions of type (2.2) for the nonlinearity of the equation in (2.1). The coefficients F

_mn

(|∇u|

²

) of these equations will be represented by the quadruple series

X

p,q,l,s

a

mnpqls

u b

pq

(t)b u

ls

(t) and X

p,q,l,s

b

mnpqls

u b

pq

(t)b u

ls

(t).

The following lemma permits us to estimate the coefficients of these series.

Let m, n, p, q, l, s be nonnegative integers, n, q, s ≥ 1, and a

mnpqls

= λ

pq

λ

ls

kJ

^m

(λ

mn

r)k

²r 1

\

0

rJ

m

(λ

mn

r)J

_p^′

(λ

pq

r)J

_l^′

(λ

ls

r) dr, m, p, l ≥0, (2.5)

b

_mnpqls

= pl kJ

^m

(λ

mn

r)k

²r

1\

0

1 r J

_m

(λ

_mn

r)J

_p

(λ

_pq

r)J

_l

(λ

_ls

r) dr, m, p, l ≥1,

where λ

kj

, k = 0, 1, . . . and j = 1, 2, . . . , are positive zeros of the Bessel

(10)

function J

k

(z) arranged in increasing order, and the prime denotes differen- tiation with respect to the argument.

Lemma 3. The following estimates are valid:

(2.6) |a

^mnpqls

| ≤ c(λ

^mn

λ

pq

λ

ls

)

^1/2

, |b

^mnpqls

| ≤ c(λ

^mn

λ

pq

λ

ls

)

^1/2

. P r o o f. Representing the derivatives of the Bessel functions in the first integral by the formulas (see [32, p. 207])

J

_k^′

(z) =

¹₂

[J

_k−1

(z) − J

k+1

(z)], k ≥ 1, J

₀^′

(z) = −J

1

(z), and using (2.3) and the inequality

(2.7) |J

^k

(z)| ≤ c/ √

z, z > 0, we deduce the first estimate in (2.6).

Making use of the formula [32, p. 207]

J

k

(z)

z = 1

2k [J

k−1

(z) + J

k+1

(z)], k ≥ 1, we have for k ≥ 1, j ≥ 1,

J

k

(λ

kj

r) r = λ

kj

2k [J

k−1

(λ

kj

r) + J

k+1

(λ

kj

r)].

Then applying (2.7) we obtain the second estimate in (2.6).

3. The main results. In this section we present several theorems con- cerning the existence, uniqueness, construction of the global in time solution of the problem (2.1), and its uniform in space long-time asymptotic expansion.

Let Ω

_δ

= {(r, θ) : r ∈ [δ, 1], θ ∈ [−π, π]}, where δ > 0 is small.

Theorem 1. Suppose that ν > 1/λ

²₀₁

, where λ

01

is the first positive zero of J

0

(z),

ϕ(r, −π) = ϕ(r, π), ∂

θ

ϕ(r, −π) = ∂

^θ

ϕ(r, π), ∂

_θ²

ϕ(r, −π) = ∂

θ²

ϕ(r, π), ϕ(r, θ) satisfies the hypotheses of Lemma 2 with m = 0, i.e.,

∂

r

ϕ(0, θ) = ϕ(1, θ) = ∂

r

ϕ(1, θ) = ∂

_r²

ϕ(1, θ) = 0,

r→0+

lim

√ r∂

_r³

ϕ(r, θ) = Φ

θ

∈ L

¹

(−π, π),

V

₀¹

( √

r∂

_r³

ϕ(r, θ)) = V

θ

∈ L

¹

(−π, π),

and ∂

_θ³

ϕ(r, θ) satisfies the hypotheses of Lemma 2 in the general case, i.e.

∂

_θ³

ϕ(0, θ) = ∂

r

∂

_θ³

ϕ(0, θ) = ∂

_r²

∂

³_θ

ϕ(0, θ) = ∂

_θ³

ϕ(1, θ)

= ∂

r

∂

_θ³

ϕ(1, θ) = ∂

_r²

∂

³_θ

ϕ(1, θ) = 0,

(11)

r→0+

lim

√ r∂

_r³

∂

_θ³

ϕ(r, θ) = e Φ

θ

∈ L

¹

(−π, π),

V

₀¹

( √

r∂

_r³

∂

_θ³

ϕ(r, θ)) = e V

θ

∈ L

¹

(−π, π).

Then there is ε

₀

such that for 0 < ε ≤ ε

0

there exists a unique solution of the problem (2.1) from the class C

¹

([0, ∞), H

r^−1−γ

(Ω)) ∩ C

⁰

([0, ∞), H

r^3−γ

(Ω)) with ∆u ∈ C

⁰

([0, ∞), H

r^1−γ

(Ω)) and ∆

²

u ∈ C

⁰

([0, ∞), H

r^−1−γ

(Ω)) for any γ > 0.

Moreover , u and ∇u are continuous and bounded in Ω × [0, ∞) and ∆u is continuous and bounded in Ω

δ

× [0, ∞).

This solution can be represented as

(3.1) u(r, θ, t) =

X

∞ N =0

ε

^{N +1}

u

^{(N )}

(r, θ, t),

where the functions u

^{(N )}

will be defined in the proof (see (4.8) and (4.12)).

The series (3.1) converges absolutely and uniformly with respect to (r, θ) ∈ Ω, t ∈ [0, ∞), ε ∈ [0, ε

0

] together with ∇u which can be calculated termwise.

In the next statement (and only there) we denote the solution of the nonlinear problem (2.1) by u

β

(r, θ, t) and the solution of the corresponding linear problem (with β = 0) by u

0

(r, θ, t). The existence and uniqueness of the latter is evident.

Corollary. Under the assumptions of Theorem 1, the following estimate holds:

sup

Ω×[0,∞)

|u

^β

(r, θ, t) − u

⁰

(r, θ, t)| ≤ C|β|, β ∈ R, where the constant C is independent of r, θ, t, and ε.

Remark 1. The parameter ε ∈ (0, ε

⁰

] which controls the initial data guarantees the absolute and uniform convergence of the series (3.1). The latter is a series of regular perturbations and can be used as an asymptotic series with respect to ε. The estimate of ε

0

will be made clear in the proof.

Remark 2. The solution presented above is a strong solution. The equation in (2.1) is satisfied in the distributional sense, i.e., in H

_r^−1−γ

(Ω), γ > 0, for each fixed t > 0. In the same sense the periodicity conditions are satisfied for ∂

t

u and ∆

²

u. The boundary conditions u|

^∂Ω

= ∆u|

^∂Ω

= 0, the initial condition, and the periodicity conditions for u and ∇u are satisfied in the classical sense. For ∆u the periodicity conditions are satisfied in H

_r^1−γ

(Ω) (and in Ω

δ

they are satisfied in the classical sense).

Remark 3. It is not difficult to construct a function ϕ(r, θ) satisfy- ing the hypothesis of Theorem 1 by separation of variables, i.e., ϕ(r, θ) = R(r)Θ(θ), where R

^(k)

(0) = R

^(k)

(1), k = 0, 1, 2; lim

r→0+

√

rR

^′′′

(r) = c

1

<

(12)

∞, V

0¹

( √

rR

^′′′

(r)) = c

2

< ∞, Θ

^(k)

(−π) = Θ

^(k)

(π), k = 0, 1, 2; Θ

^′′′

∈ L

1

(−π, π). The fact that Θ ∈ L

¹

(−π, π) follows from the absolute inte- grability of Θ

^′′′

in (−π, π) and the existence of Θ

^(k)

(−π), k = 0, 1, 2.

Theorem 2. Under the hypotheses of Theorem 1, the solution of (2.1) has the following asymptotic representation as t → ∞:

(3.2) u(r, θ, t) = exp(−κ

01

t)[A

_ε

J

₀

(λ

₀₁

r) + O(exp(−κ

01

t))],

where κ

01

= λ

²₀₁

(νλ

²₀₁

− 1) > 0, and the coefficient A

^ε

will be defined in the proof (see (7.1) and (7.3)). The estimate of the residual term is uniform with respect to (r, θ) ∈ Ω and ε ∈ [0, ε

⁰

].

Consider the problem (2.1) on a bounded time interval [0, T ], T < ∞, and denote it by (2.1

^∗

).

Theorem 3. If 0 < ν ≤ 1/λ

²01

and the remaining assumptions of Theorem 1 hold, then for any T > 0 there is ε

0

(T ) > 0 such that for 0 < ε ≤ ε

⁰

(T ) there exists a unique solution of the problem (2.1

^∗

) from the class stated in Theorem 1 with [0, ∞) replaced by [0, T ]. This solution is represented in the form (3.1), where ε

₀

(T ) → 0 as T → ∞. For any fixed ε there exists T < ∞ such that the solution of (2.1

^∗

) cannot be extended beyond this point.

The rest of the paper is organized as follows. In Section 4 we construct a solution of the problem (2.1) and verify that it belongs to the required function space. Section 5 is devoted to the proof of uniqueness. This completes the proof of Theorem 1. The Corollary is proved in Section 6. The long-time asymptotic expansion of the solution for ν > 1/λ

²₀₁

which forms the content of Theorem 2 is obtained in Section 7. Theorem 3 deals with the case 0 < ν ≤ 1/λ

²01

and is proved in Section 8. Some final remarks are given in Section 9.

4. Existence and construction of solutions: proof of Theorem 1.

In order to satisfy the boundary and periodicity conditions in (2.1) we seek a solution of this problem in the form (2.2), namely:

(4.1) u(r, θ, t) =

X

∞ n=1

X

∞ m=−∞

u b

mn

(t)χ

mn

(r, θ), where

b

u

mn

(t) = (u, χ

mn

)

r,0

kχ

^mn

k

²r,0

, m ∈ Z, n ∈ N.

Since J

−m

(z) = (−1)

^m

J

m

(z) for integer m ≥ 0, the zeros of J

^m

(z) and

J

−m

(z) coincide (λ

−m,n

= λ

mn

, m ≥ 0, n ≥ 1), and (4.1) can be rewritten

in the form of a double series with m ≥ 0. Using the fact that u(r, θ, t) is a

(13)

real-valued function we can write b u

_mn

(t) = 1

kχ

mn

k

²r,0 1

\

0

rJ

_m

(λ

_mn

r) dr

π

\

−π

e

^−imθ

u(r, θ, t) dθ,

b u

_−m,n

(t) = 1 kχ

^mn

k

²r,0

1

\

0

rJ

_−m

(λ

_−m,n

r) dr

π

\

−π

e

^imθ

u(r, θ, t) dθ

= (−1)

^m

kχ

^mn

k

²r,0

1

\

0

rJ

m

(λ

mn

r) dr

π\

−π

e

^imθ

u(r, θ, t) dθ, m ≥ 0, n ≥ 1.

Therefore, b

u

mn

(t) = (−1)

^m

u b

−m,n

(t), m ≥ 0, n ≥ 1,

where the bar denotes complex conjugation (this notation should not be confused with Ω, where the bar denotes closure).

We can rewrite the expression (4.1) as u(r, θ, t) =

X

∞ n=1

b

u

0n

(t)J

0

(λ

0n

r) (4.2)

+ X

∞ n=1

X

∞ m=1

J

m

(λ

mn

r)[b u

mn

(t)e

^imθ

+ b u

mn

(t)e

^−imθ

]

= X

∗ m,n

b

u

mn

(t)J

mn

(λ

mn

r)e

^imθ

.

Here and below the double sum with an asterisk includes the “usual” double sum P

∞

m,n=1

and P

∞ n=1

.

Expanding |∇u|

²

in the series of type (2.2) and denoting its coefficients by F

mn

(|∇u|

²

)(t), m ∈ Z, n ∈ N, we substitute this expansion and (4.1) into (2.1) and obtain the following Cauchy problem for b u

mn

(t):

(4.3) b u

^′_mn

(t) + κ

mn

u b

mn

(t) = βF

mn

(|∇u|

²

)(t), t > 0,

b u

mn

(0) = ε

²

ϕ b

mn

, m ∈ Z, n ∈ N, where

κ

mn

= λ

²_mn

(νλ

²_mn

− 1) > 0, F

mn

(|∇u|

²

)(t) = (|∇u|

²

(t), χ

mn

)

r,0

kχ

^mn

k

²r,0

, and b ϕ

mn

are the coefficients of the type (2.2) expansion of ϕ(r, θ), that is,

ϕ(r, θ) = X

m,n

b

ϕ

mn

(t)χ

mn

(r, θ), ϕ b

mn

= (ϕ, χ

mn

)

r,0

kχ

^mn

k

²r,0

.

(14)

We now prove the following estimates:

(4.4) | b ϕ

_mn

| ≤ c/λ

^7/2mn

, m ≥ 0, n ≥ 1.

For m = 0 we can write b

ϕ

0n

= 1

2πkJ

⁰

(λ

0n

r)k

²r 1

\

0

rJ

0

(λ

0n

r) dr

π

\

−π

ϕ(r, θ) dθ

= 1

2πkJ

⁰

(λ

0n

r)k

²r π

\

−π

dθ

1

\

0

rJ

0

(λ

0n

r)ϕ(r, θ) dr.

Since ϕ(r, θ) satisfies the hypothesis of Lemma 2 with m = 0 we have

1\

0

rϕ(r, θ)J

0

(λ

0n

r) dr

≤ C

^θ

/λ

^9/2_0n

,

where C

θ

∈ L

¹

(−π, π). Taking into account (2.3) we obtain the required estimate.

It remains to justify the change of the order of integration performed above. We observe that the conditions

V

₀¹

( √

r∂

_r³

ϕ(r, θ)) = V

_θ

∈ L

1

(−π, π), lim

r→0+

√ r∂

_r³

ϕ(r, θ) = Φ

_θ

∈ L

1

(−π, π)

imply that there exists N

θ

∈ L

¹

(−π, π) such that

|∂

r³

ϕ(r, θ)| ≤ N

θ

/ √

r, r ∈ (0, 1).

Therefore, using the boundary conditions ϕ(1, θ) = ∂

r

ϕ(1, θ) = ∂

_r²

ϕ(1, θ)

= 0 we deduce that

|ϕ(r, θ)| ≤

r

\

1

dr

1 r1

\

1

dr

2 r3

\

1

|∂

r³

ϕ(r

3

, θ)| dr

³

≤ cN

^θ

uniformly with respect to r ∈ [0, 1]. The change of the order of integration is justified.

Assume now that m ≥ 1. Defining φ

_m

(r) = 1

2π

π\

−π

e

^−imθ

ϕ(r, θ) dθ

we can write

ϕ b

mn

= 1 kJ

m

(λ

_mn

r)k

²r

π

\

−π

rJ

m

(λ

mn

r)φ

m

(r) dr.

Integrating three times by parts in the integral defining φ

m

(r) and using the

(15)

periodicity conditions in θ for ϕ(r, θ), ∂

θ

ϕ(r, θ), and ∂

_θ²

ϕ(r, θ) we get φ

m

(r) = i

2πm

³

π

\

−π

e

^−imθ

∂

_θ³

ϕ(r, θ) dθ, m ≥ 1.

Changing the order of integration in the integral representation of b ϕ

mn

we obtain

b

ϕ

mn

= i

2πm

³

kJ

^m

(λ

mn

r)k

²r π

\

−π

e

^−imθ

dθ

1

\

0

rJ

m

(λ

mn

r)∂

_θ³

ϕ(r, θ) dr.

Since ∂

_θ³

ϕ(r, θ) satisfies the hypothesis of Lemma 2 in r, for m ≥ 1 we have

1

\

0

rJ

m

(λ

mn

r)∂

_θ³

ϕ(r, θ) dr

≤ C

θ

(m + 1)

³

λ

^9/2mn

,

where C

_θ

∈ L

1

(−π, π). The last inequality and the estimate (2.3) imply (4.4) with m ≥ 1.

In order to show that the change of the order of integration is valid we observe that there exists P

θ

∈ L

¹

(−π, π) such that

|∂

r³

∂

_θ³

ϕ(r, θ)| ≤ P

θ

/ √

r, r ∈ (0, 1).

Therefore, using the boundary conditions ∂

_θ³

ϕ(0, θ) = ∂

r

∂

_θ³

ϕ(0, θ) =

∂

_r²

∂

³_r

ϕ(0, θ) = 0 we deduce that

|∂

θ³

ϕ(r, θ)| ≤

r

\

0

dr

1 r1

\

0

dr

2 r2

\

0

dr

3

|∂

r³

∂

_r³

ϕ(r

3

, θ)| ≤ cP

^θ

uniformly with respect to r ∈ [0, 1]. The estimates (4.4) are established for all integer m ≥ 0, n ≥ 1.

Integrating the Cauchy problem (4.3) in time we obtain b

u

mn

(t) = ε b Φ

mn

exp(−κ

^mn

t) (4.5)

+ β

t

\

0

exp[−κ

^mn

(t − τ)]F

^mn

(|∇u|

²

)(τ ) dτ,

where m ≥ 0, n ≥ 1, and b Φ

mn

= ε b ϕ

mn

. It is convenient to keep one small parameter in the coefficients b Φ

mn

in order to simplify some estimates.

In order to calculate F

mn

(|∇u|

²

) we set

(4.6) F

mn

(|∇u|

²

) = F

mn

[(∂

r

u)

²

] + F

mn

[(∂

θ

u)

²

/r

²

],

where

(16)

F

_mn

[(∂

_r

u)

²

] = 1 kχ

^mn

k

²r,0

1

\

0

rJ

_m

(λ

_mn

r) dr

π

\

−π

dθ e

^−imθ

× ∂

^r

h X

^∞

q=1

n b

u

0q

(t)J

0

(λ

0q

r) + X

∞ p=1

J

p

(λ

pq

r)[b u

pq

(t)e

^ipθ

+ b u

pq

(t)e

^−ipθ

] oi

× ∂

r

h X

^∞

s=1

n

b u

_0s

(t)J

_0s

(λ

_0s

r) + X

∞

l=1

J

_l

(λ

_ls

r)[b u

_ls

(t)e

^ilθ

+ b u

_ls

(t)e

^−ilθ

] oi

= X

′ p,q,l,s

a

mnpqls

b u

pq

(t)b u

ls

(t), where

X

′ p,q,l,s

a

_mnpqls

b u

_pq

(t)b u

_ls

(t) = X

p,l≥0; q,s≥1 p+l=m

a

_mnpqls

b u

_pq

(t)b u

_ls

(t)

+ X

p,q,l,s≥1 l−p=m

a

mnpqls

u b

pq

(t)b u

ls

(t)

+ X

p,q,l,s≥1 p−l=m

a

mnpqls

u b

pq

(t)b u

ls

(t),

and the coefficients a

_mnpqls

are defined by (2.5). Here we have used the relations

π

\

−π

e

^{i(−m+p+l)θ}

dθ =

2π, p + l = m, 0, p + l 6= m,

π

\

−π

e

^{i(−m−p+l)θ}

dθ =

2π, l − p = m, 0, l − p 6= m,

π\

−π

e

^{i(−m+p−l)}

dθ =

2π, p − l = m, 0, p − l 6= m,

π

\

−π

e

^{−i(m+p+l)θ}

dθ =

2π, m = p = l = 0, 0, m ≥ 0, p ≥ 1, l ≥ 1.

Next, we have

F

mn

[(∂

θ

u)

²

/r

²

] = 1 kχ

^mn

k

²

1

\

0

1 r J

m

(λ

mn

r)dr

π

\

−π

e

^−imθ

× X

∞ q,p=1

J

_p

(λ

_pq

r)ip[b u

_pq

(t)e

^ipθ

− b u

_pq

(t)e

^−ipθ

]

(17)

× X

∞ s,l=1

J

l

(λ

ls

r)il[b u

ls

(t)e

^ilθ

− b u

ls

(t)e

^−ilθ

] dθ

= X

′′

p,q,l,s

b

mnpqls

u b

pq

(t)b u

ls

(t), where

X

′′

p,q,l,s

b

mnpqls

u b

pq

(t)b u

ls

(t) = − X

p,l≥0; q,s≥1 p+l=m≥1

b

mnpqls

b u

pq

(t)b u

ls

(t)

+ X

p,q,l,s≥1 l−p=m

b

mnpqls

b u

pq

(t)b u

ls

(t)

+ X

p,q,l,s≥1 p−l=m

b

mnpqls

b u

pq

(t)b u

ls

(t),

and the coefficients b

mnpqls

are defined by (2.5).

The three sums with the additional conditions p + l = m, l − p = m, and p−l = m are of the convolution type. Note that in the sum P

′′

p,q,l,s

the ana- log of the term P

∞

q,s=1

a

0n0q0s

u b

0q

(t)b u

0s

(t) corresponding to p + l = m = 0 and representing the “purely radial part” is absent as a result of differentiating with respect to θ.

To solve the nonlinear integral equation (4.5) we use perturbation theory.

Representing b u

m

(t) as a formal series in ε,

(4.7) b u

_mn

(t) =

X

∞ N =0

ε

^{N +1}

v b

_mn^{(N )}

(t),

we substitute it into (4.5) and compare the coefficients of equal powers of ε. As a result, we obtain the following representations for N ≥ 0, m ≥ 0, n ≥ 1, t > 0:

(4.8)

b v

⁽⁰⁾_mn

(t) = b Φ

_mn

exp(−κ

mn

t), b

v

_mn^{(N )}

(t) = β

t

\

0

exp[−κ

^mn

(t − τ)]Q

^{(N )}mn

(bv(τ)) dτ, N ≥ 1, where

Q

^{(N )}_mn

(bv(t)) = X

′ p,q,l,s

a

mnpqls

X

N j=1

b

v

_pq^(j−1)

(t)bv

ls^{(N −j)}

(t)

+ X

′′

p,q,l,s

b

mnpqls

X

N j=1

b

v

^(j−1)_pq

(t)bv

^{(N −j)}ls

(t).

(18)

Now we have to prove that the formally constructed function (4.2), (4.7), (4.8) is really a solution of the problem (2.1) in the required function space.

To this end we study the convergence of the series (4.9) u(r, θ, t) = X

∗

m,n

h X

^∞

N =0

ε

^{N +1}

b v

_mn^{(N )}

(t) i

J

m

(λ

mn

r)e

^imθ

.

First, we establish the following estimates for N ≥ 0, m ≥ 0, n ≥ 1, t > 0:

(4.10) |bv

^{(N )}mn

(t)| ≤ c

^Nβ

(N + 1)

⁻²

λ

^−7/2_mn

exp(−κ

⁰¹

t),

where c

_β

= c|β|, β ∈ R. We use induction on N. For N = 0 and sufficiently small ε we have, from (4.4) and (4.8),

|bv

mn⁽⁰⁾

(t)| = |ε b ϕ

mn

| exp(−κ

^mn

t) ≤ λ

^−7/2mn

exp(−κ

⁰¹

t).

Assuming that (4.10) is valid for |bv

^mn^(s)

(t)| with 0 ≤ s ≤ N − 1 we prove it for s = N. For 1 ≤ j ≤ N we have

j

⁻²

(N + 1 − j)

⁻²

≤ 2

²

(N + 1)

⁻²

[j

⁻²

+ (N + 1 − j)

⁻²

].

By means of Lemma 3 we can estimate a typical term on the right-hand side of (4.8). Indeed,

|ℑ|≤c|β|

t

\

0

exp[−κ

^mn

(t−τ)] X

p,l≥0; q,s≥1 p+l=m

|a

^mnpqls

| X

N j=1

|bv

pq^(j−1)

(τ )| · |bv

^{(N −j)}ls

(τ )| dτ

≤ c

_β

√ λ

_mn

(N + 1)

²

S

mn

(t) X

∗ p,q

1 λ

³_pq

X

∗ l,s

1 λ

³_ls

X

N j=1

c

^j−1_β

c

^{N −j}_β

[j

⁻²

+ (N + 1 − j)

⁻²

], where

S

mn

(t) = exp(−κ

^mn

t)

t

\

0

exp(L

mn

τ ) dτ,

L

mn

= κ

mn

− 2κ

⁰¹

= λ

²_mn

(νλ

²_mn

− 1) − 2λ

²01

(νλ

²₀₁

− 1).

The convergence of the double series in p, q and in l, s in the last inequality follows from the asymptotics (2.4) and the fact that the series

X

∞ m,n=−∞

m,n6=0

1 (m + 2n)

^σ

converges for σ > 2 and diverges for σ ≤ 2 (see [5]). Now we have to consider three cases.

(i) If m = 0, n = 1, then L

01

= −κ

⁰¹

, and S

01

(t) = exp(−κ

⁰¹

t) 1 − exp(−κ

⁰¹

t)

κ

01

≤ exp(−κ

⁰¹

t)

λ

⁴₀₁

(ν − 1/λ

⁴01

) .

(19)

(ii) If m = 0, n ≥ 2, then for the positive zeros of J

⁰

(z) we have (see [14]) λ

²_0n

≥ λ

²02

> 2λ

²₀₁

, and therefore,

L

0n

= λ

²_0n

ν(λ

²_0n

− λ

²01

) + (νλ

²₀₁

− 1)(λ

²0n

− 2λ

²01

) > 0.

Then

S

0n

(t) = exp(−κ

⁰ⁿ

t) exp(L

0n

t) − 1 L

0n

≤ exp(−2κ

⁰¹

t)

νλ

²_0n

(λ

²_0n

− λ

²01

) + (νλ

²₀₁

− 1)(λ

²0n

− 2λ

²01

)

≤ exp(−2κ

⁰¹

t)

νλ

⁴_0n

(1 − λ

²01

/λ

²_0n

) ≤ c(ν) exp(−κ

⁰¹

t) λ

⁴_0n

.

(iii) If m ≥ 1, n ≥ 1, then λ

²mn

− 2λ

²01

≥ λ

²11

− 2λ

²01

> 0 (see [14]), and consequently,

L

mn

= νλ

²_mn

(λ

²_mn

− λ

²01

) + (νλ

²₀₁

− 1)(λ

²mn

− 2λ

²01

)

≥ νλ

²11

(λ

²₁₁

− λ

²01

) + (νλ

²₀₁

− 1)(λ

²11

− 2λ

²01

) > 0.

Therefore,

S

mn

(t) = exp(−κ

^mn

t) exp(L

mn

t) − 1 L

mn

≤ exp(−2κ

⁰¹

t)

νλ

⁴_mn

(1 − λ

²01

/λ

²_mn

) ≤ c(ν) exp(−κ

⁰¹

t) λ

⁴_mn

.

The estimates (4.10) are proved. Moreover, in items (ii) and (iii) we have established that for t > 0, N ≥ 0, m = 0, n ≥ 2 and m ≥ 1, n ≥ 1,

(4.11) |bv

^{(N )}0n

(t)| ≤ c

^Nβ

(N + 1)

⁻²

λ

^−7/2_0n

exp(−2κ

⁰¹

t),

where c

_β

= c|β|, β ∈ R (in these two cases the estimate for bv

^mn⁽⁰⁾

(t) can also be rewritten with exp(−2κ

⁰¹

t) since κ

mn

≥ 2κ

⁰¹

). The inequalities (4.11) will be used later for calculating the long-time asymptotics of the solution.

In order to obtain the representation (3.1) we interchange the order of summation in the series (4.9) to get

(4.12) u(r, θ, t) =

X

∞ N =0

ε

^{N +1}

u

^{(N )}

(r, θ, t), where

u

^{(N )}

(r, θ, t) = X

∗ l,s

bv

mn^{(N )}

(t)J

_m

(λ

_mn

r)e

^imθ

and bv

^mn^{(N )}

(t) are defined by (4.8). This interchange is possible due to the

absolute and uniform (in (r, θ) ∈ Ω, t ≥ 0, ε ∈ [0, ε

⁰

]) convergence of the

series in question, which in turn follows from (4.8) with 0 < ε ≤ ε

⁰

< c

⁻¹_β

.

(20)

Differentiating (4.8) with respect to t we find that

∂

t

b v

_mn⁽⁰⁾

(t) = − κ

^mn

Φ b

mn

exp(−κ

^mn

t),

∂

t

v b

_mn^{(N )}

(t) = β h κ

mn

t

\

0

exp[−κ

^mn

(t−τ)]Q

^{(N )}mn

(bv(τ)) dτ + Q

^{(N )}mn

(bv(t)) i

, N ≥1, where Q

^{(N )}mn

(bv(t)) is defined by (4.8).

Taking into account the expression for κ

mn

(see (4.3)) and (4.7) we deduce that for m ≥ 0, n ≥ 1, t ≥ 0, k = 0, 1,

(4.13) |∂

^t

b u

mn

(t)| ≤ cλ

^4k−7/2mn

exp(−κ

⁰¹

t).

Recalling the asymptotics (2.4) and using (4.13) with k = 0 we conclude that the series X

m,n

λ

^2s_mn

|b u

mn

(t)|

²

kJ

^m

(λ

mn

r)k

²r

with s = 3 − γ, γ > 0, converges uniformly with respect to t ≥ 0. There- fore, u ∈ C

⁰

([0, ∞), H

r^3−γ

(Ω)). Moreover, thanks to (4.13) the series (4.2) converges absolutely and uniformly with respect to (r, θ) ∈ Ω, t ≥ 0, and ε ∈ [0, ε

0

]. Therefore, u(r, θ, t) is continuous and bounded in this domain.

Calculating ∇u by means of (4.2) we can see that for m ≥ 0, n ≥ 1, t ≥ 0,

|F

^mn

(∇u)(t)| ≤ c λ

^5/2mn

exp(−κ

⁰¹

t) and, therefore, the series

X

∗ m,n

F

mn

(∇u)J

^m

(λ

mn

r)e

^imθ

converges absolutely and uniformly in (r, θ) ∈ Ω, t ≥ 0.

As regards ∆u, for s = 1 − γ, γ > 0, the series k∆uk

²r,s

= X

m,n

λ

^2s_mn

|F

^mn

(∆u)(t)|

²

kJ

^m

(λ

mn

r)k

²r

converges uniformly in t ≥0, and this implies that ∆u∈C

⁰

([0, ∞), H

r^1−γ

(Ω)).

However, the convergence of the series of type (4.2) representing ∆u in the pointed region Ω

_δ

is better than in the domain Ω. Indeed, for (r, θ) ∈ Ω

δ

, t ≥ 0 we have

(4.14) ∆u = X

∗

m,n

F

mn

(∆u)(t)J

m

(λ

mn

r)e

^imθ

, where

|F

^mn

(∆u)(t)| ≤ c λ

^3/2mn

exp(−κ

⁰¹

t), |J

^m

(λ

mn

r)| ≤ c

√ λ

_mn

.

(21)

To analyze the convergence of (4.14) we take the series X

∞

n=A

X

∞ m=B

F

mn

(∆u)J

m

(λ

mn

r)e

^imθ

with sufficiently large positive A and B and compare it with the integral I = e

∞\

A

dn

∞\

B

dm e

^imθ

λ

²_mn

.

Using the asymptotics (2.4) we integrate by parts in m to obtain I = e 1

i

∞

\

A

ie

^iBθ

B(B + 2n)

²

+

∞

\

B

3m + 2n

m

²

(m + 2n)

³

e

^imθ

dm

dn, which implies that

|e I| ≤ c B

∞\

A

dn

(B + 2n)

²

≤ c B(B + 2A) .

Therefore, the series (4.14) converges uniformly in Ω

δ

and the boundary condition ∆u|

^∂Ω

= 0 is satisfied in the classical sense.

For u

t

and ∆

²

u the corresponding norms in H

_r^s

(Ω) are ku

t

(t)k

²r,s

= X

m,n

λ

^2s_mn

|F

mn

(u

_t

)|

²

kJ

m

(λ

_mn

r)k

²r

, k∆

²

u(t)k

²r,s

= X

m,n

λ

mn

|F

^mn

(∆

²

u)(t)|

²

kJ

^m

(λ

mn

r)k

²r

, where

|F

mn

(u

_t

)(t)| ≤ cλ

^1/2mn

exp(−κ

01

t), |F

mn

(∆

²

u)(t)| ≤ cλ

^1/2mn

exp(−κ

01

t).

Hence, these series converge uniformly with respect to t ≥ 0 for s = −1 − γ, γ > 0, and this implies that u

t

and ∆

²

u belong to C

⁰

([0, ∞), H

r^−1−γ

(Ω)).

This completes the proof of the existence of a global in time solution of (2.1).

5. Uniqueness of solutions: proof of Theorem 1 (continuation).

Assume that there exist two solutions u

⁽¹⁾

and u

⁽²⁾

in the class stated in the theorem. Setting w = u

⁽¹⁾

− u

⁽²⁾

and expanding w into the series (4.2) we have

w(r, θ, t) = X

∗ m,n

w b

_mn

(t)J

_m

(λ

_mn

r)e

^imθ

,

where the estimates (4.13) with k = 0 are valid for b w

mn

(t). Since the linear

part in the expression (4.5) equals zero the coefficients b w

mn

(t) satisfy the

(22)

integral equation w b

_mn

(t) = β

t

\

0

exp[−κ

mn

(t − τ)][F

mn

(|∇u

⁽¹⁾

|

²

)(τ ) − F

mn

(|∇u

⁽²⁾

|

²

)(τ )] dτ, where F

_mn

(|∇u|

²

) are defined by (4.6). We can represent a typical term in the integrand in the last formula as follows:

X

′ p,q,l,s

a

mnpqls

u b

⁽¹⁾_pq

(t)b u

⁽¹⁾_ls

(t) − X

′ p,q,l,s

a

mnpqls

b u

⁽²⁾_pq

(t)b u

⁽²⁾_ls

(t)

= X

′ p,q,l,s

a

mnpqls

{b u

⁽¹⁾_pq

(t)[b u

⁽¹⁾_ls

(t) − b u

⁽²⁾_ls

(t)] + b u

⁽²⁾_ls

(t)[b u

⁽¹⁾_pq

(t) − b u

⁽²⁾_pq

(t)]}

= X

′ p,q,l,s

a

mnpqls

[b u

⁽¹⁾_pq

(t) b w

ls

(t) + b u

⁽²⁾_ls

(t) b w

pq

(t)].

Note that w(r, θ, t) ∈ H

r^3−γ

(Ω) for all t ≥ 0 and any γ > 0 since u

⁽¹⁾

and u

⁽²⁾

belong to the same space.

In order to estimate the integrand we shall use the Cauchy–Schwarz inequality and the relations

kw(t)k

²r,1

= X

m,n

λ

²_mn

| b w

_mn

(t)|

²

kJ

m

(λ

_mn

r)k

²r

< ∞, ku

^(k)

(t)k

²r,1

< ∞, k = 1, 2.

Since the inequalities (2.6) are valid for the coefficients a

_mnpqls

we can write

X

′ p,q,l,s

a

_mnpqls

u b

⁽¹⁾_pq

(t) b w

_ls

(t)

≤ c p λ

mn

X

∞ p,q,l,s=1

p λ

pq

|b u

⁽¹⁾_pq

(t)| p

λ

ls

| b w

ls

(t)|

≤ c p λ

mn

X

^∞

p,q=1

λ

pq

|b u

⁽¹⁾_pq

(t)|

²

1/2

X

^∞

l,s=1

λ

ls

| b w

ls

(t)|

²

1/2

≤ c p λ

mn

X

^∞

p,q=1

λ

²_pq

|b u

⁽¹⁾_pq

(t)|

²

kJ

^p

(λ

pq

r)k

²r

1/2

× X

^∞

l,s=1

λ

²_ls

| b w

ls

(t)|

²

kJ

^l

(λ

ls

r)k

²r

1/2

≤ c p

λ

mn

ku(t)k

^r,1

kw(t)k

^r,1

.

(23)

Analogous estimates hold for P

′′

p,q,l,s

b

mnpqls

u b

^(k)pq

(t) b w

ls

(t), k = 1, 2. There- fore,

| b w

mn

(t)| ≤ c p λ

mn

t

\

0

exp[−κ

^mn

(t − τ)]kw(τ)k

^r,1

dτ.

Squaring both sides, multiplying the result by λ

²_mn

kJ

m

(λ

_mn

r)k

²r

, and sum- ming in m and n we deduce that for some T

1

> 0 and t ∈ [0, T

¹

],

kw(t)k

²r,1

≤ c X

m,n

λ

³_mn

kJ

m

(λ

_mn

r)k

²r

^\^t

0

exp[−κ

mn

(t − τ)]kw(τ)k

r,1

dτ

2

≤ c( sup

t∈[0,T1]

kw(t)k

r,1

)

²

X

m,n

λ

²_mn

1 − exp(−κ

mn

t) κ

mn

2

. This implies that

( sup

t∈[0,T1]

kw(t)k

^r,1

)

²

≤ c(T

¹

)( sup

[0,T1]

kw(t)k

^r,1

)

²

,

where the constant c(T

1

) can be made less than one by the appropriate choice of T

1

. This contradiction allows to us complete the proof of uniqueness for t ∈ [0, T

1

]. Continuing this process for the integrals [T

₁

, T

₂

], [T

_2,

T

₃

], . . . . . . , [T

n

, T

n+1

], . . . with T

n

→ ∞ we obtain the same result for all t > 0.

This completes the proof of Theorem 1.

6. Asymptotics with respect to β: proof of the Corollary. First, we construct the solution of the linear problem corresponding to (2.1) with β = 0. Putting β = 0 in (4.3) we obtain the following expression for the eigenfunction expansion coefficients:

e u

_mn

(t) = ε

²

ϕ b

_mn

exp(−κ

mn

t),

where κ

mn

= λ

²_mn

(νλ

²_mn

− 1) and the estimates (4.4) are valid for b ϕ

mn

. The solution of the linear problem is

u

0

(r, θ, t) = X

∗ m,n

e u

mn

(t)J

m

(λ

mn

r)e

^imθ

, where P

∗

m,n

is defined by (4.2). The uniqueness of the solution in the corresponding function space is evident.

Note that, according to (4.7) and (4.8), b

u

mn

(β, t) = e u

mn

(t) + X

∞ N =1

ε

^{N +1}

b v

^{(N )}_mn

(β, t).

Here we have shown the dependence on β in the notation of the coefficients.

The estimates (4.10) are valid for bv

^{(N )}^mn

(β, t), N ≥ 1, that is,

|bv

mn^{(N )}

(β, t)| ≤ c

^N

|β|

^N

(N + 1)

⁻²

λ

^−7/2_mn

exp(−κ

⁰¹

t).