landau-ginzburg type equations in the sub critical case

LANDAU-GINZBURG TYPE EQUATIONS IN THE SUB CRITICAL CASE

NAKAO HAYASHI, ELENA I. KAIKINA, AND PAVEL I. NAUMKIN

Abstract. We study the Cauchy problem for the nonlinear Landau-Ginzburg equation

∂tu−α∆u+β|u|^σu= 0, x∈Rⁿ, t >0, u(0, x) =u0(x), x∈Rⁿ, (0.1)

where α, β ∈ C with dissipation condition α > 0. We are interested in the subcritical case σ ∈ 0,_n²

.We assume thatθ =

R

u0(x)dx

= 0 and δ(α, β)>0,where

δ(α, β) = β|α|^n−n2σ

(2 +σ)|α|²+σα²

n 2 .

Furthermore we suppose that the initial datau0∈L¹are such that (1 +|x|)^au0∈ L¹, with sufficiently small normε=(1 +|x|)^au0₁, wherea∈(0,1).Also we assume thatσis sufficiently close to _n².Then there exists a unique solution of the Cauchy problem (0.1) such that

u(t, x)∈C((0,∞) ;L^∞)∩C [0,∞);L¹, satisfying the following time decay estimates for larget >0

u(t)∞≤Cεt^−1σ.

Note that in comparison with the corresponding linear case the decay rate of the solutions of (0.1) is more rapid.

1. Introduction

This paper is devoted to the study of global existence and large time asymp- totic behavior of small solutions to the Cauchy problem for the complex Landau - Ginzburg equation

∂_tu−α∆u+β|u|^σu= 0, x∈Rⁿ, t >0, u(0, x) =u0(x), x∈Rⁿ

(1.1)

in the sub critical case σ∈ 0,_n²

, whereα, β ∈C andαsatisfies the dissipation conditionα >0. In the super critical caseσ > _n² the global existence result can be easily obtained for anyαand β,with α >0, since the nonlinear term decays rapidly enough |u|^σu_L1 ≤ Ct^−n2σ, when ⁿ₂σ >1. In the so-called critical case σ= _n² and sub critical σ∈

0,_n²

cases we can not expect a sufficient time decay of the nonlinear term for any α and β with α ≥ 0. Therefore we assume the

Date: April, 17, 2001.

1991Mathematics Subject Classification. 35Q35.

Key words and phrases. Dissipative Nonlinear Evolution Equation, Large Time Asymptotics, Landau-Ginzburg equation.

1

conditionδ(α, β)>0, where

δ(α, β) = β|α|^n−n2σ

(2 +σ)|α|²+σα^{2 n}₂.

We also suppose that the mean value of the initial dataθ≡ u0(x)dx = 0.The nonlinear term in equation (1.1) have no enough regularity to get smooth solutions in the higher order Sobolev spaces, so using smoothing properties of the linear evolution group we consider the initial value problem under the assumptions of rather small regularity on the initial data u0 ∈ L¹. To estimate the remainder terms in the large time asymptotic formulas we assume that the initial data decay at infinity as |x|^au0(x) ∈ L¹ with somea ∈ (0,1). Existence and uniqueness of local solutions for problem (1.1) were proved (see, e.g., [5], [6], [15]). The blow-up phenomena for positive solutions to the semilinear parabolic equation

u_t−∆u=u^1+σ (1.2)

(note that equation (1.2) is a particular case of (1.1) if α = 1 and β =−1) was obtained in paper [3] forσ∈

0,_n²

, in [8] forσ= _n², n= 1,2, and in paper [14] for σ= _n², for anyn≥1.Global in time existence of small solutions to (1.2) was proved in [3] for the super critical caseσ > _n². Large time behavior of positive solutions was studied extensively for a particular case of (1.1) withα= 1 andβ = 1

u_t−∆u=−u^1+σ, σ >0 (1.3)

(see paper [12] for the super critical case σ > _n², [4] for the critical case σ = ²_n and papers [1], [2], [7], [13] for the sub critical case σ ∈

0,_n²

). We describe in more details the results on the asymptotic behavior of solutions to the Cauchy problem for equation (1.3) in the sub critical case σ ∈

0,_n²

. In paper [7] it was proved that if the initial data are nonnegative u0 ≥ 0, u0 ∈ L¹ and decay slowly at infinity as lim_x→±∞|x|^2σu0(x) = +∞,then the solution of (1.3) has the asymptotic representation u(t, x) = t^−1σσ^−1σ +o

t^−σ1

as t → ∞ uniformly in domains

x∈Rⁿ; |x| ≤C√

t with any C >0. On the other hand in paper [1], there were considered the nonnegative initial data decaying sufficiently rapidly at infinity, i.e. 0≤u0(x)≤Ce^−bx2 for all x∈Rⁿ,with some b, C >0. Then it was shown that the main term of the asymptotic behavior of solution has a self-similar character u(t, x) = t^−σ1w0

√x t

+o

t^−σ1

as t → ∞ uniformly with respect to x∈Rⁿ,wherew0(ξ) is a positive solution of the elliptic equation

−∆w−1

2ξ∇w+w^1+σ= 1 σw (1.4)

which decays rapidly at infinity: lim_|ξ|→∞|ξ|^σ2 w0(ξ) = 0.This result was improved in paper [2], where the intermediate case was considered: if the initial data are such thatu0∈L¹, u0 = 0 and lim_|x|→∞|x|^σ2u0(x) =κ>0,then the solutions have the asymptotic representation u(t, x) =t^−1σw_κ

√x t

+o

t^−σ1

as t → ∞ uniformly with respect to x∈Rⁿ,where w_κ(ξ) is a positive solution of equation (1.4) such that lim_|ξ|→∞|ξ|^σ2w_κ(ξ) =κ. Note that in these papers there was no restriction on the size of the initial data. The methods of these papers could not be applied for the case of complexα, β∈C,α >0 and as far as we know there are no results

on the asymptotic behavior of solutions to the Cauchy problem (1.1) in the case of complex coefficientsα, β.

We propose here a different approach, developed in our previous papers [9], [10]

and [11], where we considered the critical caseσ=_n².The main idea is to transform equation (1.1) via a change of the dependent variableu=e^−ϕ+iψv to the form

v_t−α∆v+βe^−σϕ

|v|^σ−1 θ

|v|^σvdx

v= 0,

where the nonlinear term having zero mean value obtains better decay properties.

However we assume the smallness of the initial data and the restriction onσto be able to apply the contraction mapping principle.

To state our result precisely we give.

Notation and function spaces. The Fourier transformationFφ= ˆφis defined by ˆφ(ξ) = 1

(2π)^n/2

e^−ixξφ(x)dx andF⁻¹φ(x) or ˇφ(x) is the inverse Fourier transform ofφ, i.e.

φ(x) =ˇ 1 (2π)^n/2

e^ixξφ(ξ)dξ.

We denote by L^p for 1 ≤p≤ ∞, the usual Lebesgue space with a norm φ_p =

Rⁿ|φ(x)^pdx1/p

if 1≤p <∞andφ_∞= ess.sup_x∈Rn|φ(x)|.Define byW_p^0,a= {φ∈L^p:x^aφ∈L^p}weighted Sobolev space with a normφ_W^0,a_p =x^aφ_Lp, where x = √

1 +x². By C(I;B) we denote the space of continuous functions from a time interval Ito the Banach space B. Different positive constants could be denoted by the same letterC.

We defineθ= u0(x)dx>0. We suppose thatσ∈ 0,ⁿ₂

is sufficiently close to ⁿ₂.

In the present paper we prove the following result.

Theorem 1.1. Let α, β ∈C, α >0, δ(α, β) >0. We assume that the initial data u0 ∈W⁰₁^,a, a∈(0,1) have a sufficiently small norm ε=x^au0₁ and are such that θ = u0(x)dx ≥ Cε with 0 < C < 1. We assume that _n² −ε³ <

σ < ²_n. Then there exists a unique mild solution u(t, x) ∈ C

[0,∞) ;W⁰₁^,a ∩ C((0,∞) ;L^∞) of the Cauchy problem (1.1), satisfying the following time decay estimates

u(t)_L∞ ≤Cεt^−σ1

for large t >0. Furthermore there exist a number A and a functionV ∈ W₁^0,a∩ W^0,a_∞ such that the asymptotic formula

u(t, x) =At^−σ1V √x

t

e^iωlogt+O

t^−σ1−γ

, is valid fort→ ∞uniformly with respect tox∈Rⁿ, whereω=−_η¹

β|V|^σV(y)dy, γ=¹₂min

a,1−ⁿ₂σ

, V (ξ)is the solution of the integral equation V (ξ) = 1

(4πα)ⁿ²e^−ξ

2

4α − β

η(4πα)ⁿ² 1

0

dz z(1−z)ⁿ²

e⁻

(ξ−y√z)²

4α(1−z) F(y)dy, (1.5)

whereη= ₁₋^σn 2σ

β|V (y)|^σV(y)dy, F(y) =|V(y)|^σV(y)−V(y)

|V(ξ)|^σV (ξ)dξ.

We organize our paper as follows. Section 2 is devoted to the proof of preliminary estimates in Lemmas 2.1 - 2.4. In Section 3 we prove Theorem 1.1 and the existence of solution V(ξ) to integral equation (1.5).

2. Preliminaries The solution of the linear Cauchy problem

u_t−α∆u=f(t, x), x∈Rⁿ, t >0, u(0, x) =u0(x), x∈Rⁿ

can be written by the Duhamel formulau(t) =G(t)u0+_t

0G(t−τ)f(τ)dτ ,where the Green operatorG is given by

G(t)φ=F⁻¹e^−αξ2tˆφ(ξ) =

G(t, x−y)φ(y)dy, andG(t, x) = (4παt)⁻ⁿ² e^−x2/4αt. Denoteϑ= (2π)ⁿ² φˆ(0).

We first collect some preliminary estimates.

Lemma 2.1. The following estimates are true, provided that the right-hand sides are finite:

·^bG(t)φ

p≤Ct⁻ⁿ²(¹_q−¹_p)t^2b·^bφ

q

and ifφ∈L¹

|·|^b(G(t)φ−ϑG(t))

p ≤ Ct^b−a2 t⁻ⁿ²(¹q−¹_p)·^aφ_q +C|ϑ|t^b2−n2(¹−¹_p)

for allt >0, where1≤q≤p≤ ∞, b∈[0, a], a∈(0,1).

Proof. Since x^b ≤ x−y^by^b, by virtue of the Young inequality with ¹_p =

1

q +¹_r−1,we obtain ·^bG(t)φ

p ≤ 1

(4πt|α|)ⁿ² x^b

e^{−(x−y)28t} (¹_α⁺_α¹)|φ(y)|dy p

≤ Ct⁻ⁿ²

x−y^be^{−(x−y)28t} (_α¹⁺_α¹)y^b|φ(y)|dy p

≤ Ct⁻ⁿ² ·^be^−Cx2/t

r

·^bφ

q

≤ Ct⁻ⁿ²(¹_q−¹_p)t^b2·^bφ

q

for all t > 0, where b ≥ 0,1 ≤ p ≤ ∞. Hence the first estimate of the lemma is true. The second estimate for the case t∈[0,1] follows from the first one and the

inequality |·|^bG(t)

p ≤Ct^b2−n2(¹−¹_p). We now consider the caset ≥1. We write for any 0≤b≤a≤1

|x|^b(G(t)φ−ϑG(t)) = 1 (4πtα)ⁿ²

|x|^b

|y|^a

e^{−(x−y)24αt} −e^−4αtx2

|y|^aφ(y)dy.

(2.1)

Changing the dependent variablesx=ξ√

t andy=η√

t, applying the inequality

|ξ|^b

|η|^a

e^{−(ξ−η)24α} −e^−ξ

2 4α

≤C|ξ−η|^b+|η|^b

|η|^a e^{−(ξ−η)24α} +C|ξ|^b

|η|^ae^−ξ

2 4α

≤ C(1 +|ξ−η|)e^{−C(ξ−η)2}+C|ξ|e^−Cξ2 ≤Ce^{−C(ξ−η)2}+Ce^−Cξ2 for all|η| ≥1 and the estimate

e^{−(ξ−η)24α} −e^−ξ

2 4α

≤C|η|e^{−C(ξ−η)2}+C|η|e^−Cξ2 for all|η| ≤1,we obtain

y∈Rsupⁿ

|x|^b

|y|^a

e^{−(x−y)24αt} −e^−4αtx2

r

= t^{b−a2 +2rn} sup

η∈Rⁿ

|ξ|^rb

|η|^ra

e^{−(ξ−η)24α} −e^−ξ

2 4α

^rdξ ¹_r

≤ Ct^{b−a2 +2rn} sup

η∈Rⁿ e^{−C(ξ−η)2}+e^−Cξ2

dξ≤Ct^{b−a2 +2rn}, (2.2)

where r∈[1,∞).The caser=∞is considered in the same way. Substitution of (2.2) into (2.1) via the Young inequality with ¹_p =¹_q +¹_r−1,yields

|·|^b(G(t)φ−ϑG(t))

p ≤ Ct⁻ⁿ² sup

y∈Rⁿ

|x|^b

|y|^a

e^{−(x−y)24αt} −e^−4αtx2

r

|·|^aφ_q

≤ Ct^{b−a2 −n2}(¹_q−¹_p)·^aφ_q

for allt≥1 and 1≤p≤ ∞.Therefore the second estimate of the lemma is valid.

Lemma 2.1 is proved.

Lemma 2.2. Let the functionf(t, x)have the zero mean valuefˆ(t,0) = 0and the norm

sup

t>0 sup

1≤s≤pt^ν+n2(¹−¹_s)t^−λ2·^λf(t)

s=f_F

be finite, where1≤p≤ ∞, λ∈(0,1), ν∈(0,1).We also suppose that the function g(t)is such thatg(t)≥ ¹₂t^µ for allt >0,whereµ >0is such thatµ+ν <1 +^λ₂. Then the following inequality is valid

·^b _t

0 g⁻¹(τ)G(t−τ)f(τ)dτ p

≤Ct^b2t^{1−ν−µ−n2}(¹−_p¹)f_F, for allt >0, whereb∈[0, λ].

Proof. Since ˆf(t,0) = 0 we have |x|^bG(t)f = |x|^b

G(t)f−(2π)ⁿ² fˆ(0)G(t) with anyb∈[0, λ].Applying the second estimate of Lemma 2.1 we obtain

|·|^bG(t−τ)f(τ)

p≤Ct−τ^b−λ2 (t−τ)⁻ⁿ²(¹_q−¹_p)·^λf(τ) (2.3) q

for all 0< τ < t, where 0≤ b≤ λ≤1, 1 ≤q ≤p≤ ∞. Therefore by virtue of (2.3) we get

|·|^b _t

0 g⁻¹(τ)G(t−τ)f(τ)dτ p

≤ C ^t₂

0 t−τ^b−λ2 (t−τ)⁻ⁿ²(¹−¹_p)|·|^λf(τ)

1τ^−µdτ +C

_t

2t

t−τ^b−λ2 |·|^λf(τ)

pτ^−µdτ

≤ Cf_{F 2}^t

0 t−τ^b−λ2 (t−τ)⁻ⁿ²(¹−¹_p)τ^−ντ^λ2−µdτ +Cf_F

_t

t2

t−τ^b−λ2 τ^−ν−n2(¹−¹_p)τ^λ2−µdτ

≤ Cf_Ft^b−λ2 t⁻ⁿ²(¹−¹_p) ^t

2

0

τ^−ντ^λ2−µdτ +Cf_Ft^λ2 t^−ν−n2(¹−¹_p) _t

2t

t−τ^b−λ2 dτ

≤ Ct^b2t^{1−ν−µ−n2}(¹−¹_p)f_F (2.4)

for allt >0, where 1≤p≤ ∞,0≤b≤λ.Lemma 2.2 is proved.

Denoteζ= ^θσσ2

n2

(4π)ⁿ2σ(¹−ⁿ₂σ)δ(α, β).Since _n²−ε³< σ < ²_n andθ^σ≥Cε^σ≥Cεⁿ², we can suppose thatζ= ^θσσ2

n2

(4π)ⁿ2σ(¹−ⁿ₂σ)δ(α, β)≥Cεⁿ²⁻³≥1.

Lemma 2.3. We assume thatv˜∈W⁰₁^,awitha∈(0,1)and sufficiently small norm

˜v_W^0,a

1 =ε and v(0) = (2π)˜ ⁻ⁿ² θ ≥Cε > 0. Let the function v(t, x) satisfy the estimates v(t)_1+σ≤Cεt^{−2(1+σ)nσ} andv(t)− G(t) ˜v_1+σ≤Cε^1+σt^{−2(1+σ)nσ} for all t >0.Then the following inequality is valid

1 + σ θ

_t

0 dτ

β|v|^σv(τ , x)dx≥ 1

2t^1−n2σ (2.5)

for allt >0.

Proof. Applying Lemma 2.1 we get

G(t) ˜v−θG(t)_1+σ≤Cεt^{−2(1+σ)nσ} t^−a2 , then via the assumption

v(t)− G(t) ˜v1+σ≤Cε^1+σt^{−2(1+σ)nσ}

we have by the H¨older inequality |v|^σv−θ^1+σ|G|^σG

1≤ |v|^σv− |G(t) ˜v|^σG(t) ˜v₁ +|G(t) ˜v|^σG(t) ˜v−θ^1+σ|G|^σG

1

≤ C

v^σ_1+σ+G(t) ˜v^σ_1+σ

v− G(t) ˜v_1+σ +C

G(t) ˜v^σ_1+σ+θ^σG^σ_1+σ

G(t) ˜v−θG_1+σ

≤ Cε^1+2σt^−n2σ+Cε^1+σt^−n2σt^−a2 for allt >0. By a direct computation we obtain

β|G|^σG(t, x)dx =

β

(4πt)^n2σ+n2 |α|^n2σαⁿ²

e^−x8t2σ(_α¹⁺_α¹)−_4tα^x2dx

= 2ⁿ²t^−n2σ (4π)^n2σ



 β|α|^n−n2σ

(2 +σ)|α|²+σα^{2 n}₂





= 2ⁿ²t^−n2σ

(4π)^n2σδ(α, β). Therefore we get

σ θ

β|v|^σv(t, x)dx−θ^σσ2ⁿ²t^−n2σ

(4π)^n2σ δ(α, β)

=

σ θ

β|v|^σv(t, x)dx−σ θθ^1+σ

β|G|^σG(t, x)dx

≤ C

θ |v|^σv−θ^1+σ|G|^σG

1≤Cε^2σt^−n2σ+Cε^σt^−n2σt^−a2 for allt >0, whence

σ θ

_t

0 dτ

β|v|^σv(τ , x)dx− θ^σσ2ⁿ² (4π)^n2σ

1−ⁿ₂σδ(α, β)t^1−n2σ

=

σ θ

_t

0 dτ

β|v|^σv(τ , x)dx−ζt^1−n2σ

≤ Cε^2σ _t

0 τ^−n2σdτ+Cε^σ _t

0 τ^−n2στ^−a2dτ

≤ Cε^σζt^1−n2σ+Cε^σt^{1−n2σ−a2} ≤1 2ζt^1−n2σ for allt >0. Thus we get

σ θ

_t

0 dτ

β|v|^σv(τ , x)dx≥1

2ζt^1−n2σ. (2.6)

Estimate (2.6) implies (2.5), sinceζ≥1.Lemma 2.3 is proved.

Lemma 2.4. Let the function f(x) have the zero mean value fˆ(0) = 0 and the norm

·^af_p+·^af₁

be finite, where a∈(0,1), 1≤p≤ ∞.Then the following inequalities are valid

ξ^a 1

0

dz z(1−z)ⁿ²

dye⁻

(ξ−y√z)²

4α(1−z) f(y) p

≤C·^af₁+C·^af_p

and

ξ^a 1

0

dz (1−z)ⁿ²

1 z − t

tz dye^−(ξ−y

√z)²

4α(1−z) f(y) p

≤ Ct^−a2

·^af₁+·^af_p for allt >0, where1≤p≤ ∞.

Proof. By the Young inequality for convolutions we get ξ^a

e⁻

(ξ−y√z)²

4α(1−z) f(y)dy p

≤

ξ−y√ z_a

e^−C(ξ−y

√z)²

1−z |f(y)|dy p

+

e^−C(ξ−y

√z)²

1−z y^a|f(y)|dy p

≤

ξ^ae^−Cξ

2 1−z

1f_p+ e^−Cξ

2 1−z

1·^af_p≤(1−z)ⁿ² ·^bf

p

for allz∈₁

2,1 ,since

φ

· −y√ z

f(y)dy p

= z^−2pn

φ(· −y)f y

√z

dy p

≤ Cz^−2pn φ₁ f

√· z

p

≤Cφ₁f_p. Therefore

ξ^a

1

12

1 z(1−z)ⁿ²

e⁻

(ξ−y√z)²

4α(1−z) f(y)dydz p

≤ C 1

12

(1−z)⁻ⁿ² ξ^a

e^−(ξ−y

√z)²

4α(1−z) f(y)dy p

dz≤C·^af_p (2.7)

and

ξ^a 1

12

dz (1−z)ⁿ²

1 z − t

tz dye⁻

(ξ−y√z)²

4α(1−z) f(y) p

≤ Ct⁻¹ 1

12

(1−z)⁻ⁿ² ξ^a

e⁻

(ξ−y√ z)²

4α(1−z) f(y)dy p

dz

≤ Ct⁻¹·^af_p, (2.8)

where 1≤p≤ ∞.

Via the condition

f(y)dy= 0,we write ξ^a

e^−(ξ−y

√z)²

4α(1−z) f(y)dy p

=

ξ^a e^−(ξ−y

√z)²

4α(1−z) −e^{− ξ}

2 4α(1−z)

f(y)dy p

≤

ξ^a− ξ−y√

z_a e^−(ξ−y

√z)²

4α(1−z)

f(y)dy p

+

ξ−y√ z_a

e⁻

(ξ−y√ z)²

4α(1−z) − ξ^ae^{− ξ}

2 4α(1−z)

f(y)dy p

≤ Cz^a2

e^−C(ξ−y√

z)²y^a|f(y)|dy p

+Cz^a2

e^−C(ξ−y√

z)²+e^−Cξ2

y^a|f(y)|dy p

≤ Cz^a2 ·^af₁e^−Cξ2

p≤Cz^a2·^af₁ for allz∈

0,¹₂ , since

φ

· −y√ z

f(y)dy p

= z^−2pn

φ(· −y)f y

√z

dy p

≤ Cz^−2pn φ_p f

√· z

1≤Czⁿ²(¹−_p¹)φ_pf₁

≤ Cφ_pf₁.

Thus

ξ^{a 1}

2

0

1 z(1−z)ⁿ²

e^−(ξ−y

√z)²

4α(1−z) f(y)dydz p

≤ ¹

2

0

dz z(1−z)ⁿ²

ξ^a

e^−(ξ−y

√z)²

4α(1−z) f(y)dy p

≤ C ¹₂

0

dz

z^1−a2(1−z)ⁿ² ·^af₁≤C·^af₁ (2.9)

and

ξ^{a 1}₂

0

dz (1−z)ⁿ²

1 z− t

tz e⁻

(ξ−y√z)²

4α(1−z) f(y)dy p

≤ ¹

2

0

dz (1−z)ⁿ²

1 z − t

tz ξ^a

e⁻

(ξ−y√ z)²

4α(1−z) f(y)dy p

≤ C ¹₂

0

z^a2 (1−z)ⁿ²

1 z − t

tz

dz·^af₁≤Ct^−a2 ·^af₁, (2.10)

where 1 ≤ p ≤ ∞. Collecting estimates (2.7) - (2.10), we get the results of the lemma. Lemma 2.4 is proved.

3. Proof of Theorem 1.1

The local existence of solutions for the Cauchy problem (1.1) can be obtained by the standard contraction mapping principle (for the proof see [5], [6], [16]).

Theorem 3.1. Let the initial datau0∈W₁^0,a.Then for someT >0there exists a unique solutionu(t, x)∈C

[0, T);W₁^0,a

∩C((0, T) ;L^∞)of the Cauchy problem (1.1). Moreover if the initial data are sufficiently small·^au0₁ ≤ε, then there exists a time T ≥ 1 such that the solution u(t, x) of the Cauchy problem (1.1) satisfy the estimates

1≤p≤∞sup sup

b∈[0,a] sup

t∈(0,T]t^σ1−2pn t^−2b·^bu(t)

p≤2ε.

In order to get desired a-priori estimates of local solutions we make a change of the dependent variableu(t, x) =e^{−ϕ(t)+iψ(t)}v(t, x),then we get from (1.1)

v_t−α∆v+βe^−σϕ|v|^σv−(ϕ −iψ)v= 0.

(3.1)

Now let us demand that the real-valued functionsϕ(t) andψ(t) satisfy the following condition

βe^−σϕ|v|^σv−(ϕ −iψ)v dx= 0, hence via equation (3.1) we get _dt^d

v(t, x)dx= 0 for allt >0. Therefore βe^−σϕ

|v|^σvdx= (ϕ −iψ) (2π)ⁿ² v(0,0).

If we chooseϕ(0) = 0 and ψ(0) = arg ˆu0(0) we have ˆ

v(t,0) = ˆv(0,0) =e^{ϕ(0)−iψ(0)}uˆ0(0)≡ |ˆu0(0)| ≡ θ (2π)ⁿ² >0 and we obtain the equations

ϕ (t) = ¹_θe^−σϕ β

|v|^σvdx , ψ (t) =−¹_θe^−σϕ

β

|v|^σvdx (3.2) .

Thus we need to solve the following system





v_t−α∆v+βe^−σϕ

|v|^σ−¹_θ

|v|^σvdx v= 0, ϕ = ¹_θe^−σϕ

β|v|^σvdx , v(0, x) = ˜v(x), ϕ(0) = 0, (3.3)

where the initial data

˜

v(x) =u0(x) exp

−iarg

u0(x)dx

=u0(x) exp (−iarg ˆu0(0)). Multiplying the second equation of system (3.3) by the factor e^σϕ(t), then inte- grating with respect to timet >0 and making a change of the dependent variable e^σϕ(t)=g(t),we get an integral equation

v=A(v), (3.4)

where

A(v) (t) =G(t) ˜v−β _t

0 g⁻¹(τ)G(t−τ)f(τ)dτ and

f(τ) = |v|^σv(τ)−v(τ) θ

|v|^σv(τ , x)dx, g(τ) = 1 + σ

θ _τ

0

β|v|^σv(t, x)dxdt.

We define v0 = G(t) ˜v and successive approximations v_k+1 = A(v_k) for k = 0,1,2, ...We prove thatAis a contraction mapping in the set

X =

v∈C

[0,∞);W⁰₁^,a

∩C((0,∞);L^∞) :

1≤p≤∞sup sup

b∈[0,a]sup

t>0tⁿ²(¹−_p¹)t^−b2·^bv(t)

p≤2ε

.

We prove that the mapping A transforms the setX into itself, also we prove the estimates

v_k ∈X; vk(t)− G(t) ˜v_1+σ≤Cε^1+σt^{−2(1+σ)nσ} , (3.5)

g_k(t)≥1

2t^1−n2σ, (3.6)

f_k(t, x)dx= 0,

v_k(t, x)dx=θ (3.7)

for all t > 0 and k ≥ 0, where f_k(t) = |vk|^σv_k(t)− ^vk_θ^(t)

|vk|^σv_k(t, x)dx and g_k