Cauchy problem for multidimensional coupled system of nonlinear Schr¨ odinger equation and generalized IMBq equation

Cauchy problem for multidimensional coupled system of nonlinear Schr¨ odinger equation and generalized IMBq equation

Chen Guowang

Dedicated to the memory of Prof. Jan Potoˇcek and Prof. Josef Kolom´y.

Abstract. The existence, uniqueness and regularity of the generalized local solution and the classical local solution to the periodic boundary value problem and Cauchy problem for the multidimensional coupled system of a nonlinear complex Schr¨odinger equation and a generalized IMBq equation

iεt+∇²ε−uε= 0,

utt− ∇²u−a∇²utt=∇²f(u) +∇²(|ε|²) are proved.

Keywords: coupled system of nonlinear Schr¨odinger equation and generalized IMBq, multidimensional, periodic boundary value problem, Cauchy problem, generalized local solution, classical local solution

Classification: 35L35, 35K55

1. Introduction

In [1], [2] the problems of soliton solutions for the Schr¨odinger field interacting with the Boussinesq field have been studied and an approximate solution of system

iεt+εxx−uε= 0, (1.1)

∂²

∂t² − ∂²

∂x² −δ 3

∂⁴

∂x⁴

u−δ ∂²

∂x²(u²) = ∂²

∂x²(|ε|²) (1.2)

have been found, where δ > 0 is a constant. The exact soliton solutions of the above system were obtained in [3]–[5] by various techniques. In [2] author suggested that the Boussinesq equation (1.2) (which we call Bq-equation) was replaced by the IBq equation (the improved Bq-equation)

(1.3) ∂²

∂t² − ∂²

∂x² −δ 3

∂⁴

∂x²∂t²

u−δ ∂²

∂x²(u²) = ∂²

∂x²(|ε|²)

∗ The project was supported by National Natural Science Foundation of China (Grant No. 1967/075).

and the soliton solutions of system (1.1), (1.3) were obtained. A modification of IBq equation analogous to the MKdV equation yields

(1.4) ∂²

∂t² − ∂²

∂x² − ∂⁴

∂x²∂t²

u= ∂²

∂x²(u³),

which we call the IMBq equation. The IMBq equation with several variables (see [2]) is

(1.4)^′ u_tt− ∇²u_tt− ∇²u=∇²u³.

In this paper, we study the system of the multidimensional Schr¨odinger field interacting with real generalized IMBq field

iε_t+∇²ε−uε= 0, (1.5)

utt− ∇²u−a∇²utt=∇²f(u) +∇²(|ε|²), (1.6)

whereε(x, t) denotes complex unknown function of variablesx= (x₁, x₂,· · · , x_n)

∈ Rⁿ and t ∈ R+, u(x, t) denotes a real unknown function of variables x and t, ∇ = _∂x^∂

1,_∂x^∂

2,· · ·,_∂x^∂

n

, a > 0 is a constant, i = √

−1 and f(s) is a given nonlinear function.

Let Ω⊂Rⁿbe ann-dimensional cube with width 2D(D >0) in each direction, that is Ω ={x= (x1, x2,· · ·, xn)| |x_j| ≤D, j= 1,2,· · ·, n}, x+ 2Dej denotes (x₁,· · ·, x_j−1, x_j + 2D, x_j+1,· · ·, x_n) (j = 1,2,· · ·, n) and Q_T = {x∈ Ω,0 ≤ t ≤ T}. For the system (1.5), (1.6), we discuss its periodic boundary value problem in the (n+ 1)−dimensional cylindrical domainQ_T

ε(x, t) =ε(x+ 2D_ej, t), u(x, t) =u(x+ 2D_ej, t), j= 1,2,· · ·, n, (1.7)

ε(x,0) =ε0(x), u(x,0) =ϕ(x), ut(x,0) =ψ(x), (1.8)

where ε0(x), ϕ(x) and ψ(x) are given functions of n-dimensional initial value, satisfying the periodic boundary conditions (1.7). For the system (1.5), (1.6) we also study the Cauchy problem

(1.9) ε(x,0) =ε₀(x), u(x,0) =ϕ(x), u_t(x,0) =ψ(x), whereε₀(x),ϕ(x) andψ(x) are given functions defined inRⁿ.

In Section 2 the existence and uniqueness of the generalized local solution and the classical local solution of the periodic boundary value problem (1.5)–(1.8) are proved. Moreover the regularity of the classical local solution of the problem (1.5)–(1.8) is considered. In Section 3 the existence, uniqueness and regularity of the generalized local solution and the classical local solution of the Cauchy problem (1.5), (1.6), (1.9) are proved.

For simplicity, leta= 1 in (1.6). We prove only the existence, uniqueness and regularity of the generalized local solutions and the classical local solution for the 2-dimensional problem, because we can treat the n-dimensional problem by the method of the 2-dimensional case.

2. Periodic boundary value problem (1.5)–(1.8) We first establish an orthonormal base onL₂(Ω).

Lemma 2.1([6], [7]). LetΩ₁⊂Rⁿand∧ ⊂R^mbe measurable sets. If {ϕ_j}j∈J

and {χ_k}k∈K (J and K are the index sets) are orthonormal bases in L2(Ω1) andL₂(∧)respectively, then the system of functions{ϕ_j(x)χ_k(y)}j∈J,k∈K is an orthonormal base inL2(Ω1× ∧).

It is clear that Lemma 2.1 is valid for the case of any finite orthonormal bases.

Let us consider the eigenvalue problem for the ordinary differential equation y^′′+λy= 0,

y(x₁+ 2D) =y(x₁).

We can obtain eigenvalues λ_1l = α²_l = ^lπ_D2

(l = 0,1,· · ·) and the family of eigenfunctions {y_l(x₁)} = ₁

√2D,√¹

Dcosα_lx₁,√¹

Dsinα_lx₁, l = 1,2,· · · , which composes an orthonormal base. Similarly, we can obtain eigenvalues λ_2j = β²_j = ^jπ_D2

(j = 0,1,· · ·) and the family of eigenfunctions {z_j(x₂)} = ₁

√2D,√¹

Dcosβ_jx₂,√¹

Dsinβ_jx₂, j = 1,2,· · · of the periodic boundary value problemz^′′+λz= 0,z(x₂+2D) =z(x₂).{z_j(x₂)}composes an orthonormal base.

According to Lemma 2.1, the family of functions{y_l(x₁)z_j(x₂), l, j = 0,1,· · · } composes an orthonormal base inL₂(Ω).

Let (ε(x, t), u(x, t)) be the solution of the problem (1.5)–(1.8), with ε(x, t) = P_∞

l,j=0L_lj(t)y_l(x₁)z_j(x₂) and u(x, t) = P_∞

l,j=0T_lj(t)y_l(x₁)z_j(x₂). The initial value functions may be expressed ε₀(x) = P_∞

l,j=0ε_ljy_l(x₁)z_j(x₂), ϕ(x) = P_∞

l,j=0ϕ_ljy_l(x₁)z_j(x₂) and ψ(x) = P_∞

l,j=0ψ_ljy_l(x₁)z_j(x₂), where ε_lj are com- plex numbers,ϕ_lj,ψ_lj are real numbers.

Substituting the expressions ofε(x, t) andu(x, t) into the system (1.5), (1.6), mul- tiplying both sides of (1.5) and (1.6) byy_l(x1)z_j(x2) respectively and integrating over Ω, we get

(2.1) iL˙_lj−(α²_l +β²_j)L_lj− Z

Ω

uεy_l(x₁)z_j(x₂)dx= 0,

(2.2) (1 +α²_l +β_j²) ¨T_lj+ (α²_l +β²_j)T_lj− Z

Ω

(∇²f(u) +∇²(|ε|²))· y_l(x₁)z_j(x₂)dx= 0, l, j= 0,1,· · ·.

Substituting the expressions ofε(x, t), u(x, t),ε0(x),ϕ(x) andψ(x) into the initial conditions (1.8), we obtain

(2.3) L_lj(0) =ε_lj, T_lj(0) =ϕ_lj, T˙_lj(0) =ψ_lj, l, j= 0,1· · ·,

where ˙T_lj(t) = _dt^dT_lj(t), etc. HenceL_lj(t) andT_lj(t) satisfy the infinite system of ordinary differential equations (2.1)–(2.3). Let us now prove the existence of the solution for the initial value problem (2.1)–(2.3). For this purpose, we shall first consider the existence of the solution for the following initial value problem of a finite system of ordinary differential equations

iL˙_{N lj}−(α²_l +β_j²)L_{N lj} = Z

Ωu_Nε_Ny_l(x1)z_j(x2)dx, (2.4)

(1 +α²_l +β_j²) ¨T_{N lj} + (α²_l +β²_j)T_{N lj} (2.5)

= Z

Ω

(∇²f(u_N) +∇²(|ε_N|²))y_l(x₁)z_j(x₂)dx, L_{N lj}(0) =ε_lj, T_{N lj}(0) =ϕ_lj, T˙_{N lj}(0) =ψ_lj, l, j= 0,1· · · , N, (2.6)

whereε_N(x, t) =PN

l,j=0L_{N lj}(t)y_l(x1)z_j(x2) andu_N(x, t) = PN

l,j=0T_{N lj}(t)y_l(x1)z_j(x2). In order to use Leray-Schauder’s fixed-point argu- ment we are also going to consider the following initial value problem of the finite system of ordinary differential equations with the parameterθ

iL˙_{N lj}−(α²_l +β_j²)L_{N lj} =θ Z

Ω

u_Nε_Ny_l(x1)z_j(x2)dx, (2.7)

(1 +α²_l +β_j²) ¨T_{N lj}+ (α²_l +β_j²)T_{N lj} (2.8)

=θ Z

Ω

(∇²f(u_N) +∇²(|ε_N|²))y_l(x₁)z_j(x₂)dx, L_{N lj}(0) =θε_lj, T_{N lj}(0) =θϕ_lj,

(2.9)

T˙_{N lj}(0) =θψ_lj, l, j= 0,1· · · , N, 0≤θ≤1.

Lemma 2.2. Suppose that the following conditions are satisfied.

1. f ∈C⁵,|f^(j)(s)| ≤K_j|s|^P+(5−j)(j= 1,2,3,4,5), where

K_j(j= 1,2,3,4,5)are constants,p≥1 is a natural number and(j)denotes the order of derivatives.

2. (L(t), T(t))is the solution of the system(2.4),(2.5), where

L(t) = (L_{N lj}(t), l, j= 0,1,· · ·, N)and T(t) = (T_{N lj}(t), l, j= 0,1,· · ·, N).

Let

E_N(t) = XN

l,j=0

(1 +α²_l +β²_j + X

h+m=8 h,m=0,2,4,6,8

α^h_lβ_j^m (2.10)

+2 X

h+m=10 h,m=2,4,6,8

α^h_lβ_j^m+α¹⁰_l +β_j¹⁰) ˙T_{N lj}²

+ XN

l,j=0

(1 +α²_l +β_j²+ 2 X

h+m=10 h,m=2,4,6,8

α^h_lβ_j^m+α¹⁰_l +β¹⁰_j )T_{N lj}²

+ XN

l,j=0

(1 + X

h+m=10 h,m=0,2,4,6,8,10

α^h_lβ_j^m)|L_{N lj}|²+ 1.

Then there is

(2.11) dE_N(t)

dt ≤K₆(E_N(t))^P+62 , whereK₆>0is a constant independent of θ, Dand N.

Proof: Multiplying both sides of (2.7) by (1 +α⁴_lβ_j⁶+α⁶_lβ_j⁴+α²_lβ⁸_j +α⁸_lβ_j²+ α¹⁰_l +β¹⁰_j )L_{N lj}, summing up the products forl, j = 0,1,· · ·, N and taking the imaginary part, we get

(2.12)

d dt

n X^N

l,j=0

(1 + X

h+m=10 h,m=0,2,4,6,8,10

α^h_lβ_j^m)|L_{N lj}|²o

= 2θI_mhu_Nε_N,− X

h+m=10 h,m=0,2,4,6,8,10

ε_{N x}h 1x^m₂ i,

whereL_{N lj} is the conjugate number ofL_{N lj} andhu, vi=R

Ωu(x)v(x)dx.

Multiplying both sides of (2.8) by (1 +α⁴_lβ_j⁴+α⁶_lβ_j²+α²_lβ_j⁶+α⁸_l +β⁸_j) ˙T_{N lj}, summing up the products for l, j = 0,1,· · ·, N, adding _dt^d PN

l,j=0T_{N lj}² to both sides of the obtained equation and adding the equation (2.12), we have

(2.13)

dE_N(t)

dt = 2θh∇²f(u_N) +∇²(|ε_N|²), u_{N t}+ X

h+m=8 h,m=0,2,4,6,8

u_{N x}h 1x^m₂ ti

+d

dthu_N, u_Ni+ 2θImhu_Nε_N,− X

h+m=10 h,m=0,2,4,6,8,10

ε_Nx^h₁x^m₂ i.

k · kL2(Ω)=k · k,k · kL∞(Ω) andk · kH^m(Ω) denote the norm of the spaceL₂(Ω), L_∞(Ω) and Sobolev spaceH^m(Ω) respectively.

Since by (2.10) we have

(2.14)

E_N(t) = X2 j=1

X

h=0,1,4,5

hu_{N x}h jt, u_{N x}h

jti+ X

h+m=4 h,m=1,2,3

hu_{N x}h

1x^m₂ t, u_{N x}h 1x^m₂ ti

+2 X

h+m=5 h,m=1,2,3,4

hu_{N x}h

1x^m₂ t, u_{N x}h

1x^m₂ti+hu_N, u_Ni+ X2 j=1

hu_{N xj}, u_{N xj}i

+2 X

h+m=5 h,m=1,2,3,4

hu_{N x}h

1x^m₂ , u_{N x}h 1x^m₂ i+

X2 j=1

hu_{N x}5 j, u_{N x}5

ji +hε_N, ε_Ni+ X

h+m=5 h,m=0,1,2,3,4,5

hε_{N x}h

1x^m₂ , ε_{N x}h

1x^m₂ i+ 1,

by using the Gagliardo-Nirenberg theorem [8] and (2.14) we obtain ku_NkL∞(Ω)≤C₁ku_Nk⁴⁵ku_Nk

1 5

H⁵(Ω)≤C₂(E_N(t))¹², (2.15)

ku_{N xj}kL∞(Ω)≤C₃ku_Nk³⁵ku_Nk

2 5

H⁵(Ω)≤C₄(E_N(t))¹², j= 1,2, (2.16)

ku_{N xjxk}k,ku_{N xjxk}kL∞(Ω)≤C₅(E_N(t))¹², j, k= 1,2, (2.17)

ku_{N x}j

1x^k₂k,ku_{N x}j

1x^k₂kL∞(Ω)≤C₆(E_N(t))¹², (2.18)

j+k= 3, j, k= 0,1,2,3, ku_{N x}j

1x^k₂k,ku_{N x}j

1x^k₂kL∞(Ω)≤C₇(E_N(t))¹², (2.19)

j+k= 4, j, k= 0,1,2,3,4.

Similarly we can obtain

kε_NkL∞(Ω),kε_{N xj}k,kε_{N xj}kL∞(Ω)≤C8(E_N(t))¹², j = 1,2, (2.20)

kε_{N xjxk}k,kε_{N xjxk}kL∞(Ω)≤C₉(E_N(t))¹², j, k= 1,2, (2.21)

kε_{N x}j

1x^k₂k,kε_{N x}j

1x^k₂kL∞(Ω)≤C₁₀(E_N(t))¹², (2.22)

j+k= 3, j, k= 0,1,2,3, kε_{N x}j

1x^k₂k,kε_{N x}j

1x^k₂kL∞(Ω)≤C₁₁(E_N(t))¹², (2.23)

j+k= 4, j, k= 0,1,2,3,4.

Using the assumptions of Lemma 2.2, H¨older inequality and the inequalities (2.15)–(2.23) and (2.10), we have

(2.24)

|h∇²f(u_N), u_{N t}i|

=| Z

Ω[f^′′(u_N)(u²_{N x1}+u²_{N x2}) +f^′(u_N)(u_{N x}2

1+u_{N x}2

2)]u_{N t}dx|

≤C₁₂(E_N(t))^p+42 X2 j=1

Z

Ω

X2 j=1

(|u_{N xj}|+|u_{N x}2

j|)|u_{N t}|dx

≤C₁₂(E_N(t))^p+42 X2 j=1

(ku_{N xj}k+ku_{N x}2

jk)ku_{N t}k

≤C₁₃(E_N(t))^p+62 .

By means of integration by parts, H¨older inequality, inequalities (2.15)–(2.19) and the assumptions of the lemma, we have

(2.25) |h∇²f(u_N), u_{N x}6

1x²₂ti|=h ∂³

∂x³₁∇²f(u_N), u_{N x}3

1x²₁ti≤C14(E_N(t))^p+62 . Similarly we have

|h∇²f(u_N), u_{N x}4

1x⁴₂t+u_{N x}2

1x⁶₂t+u_{N x}8

1t+u_{N x}8

2ti| ≤C₁₅(E_N(t))^p+62 , (2.26)

|h∇²(|ε_N|²), u_{N t}+ X

j+k=8 j,k=0,2,4,6,8

u_{N x}^j

1x^k₂ti| ≤C16(E_N(t))³². (2.27)

Integrating by parts and using inequalities (2.15)–(2.23) we get

(2.28)

|I_mhu_Nε_N, ε_{N x}4

1x⁶₂i|=|I_m Z

Ω

(u_Nε_N)_x2 1x³₂ε_{N x}2

1x³₂dx|

=|I_m Z

Ω

(u_{N x}2

1x³₂ε_N+ 3u_{N x}2

1x²₂ε_{N x2}+ 3u_{N x}2 1x2ε_{N x}2

2

+u_{N x}2 1ε_{N x}3

2+ 2u_{N x1x}3

2ε_{N x1}+ 6u_{N x1x}2

2ε_{N x1x2}+ 6u_{N x1x2}ε_{N x1x}2 2

+ 2u_{N x1}ε_{N x1x}3

2+u_{N x}3 2ε_{N x}2

1+ 3u_{N x}2 2ε_{N x}2

1x2

+ 3u_{N x2}ε_{N x}2

1x²₂)ε_{N x}2

1x³₂dx| ≤C17(E_N(t))³². Similarly, we have

(2.29) |I_mhu_Nε_N, ε_{N x}6

1x⁴₂+ε_{N x}8

1x²₂+ε_{N x}2

1x⁸₂+ε_{N x}10

1 +ε_{N x}10

2 i| ≤C₁₈(E_N(t))¹². Observe thatp≥1 and it follows (2.11) from (2.13) and (2.24)–(2.29). The proof

is thus completed.

It is easy to prove the following lemma by (2.11)

Lemma 2.3. Under the conditions of Lemma2.2, if

Nlim→∞E_N(0) =b= X∞ l,j=0

1 +α²_l +β_j²+ X

h+m=8 h,m=0,2,4,6,8

α^h_lβ^m_j

+ 2 X

h+m=10 h,m=2,4,6,8

α^h_lβ_j^m+α¹⁰_l +β_j¹⁰ ψ_lj²

+ XN

l,j=0

1 +α²_l +β_j²+ 2 X

h+m=10 h,m=2,4,6,8

α^h_lβ^m_j +α¹⁰_l +β_j¹⁰ ϕ²_lj

+ XN

l,j=0

1 + X

h+m=10 h,m=2,4,6,8

α^h_lβ_j^m+α¹⁰_l +β_j¹⁰

|ε_lj|²+ 1<∞,

thenE_N(t)≤b/(1−^K6(p+4)₂ b^(p+4)/2t)^2/(p+4)is uniformly bounded(letM be the bound) and independent ofN and D in any closed subinterval0≤t ≤t1 < t_b, wheret_b = 2/[K₆(p+ 4)b^p+42 ].

It is easy to prove the following lemma by Lemma 2.3 and Leray-Schauder’s fixed-point theorem[9]as in[10].

Lemma 2.4. Under the conditions of Lemma2.3, there is a solution of the initial value problem(2.4)–(2.6)of the finite system of ordinary differential equations in [0, t₁], here0< t₁< t_b.

From the Ascoli-Arzel`a theorem we have

Corollary 2.1. Under the conditions of Lemma2.4 and for the sequences {L_{N lj}}^N_l,j=0 and {T_{N lj}}^N_l,j=0 (N = 1,2,· · ·) of the solution for the initial value problem(2.4)–(2.6), there are convergent subsequences{L_Nslj}^N_l,j=0^s and

{T_Nslj}^N_l,j=0^s respectively. AsNs→ ∞, then

L_Nslj →L_lj, T_Nslj→T_lj, T˙_Nslj →T˙_lj (l, j= 0,1,· · ·) uniformly in[0, t1].

Lemma 2.5. Under the conditions of Lemma2.3, the seriesP_∞

l,j=0|L_lj|², P_∞

l,j=0α^h_lβ^m_j |L_lj|² (h, m= 2,4,6,8, h+m= 10),P_∞

l,j=0α¹⁰_l |L_lj|², P_∞

l,j=0β_j¹⁰|L_lj|²,P_∞

l,j=0T˙_lj²,P_∞

l,j=0α^h_lT˙_lj² (h= 2,8,10),P_∞

l,j=0β_j^mT˙_lj² (m = 2,8,10), P_∞

l,j=0α^h_lβ_l^mT˙_lj² (h, m = 2,4,6, h+l = 8), P_∞

l,j=0α^h_lβ_j^mT˙_lj² (h, m= 2,4,6,8,h+m= 10),P_∞

l,j=0T_lj²,P_∞

l,j=0α^h_lT_lj² (h= 2,10),P_∞

l,j=0β_j^mT_lj²

(m= 2,10)and P_∞

l,j=0 α^h_lβ_j^mT_lj² (h, m = 2,4,6,8, h+m= 10) are convergent and uniformly bounded(letM be the bound)in [0, t1].

Proof: Let us denote S_J = P_J

l,j=0α⁴_lβ⁶_j|L_lj|² (J is a natural number). Ob- viously S_J+1 > S_J. Fixing J, taking Ns(> J) sufficiently large and using the Corollary 2.1 we obtain

S_J ≤S_J− XJ

l,j=0

α⁴_lβ_j⁶|L_Nslj|²+

Ns

X

l,j=0

α⁴_lβ⁶_j|L_Nslj|² ≤1 +E_Ns(t).

Since the functionsE_Ns(t) are bounded and independent ofNsin [0, t1], thenS_J is bounded. Consequently the seriesP_∞

l,j=0α⁴_lβ_j⁶|L_lj|²is convergent and bounded in [0, t1]. Using the same method we can prove the other conclusions. The lemma

is proved.

Corollary 2.2. Under the conditions of Lemma 2.3, there exists a constant M1 >0, such that

kεkH⁵(Ω)+kukH⁵(Ω)+kutkH⁵(Ω)≤M₁, kεk_C^3,λ_(Ω)+kuk_C^3,λ_(Ω)+kutk_C^3,λ_(Ω)≤M₁ in[0, t₁], hereε(x, t) =P_∞

l,j=0L_lj(t)y_l(x₁)z_j(x₂)and u(x, t) =P_∞

l,j=0T_lj(t)y_l(x₁)z_j(x₂).

Lemma 2.6. Under the conditions of Lemma 2.3, ε_Ns → ε, ε_Nsxl → ε_xl (l = 1,2), u_Ns → u, u_Nsxl → u_xl (l = 1,2) and u_Nst → u_t(N_s → ∞) uni- formly in Qt1, where ε_Ns(x, t) = PNs

l,j=0L_Nslj(t)y_l(x1)z_j(x2) and u_Ns(x, t) = P_Ns

l,j=0T_Nslj(t)y_l(x₁)z_j(x₂).

Proof: LetL_Nslj ≡0 (l, j > Ns). From Lemma 2.5 it follows that

|ε_Ns−ε| ≤ Xm

l,j=1

(L_lj−L_Nslj)y_l(x₁)z_j(x₂)

+ X∞ l,j=m+1

L_ljy_l(x₁)z_j(x₂)+

Ns

X

l,j=m+1

L_Nsljy_l(x₁)z_j(x₂)

≤ 1 D

Xm

l,j=1

|L_lj−L_Nslj|+2√

√M D

X∞ l,j=m+1

1 α²_lβ³_j .

Observe Corollary 2.1. We can make the right side of the above inequality small by first choosing mand then choosing Ns, thenε_Ns(x, t)→ε(x, t) uniformly in Qt1, as Ns→ ∞. u_Ns →u,u_Nsxl→uxl (l= 1,2) andu_Nst→ut(Ns→ ∞) are

proved in a similar manner. Lemma 2.6 is proved.

Theorem 2.1. Under the conditions of Lemma2.3,(L_lj(t), T_lj(t))is a solution of the initial value problem (2.1)–(2.3) in 0 ≤ t ≤ t₁ < t_b, where L_lj(t) = lim_Ns→∞L_Nslj(t), T_lj(t) = lim_Ns→∞T_Nslj(t) and (L_Nslj(t), T_Nslj(t), l, j = 0,1,· · · , Ns)being solution of the initial value problem(2.4)–(2.6).

Proof: The functions L_Nslj(t) andT_Nslj(t) satisfy the system of the Volterra integral equations

(2.30) iL_Nslj =iε_lj+ Z t

0 {(α²_l +β_j²)L_Nslj +hu_Nsε_Ns, y_l(x₁)z_j(x₂)i}dτ ≡iε_lj+H_Nslj,

(1 +α²_l +β_j²)T_Nslj= (1 +α²_l +β_j²)(ϕ_lj+ψ_ljt) (2.31)

− Z _t

0

(t−τ)[(α²_l +β_j²)T_Nslj− h∇²f(u_Ns) +∇²(|ε_Ns|²), y_l(x₁)z_j(x₂)i]dτ

≡(1 +α²_l +β_j²)(ϕ_lj +ψ_ljt)−F_Nslj, l, j= 0,1,· · ·, Ns.

The aim is to show that the functionsε_lj(t) and T_lj(t) satisfy a similar system, indeed, by using of (2.30) and (2.31) we have

(2.32)

|iL_lj−iε_lj−H_lj| ≤ |iL_lj−iL_Nslj|+|H_Nslj−H_lj|

≤ |L_lj−L_Nslj|+ (α²_l +β_j²)t1kL_lj−L_NsljkC[0,t1]

+t₁khuε−u_Nsε, y_l(x₁)z_j(x₂)ikC[0,t1]

+t1khu_Nsε−u_Nsε_Ns, y_l(x1)z_j(x2)ikC[0,t1],

(2.33)

|(1 +α²_l +β_j²)T_lj−(1 +α²_l +β_j²)(ϕ_lj+ψ_ljt) +F_lj|

≤(1 +α²_l +β²_j)|T_lj−T_Nslj|+|F_lj−F_Nslj|

≤[1 +α²_l +β_j²+ (α²_l +β_j²)t²₁]kT_lj−T_NsljkC[0,t1]

+t²₁[khf(u)−f(u_Ns), y_l^′′z_j+y_lz_j^′′ikC[0,t1]

+kh|ε|²− |ε_Ns|², y_l^′′z_j+y_lz_j^′′ikC[0,t1](l, j= 0,1,· · ·, N_s), where

H_lj = Z t

0 {(α²_l +β²_j)L_lj+huε, y_l(x1)zj(x2)i}dτ, F_lj =

Z _t

0

(t−τ)[(α²_l +β_j²)T_lj− h∇²f(u) +∇²(|ε|²), y_l(x₁)z_j(x₂)i]dτ.

From Corollary 2.1 and Lemma 2.6 it follows that the right side of (2.32) and (2.33) approaches zero asN_s→ ∞. Therefore (L_lj(t), T_lj(t)) is a solution of the integral equations

iL_lj =iε_lj+H_lj,

(1 +α²_l +β_j²)T_lj= (1 +α²_l +β_j²)(ϕ_lj+ψ_ljt)−F_lj, l, j= 0,1,· · · . Differentiating the above first formula with respect to t and differentiating the above second formula twice with respect totwe get the conclusion of the theorem.

Theorem 2.1 is proved.

Lemma 2.7[11]. Suppose thatH(z0, z1,· · · , z_l)isk-times(k≥1)continuously differentiable with respect to variables z₀, z₁,· · ·, z_l and z_j(x, t) ∈ L_∞(Qe_T)∩ L₂([0, T];H^k(Ω)), j= 0,1,· · ·, l. Then we have

∂^ekH

∂x^k₁¹· · ·∂x^k_nⁿ

2

L2(Ω)^e

≤C(M, k, l) Xl

j=1

kz_jk²_Hk(Ω)^e ,

where M = max_{j=0,1,···,l}max_(x,t)

∈Q^eT|z_j(x, t)|, Qe_T ={x= (x₁,· · ·, xn)∈Ωe ⊂ Rⁿ,t∈[0, T]},Ωeis a bounded domain inRⁿ,ek= (k1,· · ·, kn),k_j ≥0,|ek|=k= P_n

j=1k_j.

Theorem 2.2. Suppose that the conditions of Lemma 2.3 are satisfied and ε₀(x), ϕ(x), ψ(x) ∈ H⁶(Ω). Then there exists a classical solution ε(x, t) = P_∞

l,j=0L_lj(t)y_l(x1)z_j(x2), u(x, t) = P_∞

l,j=0T_lj(t)y_l(x1)z_j(x2) of the problem (1.5)–(1.8)inQ_t1.

Proof: It follows from the assumptions

(2.34)

ε_lj = Z

Ω

ε₀(x)y_l(x₁)z_j(x₂)dx, ϕ_lj = Z

Ω

ϕ(x)y_l(x₁)z_j(x₂)dx, ψ_lj =

Z

Ω

ψ(x)y_l(x₁)z_j(x₂)dx, l, j= 0,1,· · ·

and they satisfy the conditionb <∞. In this case Theorem 2.1 guarantees the existence of a solution of the initial value problem (2.1)–(2.3) in 0≤t≤t₁ < t_b and we have

(2.35) iL˙_lj = (α²_l +β_j²)L_lj+huε, y_lz_ji,

(2.36) (1 +α²_l +β_j²) ¨T_lj =−(α²_l +β²_j)T_lj +hf(u) +|ε|², y^′′_lz_j+y_lz^′′_ji, l, j= 0,1,· · ·

in [0, t₁]. It follows from the finite system (2.4)–(2.6) of ordinary differential equations that

(2.37) iL˙_Nslj = (α²_l +β_j²)L_Nslj+hu_Nsε_Ns, y_lzji, (2.38) (1 +α²_l +β²_j) ¨T_Nslj =−(α²_l +β²_j)T_Nslj

+hf(u_Ns) +|ε_Ns|², y_l^′′z_j+y_lz_j^′′i, l, j= 0,1,· · · , Ns.

SinceL_Nslj andT_Nslj converge uniformly toL_lj andT_lj (l, j= 0,1,· · ·) in [0, t₁] as N_s → ∞, and ε_Ns, u_Ns, u_Nsxj (j = 1,2) andu_Nst converge uniformly to ε, u, uxj (j = 1,2) andut respectively inQt1, as Ns → ∞, it follows from (2.35), (2.36) and (2.37), (2.38) that ˙L_Nslj →L˙_lj and ¨T_Nslj→T¨_lj uniformly in [0, t₁] as Ns→ ∞.

Multiplying both sides of (2.4) by (α¹²_l +α¹⁰_l β_j² +α⁸_lβ_j⁴ +α⁶_lβ_j⁶ +α⁴_lβ⁸_j + α²_lβ_j¹⁰+β_j¹²)L_{N lj}, summing up the products forl, j= 0,1,· · · , Nand taking the imaginary part, we have

(2.39) d dt

X

h+m=6 h,m=0,1,···,6

kε_{N x}h 1x^m₂ k²

= 2I_mh∂⁶

∂x⁶₁(u_Nε_N), ε_{N x}6

1+ε_{N x}4 1x²₂

+ε_{N x}2

1x⁴₂+ε_{N x}6

2i+ 2Imh ∂⁶

∂x⁶₂(u_Nε_N), ε_{N x}4

1x²₂+ε_{N x}2

1x⁴₂+ε_{N x}6 2i. Multiplying both sides of (2.5) by (α¹⁰_l +α²_lβ_j⁸+α⁴_lβ_j⁶+α⁶_lβ⁴_j+α⁸_lβ²_j+β_j¹⁰) ˙T_{N lj} and summing up the products forl, j= 0,1,· · ·, N, we obtain

(2.40) d dt

X

h+m=5 h,m=0,1,···,5

ku_{N x}h

1x^m₂tk²+ 2 X

h+m=6 h,m=0,1,···,5

ku_{N x}h 1x^m₂tk²

+ku_{N x}6

1tk²+ku_{N x}6

2tk²+ku_{N x}6

1k²+ku_{N x}6 2k²

+ 2 X

h+m=6 h,m=1,2,···,5

ku_{N x}h 1x^m₂ k²

≤2h ∂⁴

∂x⁴₁[∇²f(u_N) +∇²(|ε_N|²)], u_{N x}6

1t+u_{N x}6

2t+u_{N x}2 1x⁴₂t

+u_{N x}4

1x²₂ti|+ 2|h ∂⁴

∂x⁴₂[∇²f(u_N) +∇²(|ε_N|²)], u_{N x}2

1x⁴₂t+u_{N x}6 2ti. By using the H¨older inequality and Lemma 2.7, we obtain

(2.41) 2h ∂⁴

∂x⁴₁[∇²f(u_N) +∇²(|ε_N|²)], u_{N x}6

1t+u_{N x}6

2t+u_{N x}2

1x⁴₂t+u_{N x}4 1x²₂ti

≤C₁₉(ku_Nk²_H6(Ω)+ku_{N t}k²_H6(Ω)+kε_Nk²_H6(Ω)),