The Initial Boundary Value Problem of a Parabolic System

2016 International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2016) ISBN: 978-1-60595-385-4

The Initial Boundary Value Problem of a Parabolic System

Qitong OU* and Huashui ZHAN

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, P.R. China

*Corresponding author

Keywords: Existence, Uniqueness, Evolution, P-Laplacian, Parabolic system.

Abstract. _{In this paper, the existence and uniqueness of local solutions to the initial and boundary} value problem of the system

2

1 2

( pi ) ( , , , , , ), ( , ) ,

it i i i n T

u −div ∇u − ∇u = f x t u u … u x t ∈ Ω

are studied. The regularization method is used to construt a sequence of approximation solutions, with the help of monotone iterattion technique, then we get the the existence of solution of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions to the system is obtained.

Introduction

The objective of the paper is to study the existence and uniqueness of local solutions to the initial and boundary value problem of the parabolic system

2

1 2

( pi ) ( , , , , , ), ( , ) ,

it i i i n T

u −div ∇u − ∇u = f x t u u … u x t ∈ Ω

(1.1)

0

( , 0) ( ), , i i

u x =u x x∈ Ω

(1.2)

( , ) 0, ( , ) (0, ), i

u x t = x t ∈ ∂Ω × T

(1.3) where p_i>2,i=1, 2,…, ,n (0, ), n

T T R

Ω = Ω × Ω ⊂ is a bounded domain with smooth boundary

∂Ω. The conditions of f_i and u_i0 will be given later.

System (1.1) models such as non-Newtonian fluids [1,6] and nonlinear filtration[5],etc. In the non-Newtonian fluids theory, p i_i( =1, 2,…, )n are all characteristic quantity of the medium. Media

with p_i >2 are called dilatant fluids and those with p_i <2 are called pseudoplastics. If p_i=2, they are Newtonian fluids.

Some authors have studied the global finiteness of the solutions[3,4] and blow up properties of the solutions[9] with various boundary conditions to the equations and systems of evolutionary Laplacian equations. Zhao[11] and Wei-Gao[10] studied the existence and blowing-up property of the solutions to a single equation and the systems of two equations. We found that the method of [10] can be extended to the general systems of n equations. In this paper, we consider some special cases by stating some method of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and hence obtain the existence of solutions to a regularized system of equations. Then we obtain the existence of solutions to the system (1.1)-(1.3) by a standard limiting process. Systems (1.1) degenerates when u_i=0 or ∇u_i=0. In general, there would be no classical solutions and hence we have to study the generalized solutions to the problem (1.1)-(1.3). The definition of generalized solutions in this work is the following.

Definition 1.1 _Function

1 2

( , , , _n)

u= u u … u is called a generalized solution of the systems

(1.1)-(1.3) if ( ) pi(0, ; ₀1,pi( )) i T

2

0

( i ) ( ( , 0)

T

p

i it i i i i i

uϕ u − u ϕ dxdt u ϕ x dx

Ω − + ∇ ∇ ∇ − Ω

∫ ∫

∫

( , , ,1 2, , ) ,

T i n i

f x t u u u ϕdxdt

Ω

=

∫ ∫

…

(1.4) for any 1

( ), i C T

ϕ ∈ Ω ϕ_i( , )x T =0,ϕ_i( , )x t =0,( , )x t ∈ ∂Ω ×(0, ),T i=1, 2,…, .n

Equations (1.4) implies that

2

0

0 ( ) ( , ) ( , ) ( , 0)

i

t _p

i it i i i i i i i

uϕ u − u ϕ dxdt u x t ϕ x t dx u ϕ x dx

Ω − + ∇ ∇ ∇ + Ω − Ω

∫ ∫

∫

∫

1 2

0 ( , , , , , ) ,

t

i n i

f x t u u u ϕdxdt

Ω

=

∫ ∫

…

a.e. t∈(0, ).T (1.5) The following are the constrains to the nonlinear functionsf_i, i=1, 2,…, ,n involved in this paper.

Definition 1.2 _{A function}

1 2

( , , , _n)

f = f u u … u is said to be quasimonotone nondecreasing

(resp.,nonincreasing) if for fixed u_j(j≠i), f is nondecreasing (resp., nonincreasing) in u_i

(i=1, 2,…, )n .

Our main existence result is following:

Theorem 1.3 _{If there exist nonegative functions}

1 2

( , , , , , )

i n

f x t u u … u _∈C(_{Ω ×}[0, ]_{T ×R}n)_{, which}

are quasimonotonically nondecreasing for u u₁, ₂,…,u_n, and a nonnegative functiong s( )∈C R1( ) such that

1 2 1 2

( , , , , , ) min{ ( ), ( ), , ( )},

i n n

f x t u u … u ≤ g u g u … g u and ₀ ( ) ₀1,pi( ). i

u ∈L∞ Ω ∩W Ω

(i=1, 2,…, )n .Then there exists a constant T' [0, ]∈ T such that (1.1)-(1.3) has a solution

1 2

( , , , _n)

u= u u … u in the sence of Definition 1.1 with T replaced by T'. Theorem 1.4 _{Assume the}

1 2

( , , , n)

f = f f … f is Lipschitz continuous in ( ,u u_{1 2},…,un), then the

solution of (1.1)-(1.3) is unique.

The proof of Th1.4 is simple, so we omit it.

Proof of Theorem 1.3

To prove the theorem, we consider the following regularized problem

2

2 ₂

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

it i i i n T

u div u ε u f_ε x t u u u x t

−

− ∇ + ∇ = … ∈ Ω

(2.1)

0

( , 0) ( ), , i i

u x =u _ε x x∈ Ω _(2.2)

( , ) 0, ( , ) (0, ), i

u x t = x t ∈ ∂Ω × T _(2.3)

where 1_{( [0, ]} n₎ i

f_ε ∈C Ω × T ×R , f_iε is quasimonotone nondecreasing and f_iε → f_i uniformly on

bounded subsets of _{Ω ×}[0, ]_{T ×R}n_{, also}

1 2 1 2

( , , , , , ) min{ ( ), ( ), , ( )},

i n n

f_ε x t u u … u ≤ g u g u … g u

1

0 0( ),

i

u _ε∈C Ω ui0ε _Lpi ≤ ui0_Lpi , ∇ui0ε _Lpi ≤C ∇ui0_Lpi,ui0ε →ui0 strongly in 1, 0 ( ).

i

p

W Ω

Lemma 2.1 _{The regularized problem (2.1)-(2.3) has a generalized solution.}

Proof: Starting from a suitable initial iteration (u₁(0)_ε ,u(0)_2ε ,…,u_n(0)_ε ), we construct a sequence

( ) ( ) ( )

1 2

{( k , k , , k )} n

u_ε u_ε … u_ε from the iteration process

2 2

( ) ( ) ₂ ( ) ( 1) ( 1) ( 1)

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

k k k k k k

i t i i i n T

u_ε div u_ε ε u_ε f_ε x t u_ε u _ε u_ε x t

−

− − −

− ∇ + ∇ = … ∈ Ω

( )

0

( , 0) ( ), , k

i i

u_ε x =u _ε x x∈ Ω _(2.5)

( )k _{( , ) 0, ( , ) (0, ),} i

u_ε x t = x t ∈ ∂Ω × T _(2.6)

where i=1, 2,…, .n It is clear that for each k =1, 2,…, the above system consists of n

nondegenerated and uncoupled initial boundary-value problems.

By classical results(see, [8]) for fixed ε and k the problem (2.4)-(2.6) has a classical solution

( ) ( ) ( )

1 2

(u_εk ,uk_ε ,…,u_nεk ) if (u₁(_εk−1),u₂(_εk−1),…,u_n(k_ε−1)) is smooth.

To ensure that this sequence converges to a solution of (2.4)-(2.6), it is necessary to choose a suitable initial iteration. The choice of this function depends on the type of quasimonotone property of ( ,f₁ f₂,…,f_n). In the following, we establish the monotone property of the sequence.

Set ui(0)ε ( , )x t =sup {_Ω u_i0ε( )},x i=1, 2,…, .n Let

(1)

i

uε be a classical solution of the following

problem.

2 2

(1) (1) (1) (0) (0) (0)

2

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

i t i i _i n _T

uε div uε ε uε f_ε x t uε u ε u ε x t

−

− ∇ + ∇ = … ∈ Ω

(1) (0)

0

( , 0) , ,

i _i i

uε x =u _ε ≤uε x∈ Ω

(1)

( , ) 0, ( , ) (0, ), i

uε x t = x t ∈ ∂Ω × T

By (0) (0) (0) (0) (0) (0)

1 2

1 2

( , , , , , ) ( , , , , , n )

i n i

f_ε x t u_ε u _ε … u_ε ≤ f_ε x t uε u ε … u ε and the comparison therorem (see [7]),

we have that

(1) (0)

1 1 ,

uε ≤uε

(1) (0)

2 2 , ,

u ε ≤u ε …

(1) (0)

. n n

u ε ≤u ε

And hence by the quasimonootone nondecreasing property of f_iε, we have

(1) (1) (1) (0) (1) (1) (0) (0) (1) (0) (0) (0)

1 2 1 2 1 2 1 2

( , , , , , n ) ( , , , , , n ) ( , , , , , n ) ( , , , , , n )

i i i i

f_ε x t uε u ε … u ε ≤ f_ε x t uε u ε … u ε ≤ f_ε x t uε u ε … u ε ≤ f_ε x t uε u ε … u ε

for i=1, 2,…, .n

By induction, we may obtain a nondecreasing sequence of smooth functions

(0) (1) (2) ( )k

i i i i

uε ≥uε ≥uε ≥≥uε ≥ _(2.7)

In a similar way, by setting u_i(0)ε ( , )x t =inf {Ω u_i0ε( )},x i=1, 2,…, ,n we can get a solution

(1) (1) (1)

1 2

(u_ε,u _ε,…,u_nε) of

2 2

(1) (1) ₂ (1) (0) (0) (0)

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

i t i i i n T

u_ε div u_ε ε u_ε f_ε x t u_ε u _ε u _ε x t

−

− ∇ + ∇ = … ∈ Ω

(1) (0)

0

( , 0) , ,

i i i

u_ε x =u _ε ≥u_ε x∈ Ω

(1)

( , ) 0, i

u_ε x t = _{( , )}x t ∈ ∂Ω ×(0, ),T

with

(1) (0) 1 ,1

u_ε ≥u _ε u(1)2ε ≥u2(0)ε,…,

(1) (0)

. n n

u _ε ≥u _ε

In the same way as above, we obtain a nonincreasing sequence of smooth functions

(0) (1) (2) ( )k

i i i i

u_ε ≤u_ε ≤u_ε ≤≤u_ε ≤

(2.8)

It is obvious that u_i(0)_ε ≤u(0)iε . By induction, we assume that

( ) ( )k k

i i

u_ε ≤uε . Since f_iε is

quasimonotone nondecreasing, we have

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 2 1 2

1 2 3 2

( , , k , k , k ) ( , , k , k , , _nk ) ( , , k , k , , _nk ) ( , , k , k , , nk )

i i i i

for i=1, 2,…, .n

2 2

( 1) ( 1) ₂ ( 1) ( ) ( ) ( )

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

k k k k k k

i t i i i n T

u_ε div u_ε ε u_ε f_ε x t u_ε u _ε u _ε x t

−

+ + +

− ∇ + ∇ = … ∈ Ω

2 2

( 1) ( 1) ( 1) ( ) ( ) ( )

2

1 2

(( ) ) ( , , , , , ), ( , ) ,

i

p

k k k k k k

i t i i _i n _T

uε div uε ε uε f_ε x t uε u ε u ε x t

−

+ + +

− ∇ + ∇ = … ∈ Ω

( 1) ( 1)

0

( , 0) k ( , 0), ,

k

i i i

u_ε x u _ε uε x x

+ +

= = ∈ Ω

( 1) ( 1)

( , ) 0 k ( , ), k

i i

u_ε x t uε x t

+ +

= = ( , )x t ∈ ∂Ω ×(0, ),T

By the comparison principle, we have u(_iεk 1) u(iεk 1)

+ +

≤ . Therefore

( 1) ( ) (2) (1) (0)

(0) (1) (2) ( ) ( 1)

. k k

k k

i i i i i i i i i i

u_ε u_ε u_ε u_ε u_ε uε uε uε uε uε

+ +

≤ ≤ ≤≤ ≤ ≤ ≤ ≤≤ ≤ ≤

(2.9) Taking u_i( )_εk =u_i( )_εk (i=1, 2,…, ),n we get a nondecreasing bounded sequence

( ) 1

{u_iεk }_k∞= (i=1, 2,…, ).n Hence there exist functions uiε(i=1, 2,…, )n such that

( )

lim _ik _i ,

k→∞uε =uε _{a.e. in} ΩT _(2.10) By the continuity of f_iε(i=1, 2,…, ),n we have

( ) ( ) ( )

1 2 1 2

lim _i ( , , k , k , , _nk ) _i ( , , , , , _n )

k→∞ fε x t uε u ε … uε = fε x t uε u ε … uε _{a.e. in} ΩT _(2.11) We now prove that there exist a T'∈(0, ]T and a constant M(independent of k and ε ) such that for all k, we have

1

( )

( T) ,

k i _L

u_ε ∞ _Ω ≤M _{i=1, 2,…, .n}

(2.12) Let v t_i±( ) be the solutions of the ordinary differential equations

( ) i i dv g v dt ±

= ± , v_i (0) u_i0L∞_{( )}, (i 1, 2, , )n

±

Ω

= ± = … .

By standard results in [2], there exist T_i∈(0, ),T i=1, 2,…, ,n such that v_i± exists on [0, ]T_i

with T_i depends only on u_i0L∞_{( )}

Ω . By the comparison theorem

( )k _{( , ) max{ ( ), ( )},}

i i i

u_ε x t ≤ v t v t+ − (i=1, 2,…, ).n

Setting T' 1min{ ,T T_{1 2}, ,T_n}

n

= … , M =max{ ( '),v T_i+ −v T_i−( ')}, we obtain (2.12).

We now claim that u_i( )_εk w→u_iε, as k → ∞, weakly in _Lpi(0, ';_{T W0}1,pi( ))_{Ω (i=1, 2,…, ).n}

where w→ stands for weak convergence. Multiplying (2.4) by ( )k

i

u_ε and integrating over Ω_T', we obtain that

(

)

' '

2

2 ₂ 2

( ) ( ) ( ) ( ) , i i pi T T p p

k k k

i _L i i

u_ε u_ε ε u_ε dxdt C

−

Ω Ω

∇ ≤

∫ ∫

∇ + ∇ ≤

(2.13) where C is a constant independent of ε, .k

Multiplying (2.4) by u_{i t}( )_εk and integrating over Ω_T', by Cauchy inequality and integrating by parts, we obtain

(

)

(

(

)

)

' '

2

( ) ( ) 2 ( 1) ( 1) ( 1)

1 0 ( , , 1 , 2 , , ) .

i

T T

p

k k k k k

i t i i n

uε dxdt C u ε dx fε x t uε uε uε dxdt C

− − −

Ω ≤ Ω ∇ + Ω ≤

∫ ∫

∫

∫ ∫

…

By (2.13) and (2.14), we obtain that there exists a subsequence of u_i( )_εk converging weakly in following sense as j→ ∞.

( ) ( )

, w

kj k i i

u_ε u_ε

∇ → ∇

weakly in ( ') i

p T

L Ω

, (2.15)

2

( )_kj pi ( )_{kj w} ( )_{k ,}

i i xl i xl

u_ε − u_ε w_ε

∇ →

weakly in

1 '

( )

i i

p p

T

L − Ω

, for some

( )k i xl

w_ε

, (2.16)

(kj) w ( )k _, i t i t

u_ε →u_{ε weakly in} 2 '

( _T)

L Ω _{, (2.17)}

Similar as [11], we can prove that pi 2 , i xl i i xl

w_ε = ∇u_ε − u_ε i=1, 2,…, .n

Following (2.10), (2.11), (2.13) and (2.14) and the uniqueness of the weak limits, it is easy to know that, as k→ ∞,

( )k w _,

i i

u_ε u_ε

∇ →∇ _{weakly in} pi( _') T

L Ω _{, (2.18)}

( )

, k

i i

u_ε →u_ε ( ) ( ) ( )

1 2 3 1 2 3

( , , k , k , k ) ( , , , , )

i i

f_ε x t u_ε u_ε u_ε → f_ε x t u_ε u_ε u_{ε , a.e. in} ΩT'_{, (2.19)}

( )

, w k

i t i t

u_ε →u_{ε weakly in} 2 '

( _T )

L Ω _{. (2.20)}

2 2

( ) ( )

,

i i

p _p

k k

i i xl i i xl

u_ε − u_ε u_ε − u_ε

∇ → ∇

weakly in

1 '

( )

i i

p p

T

L − Ω

, i=1, 2,…, .n (2.21) Combining the above results, we have proved that u_ε =(u_1ε,u_2ε,…,u_nε), ( , )x t ∈ Ω_T' is a generalized solution of (2.1)-(2.3).

Proof of theorem 1.3

Since u_iε satisfy similar estimates as (2.12)-(2.14), combining the property of f_iε, we know that

there are functions pi(0, '; ₀1,pi( ))( 1, 2, , ) i

u ∈L T W Ω i= … n as ε →0.

By a standard limiting process, (u_1ε,u_2ε,…,u_nε)→( ,u u_{1 2},…,un).Thus u=( ,u u1 2,…,un) is a generalized solution of (1.1)-(1.3).

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2

( p )

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