On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

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TECHNICAL TRANSACTIONS 1/2017

CZASOPISMO TECHNICZNE 1/2017

MATHEMATICS

DOI: 10.4467/2353737XCT.17.013.6110

Ludwik Byszewski (lbyszews@pk.edu.pl)

Institute of Mathematics, Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology

Tadeusz Wacławski

Institute of Electrical Engineering and Computer Science, Faculty of Electrical and Computer Engineering, Cracow University of Technology

On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

O jednoznaczności rozwiązań parabolicznych zagadnień z nielokalnymi warunkami z całkami

Abstract

The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial conditions with integrals, for the operator ∂

∂ ( ) ^∂

∂



 

+( )−∂

∂ =

∑

= _x ^{a x t} _x ^{c x t} _t ^{x x x} i j i

n ij

j

n ,

, , , ( ,..., )

1 1 , in the

cylindrical domain D D:= ₀×( ,t t T_{0 0}+ )⊂ ℜⁿ⁺¹, where t, ₀ ∈ ℜ, 0 < T < ∞, are studied. The result requires that the nonlocal conditions with integrals be introduced.

Keywords: parabolic problems, semilinear equation, nonlocal initial condition with integral, cylindrical domain, uniqueness of solutions

Streszczenie

W artykule omówiono jednoznaczność klasycznych rozwiązań parabolicznych semiliniowych zagadnień

z nielokalnymi początkowymi warunkami z całkami dla operatora

∂

∂ ^ ( )_∂^∂

 

+( )−∂

∂ =

∑

= _x ^{a x t} _x ^{c x t} _t ^{x x x} i j i

n ij

j

n ,

, , , ( ,..., )

1 1 , w walcowym obszarze D D:= ₀×( ,t t T_{0 0}+ )⊂ ℜⁿ⁺¹, , gdzie t₀ ∈ ℜ, 0 < T < ∞. Wynik polega na tym, że zostały wprowadzone warunki nielokalne z całkami.

Słowa kluczowe: zagadnienia paraboliczne, równanie semiliniowe, nielokalny warunek początkowy z całką, obszar walcowy, jednoznaczność rozwiązań

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1. Introduction

In this paper we prove two theorems on the uniqueness of classical solutions to parabolic semilinear problems, for the equation

∂



 

+ −∂

∑

= _x ^{a x t} ^{u x t}_x c x t u x t u x t

i j i n

ij , j

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

1 ∂∂ =

t f x t u x t( , , ( , )), (1)

= f x t u x t( , , ( , )),

( , )x t D D∈ =: ₀×( ,t t T₀ ₀+ )⊂ ℜⁿ⁺¹,

where t₀ ∈ ℜ, 0 < T < ∞. The coefficients a_ij (i, j = 1, ...n), c and the function f are given.

The nonlocal initial condition considered in the paper is of the form u x t h x

T u x d f x x D

t t T

( , )₀ ( ) ( , ) ₀( ), ₀,

0

+ 0 = ∈

∫

+ ^{τ τ}

where |h(x)| ≤ 1 for x ∈ D₀.

The result obtained is a continuation of the results given by Rabczuk in [5], by Chabrowski in [3], by Brandys in [1] and by the first author in [1] and [2].

In monograph [5], Rabczuk gives two uniqueness criteria for classical solutions for initial – boundary problems to the equation

∂

∂ −∂

∂ = ∈ ⊂ ℜ >

∑

= ² 2

1 u x t 0 0

x

u x t

t f x t u x t x D t

i i

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

In paper [3], Chabrowski studies nonlocal problems for the equation a x t u x t

x x b x t u x t

x c x t

i j ij n

i j i i i

n

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

= =

∑

^∂_{∂ ∂} ⁺

∑

^∂_∂ ⁺

1

2

1 )) ( , )u x t u x t( , )

−∂ t

∂ =

=f x t x D( , ), ∈ 0⊂ ℜⁿ, t∈( , ).0T The nonlocal initial condition, considered in [3], is of the form

u x i x u x T x x D

i i

( , )0 +

∑

^β( ) ( , )=^ψ( ), ∈ 0,

where t₀ ∈ (0, T) and was introduced in this form as the first by Chabrowski.

In publication [2], two uniqueness criteria for classical solutions for equation (1) together with the nonlocal condition u x t( , ) ( ) ( ,₀ +h x u x t T₀+ )= f x₀( ), x D∈ ₀, are studied.

2. Preliminaries

The notation, definitions and assumptions from this section are valid throughout this paper.

We will need the set ℜ_ := (–∞, 0].

Let t₀ be a real number, 0 < T < ∞ and x =(x₁, ..., x_n) ∈ ℜⁿ.

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Define the domain (see [1] or [2])

D D:= ₀×( ,t t T₀ ₀+ ),

where D₀ is an open and bounded domain in ℜⁿ such that the boundary ∂D₀ satisfies the following conditions:

(i) If n ≥ 2 then ∂D₀ is a union of a finite number of surface patches of class C¹ which have no common interior points but have common boundary points.

(ii) If n ≥ 3 then all the edges of ∂D₀ are sums of finite numbers of (n – 2) – dimensional surface patches of class C¹.

Assumption (A₁). a a

x C D i j s n

ij ij s

, ∂ ( , ) ( , , ,..., ),

∂ ∈ ℜ =1 where a_ij = a_ij(x, t) for ( , )x t D∈ ( ,i j=1,..., ); ( , )n a x t a x tij = ji( , ) for ( , )x t D i j∈ 1( , = ,..., )n and a x tij i

i j n

( , ) j ,

λ λ

∑

= ^≥

1

0 for arbitrary (x, t) ∈ D and (λ₁, ..., λ_n) ∈ ℜⁿ; c C D∈ ( ,ℜ₋).

Assumption (A₂).

(i) f D: ×ℜ ∋( , , )x t z → f x t z( , , )∈ℜ, f C D∈ ( ×ℜ ℜ, ), ∂

∂f ∈ ×ℜ ℜ

z C D( , ) and

∂

∂ >

f x t z z

( , , ) 0 for ( , )x t D∈ , z ∈ ℜ;

(ii) f₁:∂ ×D₀ [ , ]0T →ℜ; (ii′) k C D∈ ∂ ×( 0 [ , ],0T ℜ₋);

(iii) f D0: 0→ℜ.

Assumption (A₃). h C D∈ ( , )0 ℜ and |h(x)| ≤ 1 for x ∈ D₀. Let C D^{2 1,}( , )ℜ be the space of all w C D∈ ( , )ℜ such that ∂

∂

∂ ∂ ∈ ℜ

w x

w

x x C D

i i j

, ² ( , ) for i, j = 1, ..., n and ∂

∂w∈ ℜ t C D( , ).

The symbols L and P are reserved for two operators given by the formulas ( )( , ): ( , ) ( , )

,

Lw x t

x a x t w x t

i x

i j n

ij

j

= ∂

∂



 



∑

=

1 (2)

and

( )( , ): ( )( , ) ( , ) ( , )Pw x t Lw x t c x t w x t w x t( , )

= + −∂ t

∂ (3)

for w C D∈ ^{2 1,}( , ), ( , )ℜ x t D∈ .

(4)

By n_x where x ∈ ∂D₀, we denote the interior normal to ∂D₀ at x. In short, we denote, also, n_x by n.

Let C D^{2 1,}( , ),ℜ x ∈ ∂D₀ and t ∈ [t₀, t₀ + T]. The expression du x t

d x t

u x t

x a x t n x

i i n

j ij n

x j

( , )

( , ): ( , ) ( , )cos( , )

υ ₀ = ₁∂ ⁰ ₁ ⁰ ⁰

= ∂ =

∑ ∑

⁽⁴⁾

is called the transversal derivative of the function u at the point (x₀, t). If it does not lead to misunderstanding the transversal derivative du x t

d x t ( , ) ( , )

υ ₀ will be denoted by d

dυu x t( , )0 or by du dυx

0

. For the given functions a_ij(i, j = 1, ..., n) and c satisfying Assumption (A₁) and for the given functions f, f₁, f₀ and h satisfying Assumptions (A₂) (i)–(iii) and (A₃) the first Fourier’s semilinear nonlocal problem in D consists in finding a function u C D∈ ^{2 1,}( , )ℜ satisfying the equation

( )( , )Pu x t =f x t u x t( , , ( , )) for ( , )x t D∈ , (5) the nonlocal initial condition

u x t h x

T u x d f x x D

t t T

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

0

+ 0 = ∈

∫

+ ^{τ τ} ^for ⁽⁶⁾

and the boundary condition

u x t( , )=f x t₁( , ) for x∈∂ ×D₀ [ ,t t T₀ ₀+ ]. (7) A function u possessing the above properties is called a solution of the first Fourier semilinear nonlocal problem (5)–(7) in D.

If condition (7) from the first Fourier semilinear nonlocal problem (5)–(7) is replaced by the condition

d

d u x t k x t u x t f x t x D t t T

υx ( , ) ( , ) ( , )+ = ₁( , ) for ∈∂ ×₀ [ ,₀ ₀+ ], (8) where k is the given function satisfying Assumption (A₂)(ii′) then problem (5), (6) and (8) is said to be the mixed semilinear nonlocal problem in D. A function u C D∈ ^{2 1,}( , )ℜ satisfying equation (5) and conditions (6), (8) is called a solution of the mixed semilinear nonlocal problem (5), (6) and (8) in D.

Assumption (A₄). For each two solutions w₁ and w₂ of problem (5)–(7) or of problem (5), (6) and (8) the following inequality

1 1 2

2

1 0 2 0

0 0

T w x w x d w x t T w x t T

t t T

( ( , )τ − ( , ))τ τ ( , ) ( ,











 ≤ + − +

∫

+

^[

⁾⁾

^]

²^{for x D}^∈ ⁰

is satisfied.

(5)

Remark 2.1. The reason for which Assumption (A₄) is introduced is that the problems considered are nonlocal.

3. Theorems about uniqueness

In this section we shall prove two theorems about the uniqueness of solutions of parabolic semilinear problems together with nonlocal initial conditions with integrals.

Theorem 3.1. Suppose that the coefficients a_ij (i, j = 1, ..., n) and c of the differential equation (5) satisfy Assumption (A₁) and the functions f, f₁, f₀ and h satisfy Assumptions (A₂)(i)–(iii) and (A₃). Then the first Fourier semilinear nonlocal problem (5)–(7) admits at most one solution in D in the class of the solutions satisfying Assumption (A₄).

Proof. Suppose that u₁ and u₂ are two solutions of problem (5)–(7) in D and let v u u:= −₁ ₂ in D. (9) Then the following formulas hold:

( )( , )Pv x t =f x t u x t( , , ( , ))₁ −f x t u x t( , , ( , ))₂ for (x t D, )∈ , (10) v x t h x

T v x d x D

t t T

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

0

+ 0 = ∈

∫

+ ^{τ τ} ^for ⁽¹¹⁾

v x t( , )=0 for (x t, )∈∂ ×D0 [ ,t t T0 0+ ]. (12) From the assumption that u u1, 2 ∈C D2 1^,( , ),ℜ from the second and third part of Assumption (A₂)(i) and from the mean value theorem, there exists θ ∈ (0, 1) such that

f x t u x t( , , ( , ))1 −f x t u x t( , , ( , ))2 (13)

= ∂ +

∂ ∈

v x t f x t u x t v x t

z x t D

( , ) ( , , ( , )² θ( , )) , ) . for (

By (13), (10), by Assumption (A₁) by (2) and (3) and by [4] (Section 17.11), v f x t u v

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂













∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁴⁾

= 









∫



∫

⁺ ^{vPvdx dt}

D t t T

0 0 0

= 









 + 









∫



∫

⁺ ^{vLvdx dt}

∫

⁺

∫

^{cv dx dt}

D t t T

0 0 0

0 0

0 2

(6)

− ∂

∂











∫



∫

⁺ ^v_t^{vdx dt}

D t t T

0 0 0

= − ∂

∂













=

∂ =

+

∫ ∑ ∑

∫

^v ^{n x}ⁱ ^a^ij _x^v ^d ^dt

j j x

n i

n D t t T

cos( , ) σ

1

0 1

0 0

− ∂

∂













=

+

∫ ∑

∫

^a^ij _x^v _x^v^{dx dt}

i j

i j n D t t T

0 , 1

0 0

+ 









 − ∂

∂











∫



∫

⁺ ^{cv dx dt}

∫

⁺

∫

^v_t^{vdx dt}

D t t T

D t t T 2

0 0 0

,

where dσ_x is a surface element in ℜⁿ.

From (14), (12) and from Assumption (A₁), v f x t u v

z dx dt v

tvdx

D t t T

D

2 2

0 0 0

0

∂ +

∂











 ≤ − ∂

∂









∫



∫

⁺ ^{( , ,} ^θ ⁾

∫

⁺

∫

^_

t t T

dt

0 0

. (15)

Using integration by parts, it is easy to see that

∂











 = + −

∫

⁺ ^v_t^{vdx dt}

∫

^{v x t T dx}

∫

^{v x t}

D t t T

D D

0 0 0

0 0

1 2

2

0 2

( , ) ( ,00) .dx (16)

Formulae (15) and (16) imply the inequality v f x t u v

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁷⁾

≤ −¹₂

∫

² ⁰+ +¹₂

∫

² ⁰

0 0

v x t T dx v x t dx

D D

( , ) ( , ) .

From (17) and (11), we have

v f x t u v

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂













∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁸⁾

≤ − + + 









∫ ∫ ∫

⁺ 

1 2

2 0

2

0 0

0

v x t T dx h x

T v x d dx

D t

t T D

( , ) ( ) ( , )τ τ .

By (18) and Assumption (A₄),

v f x t u v

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁹⁾

(7)

≤ − + + 









∫ ∫ ∫

⁺ 

1 2

2 1

0 2

2

0 0 0

0

v x t T dx h x

T v x d dx

D D t

t T

( , ) ( ) ( , )τ τ

≤ −¹₂

∫

² ⁰+ +¹₂

∫

² ² ⁰+

0 0

v x t T dx h x v x t T dx

D D

( , ) ( ) ( , )

= −¹₂

∫

² ⁰+ ¹− ²

0

v x t T h x dx

D

( , )[ ( )] .

From (19) and from Assumption (A₃) we obtain v f x t u v

z dx dt

D t t T

2 2

0 0 0

∂ + 0

∂











 ≤

∫

⁺ ^{( , ,} ^θ ⁾ ^.

By the above inequality and by Assumption (A₂)(i), we obtain v x t²( , )≤0 for ( , )x t D∈ and therefore

v x t( , )=0 for ( , )x t D∈ . The proof of Theorem 3.1 is thereby complete.

Theorem 3.2. Suppose that the assumptions of Theorem 3.1, concerning to the coefficients a_ij (i, j = 1, ..., n), c and the functions f, f₁, f₀ and h, are satisfied and that the function k satisfies Assumption (A₂)(ii′). Then the mixed semilinear nonlocal problem (5), (6) and (8) admits at most one solution in D in the class of the solutions satisfying Assumption (A₄).

Proof. Suppose that u₁ and u₂ are two solutions of problem (5), (6) and (8) in D, and let v u u:= −₁ ₂ in D. (20) Then the following formulas hold:

( )( , )Pv x t =f x t u x t( , , ( , ))₁ −f x t u x t( , , ( , ))₂ for ( , )x t D∈ , (21) v x t h x

T v x d x D

t t T

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

0

+ 0 = ∈

∫

+ ^{τ τ} ^for ⁽²²⁾

d

d v x t k x t v x t x t D t t T

υx ( , ) ( , ) ( , )+ =0 for ( , )∈∂ ×0 [ , , ].0 0 (23) Applying a similar argument as in the proof of Theorem 3.1 and using the definition of du

dυx (see (4)), we have

v f x t u v z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽²⁴⁾

(8)

= − 











∂

+

∫

^v_d^dv^d ^x ^dt

D t t T

υ σ

0 0 0

− ∂

∂













=

+

∫ ∑

∫

^a^{i j} _x^v _x^v^{dx dt}

i j

i j n D t t T

, ,

0 1

0 0

+ 









 − 









∫



∫

⁺ ^{cv dx dt}

∫

⁺

∫

^dv_dt^{vdx dt}

D t t T

D t t T 2

0 0 0

.

From (24), (23), and as in the proof of Theorem 3.1 v f x t u v

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽²⁵⁾

≤ 









 − + −

∂

+

∫

^{kv d} ^x ^dt

∫

^{v x t T} ^{h x dx}

D t t T

D

2 2

0 2

0 0 0

0

1

2 1

σ ( , )[ ( )] .

By (25) and Assumptions (A₂)(ii′) and (A₃) we obtain the inequality v f x t u v

z dx dt

D t t T

2 2

0 0 0

∂ + 0

∂











 ≤

∫

⁺ ^{( , ,} ^θ ⁾ ^.

Consequently, as in the proof of Theorem 3.1,

v x t( , )=0 for (x t D, )∈ and the proof of Theorem 3.2 is complete.

4. Physical interpretation of the nonlocal condition (6)

Theorems 3.1 and 3.2 can be applied to descriptions of physical problems in heat conduction theory for which we cannot measure the temperature at the initial instant but we can measure the temperature in the form of the nonlocal condition (6).

Observe, also, that in Theorem 3.1 and 3.2, the nonlocal condition (6) considered is more general than the classical initial condition and the integral periodic condition and the integral anti-periodic condition. Namely, if the function h from condition (6) satisfies the relation h x( )=0 for x D∈ 0 then condition (6) is reduced to the initial condition

(9)

u x t( , )0 = f x0( ) for x D∈ 0. Instead if the function h and f in (6) satisfy the conditions

h x( )= −1[ ( ) ]h x =1 for x D∈ 0, f x₀( )=0 for x D∈ ₀,

then condition (6) is reduced, respectively, to the integral periodic [antiperiodic] initial condition:

u x t

T u x d u x t

T u x d

t t T

( , )0 1 ( , ) ( , )0 1 ( , ) ]

0 0

0

= = − 0

+ +

∫

^{τ τ} [ forr x D

∫

^{τ τ} ∈ ₀.

References

[1] Brandys J., Byszewski L., Uniqueness of solutions to inverse parabolic problems, Comment.

Math. Prace Matem. 42.1, 2002, 17–30.

[2] Byszewski L., Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems, J. Math. Anal. Appl. 165.2, 1992, 472–478.

[3] Chabrowski J., On nonlocal problems for parabolic equations, Nagoya Math. J. 93, 1984, 109–131.

[4] Krzyżański M., Partial Differential Equations of Second Order, Vol. 1, PWN, Warszawa 1971.

[5] Rabczuk R., Elements of Differential Inequalities, PWN, Warszawa 1976 (in Polish).