Uniqueness of solutions to inverse parabolic semilinear problems under nonlocal conditions with integrals

(1)

Uniqueness of solutions to inverse parabolic semilinear problems under nonlocal conditions with integrals

Jednoznaczność rozwiązań odwrotnych parabolicznych semiliniowych zagadnień z nielokalnymi warunkami z całkami

Abstract

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

∂ ( ) ^∂

∂



 

 −∂

∂ =

∑

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

n ij , j

, ( , ) ( ,...,

1

+ , 1 xxn), in the cylindrical domain D D:= ₀×( ,t t T_{0 0}+ )⊂ ℜⁿ⁺¹, where t0∈ℜ, 0 < T < ∞ are studied. The result consists in the introduction of nonlocal conditions with integrals.

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

Streszczenie

W artykule studiowana jest jednoznaczność klasycznych rozwiązań odwrotnych parabolicznych semiliniowych zagadnień z nielokalnymi początkowymi warunkami z całkami dla operatora

∂

∂ ( ) ^∂

∂



 

 −∂

∂ =

∑

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

n ij , j

, ( , ) ( ,...,

1

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

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

TECHNICAL TRANSACTIONS 8/2017

CZASOPISMO TECHNICZNE 8/2017

MATHEMATICS

DOI: 10.4467/2353737XCT.17.138.6889

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

(2)

1. Introduction

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

∂



 

+ −∂

∑

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

i j i n

ij , j

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

1 ∂∂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) and the function f are given. By the solution of the inverse problem, for equation (1), we mean a pair of functions (u,v) satisfying equation (1) and suitable conditions. 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 obtained result is a continuation of the results given by Rabczuk in [6], by Beznohchenko and Prilenko in [1], by Chabrowski in [4], by Brandys in [2] and by the first author in [2] and [3].

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 finite number, 0 < T< ∞ and x=(x₁, ..., x_n) ∈ℜⁿ. Define the domain (see [2] or [3])

D:=D₀×(t₀, t₀+T),

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

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.

(3)

If n ≥ 3 then all the edges of ∂D₀ are sums of a 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_ji(x,t)= a_ij(x,t) 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. Assumption (A₂).

(i) f: D∈ℜ (x,t,z) →f (x,t,z)∈ℜ, f∈C(D×ℜ,ℜ), f(x,t,0)≠0 for (x,t)∈D,

∂

∂f

z∈C(D×ℜ,ℜ) and ∂

∂ f x t z

z ( , , )

>0 for (x,t)∈D, z ∈ ℜ;

(ii) f₁:∂D₀ × [0,T]→ℜ;

(ii’) k∈C(∂D₀ × [0,T],ℜ) and k(x,t)≤0 for (x,t)∈∂D₀ × [0,T];

(iii) f₀:D₀→ℜ.

Assumption (A₂). h∈C(D₀,ℜ) and |h(x)|≤1 for x∈D₀. Let C^2,1(D,ℜ) be the space of all w∈(D,ℜ) such that ∂

∂

∂ ∂ w x

w

i x xi j

, ² ∈ C(D,ℜ) for i, j=1, ..., n and ∂

∂w

t ∈ C(D,ℜ).

The symbol L is reserved for the operator given by the formula:

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

,

Lw x t

x a x t w x t x

i j i n

ij

j

= ∂

∂



 



∑

=

1 (2)

for w∈ C^2,1(D,ℜ), (x,t)∈D.

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

Let u∈C^2,1(D,ℜ), x₀∈∂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

ij j

n

x j

( , )

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

υ ₀ = ₁ ∂ ⁰ ₁ ⁰ ⁰

∂

= =

∑ ∑

(3)

is called the transversal derivative of the function u at the point (x₀,t). Shortly, we denote, also du x t

d x t ( , ) ( , ) υ ₀ by d

dυu x t( , )0 or by du dυx

0

.

For the given functions a_ij(i, j=1, ..., n) satisfying Assumption (A₁) and for the given functions f, f₁, f₀ and h satisfying Assumptions (A₂) (i) – (iii) and (A₃), the first Fourier’s inverse semilinear problem in D together with a nonlocal initial condition with integral consists in finding a pair of functions u∈C^2,1(D,ℜ), v∈C(D,ℜ_–) satisfying the equation

(4)

( )( , ) ( , ) ( , )Lu x t v x t u x t u x t( , ) ( , , ( , )) (

t f x t u x t x

+ −∂

∂ = for ,, )t D∈ , (4) 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 ⁽⁵⁾

the boundary condition:

u(x,t)=f₁(x,t) for x∈∂D₀ × [t₀, t₀+T] (6) and the condition:

v x t u x t dx dt C

D t t T

( , ) ( , )² ,

0 0

0

∫

^

∫









 =

+

(7) where C is a negative constant independent of u and v.

A pair (u,v) of functions possessing the above properties is called a solution of the first Fourier’s inverse semilinear problem (4)–(7) in D.

Remark 2.1. The assumption that f(x,t,0)≠0 for (x,t)∈D (see Assumption (A₂)(i)) implies that u = 0 cannot satisfy equation (4). Consequently, the above assumption implies that only u ≠ 0 is considered in the paper.

If condition (6) from the first Fourier’s inverse semilinear nonlocal problem (4)–(7) is replaced by the condition

d

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

υx ( , ) ( , ) ( , )+ = ₁( , ) for ( , )∈∂ ×₀ [ ,₀ ₀+ ]], (8)

where k is the given function satisfying Assumption (A₂)(ii') then problem (4), (5), (8) and (7) is said to be the mixed inverse semilinear problem in D together with a nonlocal initial condition with integral. A pair of functions u∈C^2,1(D,ℜ), v∈(D,ℜ_–) satisfying equation (4) and conditions (5), (8), (7) is called a solution of the mixed inverse semilinear problem (4), (5), (8) and (7) in D.

Assumption (A₄). For every two solutions (u₁,v₁) and (u₂,v₂) of problem (4) – (7) or of problem (4), (5), (8) and (7) the following formulas hold:

v x t u x t dx dt C i ji j i j

D t t T

( , ) ( , )² ( , , ; ).

0 0 0

∫

1 2

∫

^









 = = ≠

+

Remark 2.2. The reason for which Assumption (A₄) is introduced is that the considered problems are inverse.

(5)

Assumption (A₅). For each two solutions (u₁,v₁) and (u₂,v₂) of problem (4)–(7) or of problem (4), (5), (8) and (7) the following inequality:

1

1 2 1 0 2 0

0 0

2

T u x u x d u x t T u x t T

t t T

( , )τ − ( , )τ τ ( , ) ( ,

( )











 ≤ + − +

∫

+

^[

⁾⁾

^]

² for x D ^∈ ⁰

is satisfied.

Remark. 2.3. The reason for which Assumption (A₅) is introduced is that the considered problems are nonlocal.

3. Theorems about uniqueness

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

Theorem 3.1. Suppose that coefficients a_ij(i, j=1, ..., n) of the differential equation satisfy Assumption (A₁) and the functions f, f₁, f₀ and h satisfy Assumptions (A₂) (i) – (iii) and (A₃).

Then, the first Fourier’s inverse semilinear problem (4) – (7) admits at most one solution in D in the class of the solutions satisfying Assumptions (A₄) and (A₅).

Proof. Suppose that (u₁,v₁) and (u₂,v₂) are two solutions of problem (4) – (7) in D and let

w:= u₁ – u₁ in D. (9)

Then, the following formulas hold:

( )( , )Lw x t v x t u x t v x t u x t( , ) ( , ) ( , ) ( , ) w x t( , )

+ − −∂ t

∂

1 1 2 2 (10)

= f x t u x t( , , ( , ))1 −f x t u x t( , , ( , ))2 for ( , )x t D∈ ,

w x t h x

T w x d x D

t t T

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

0

+ 0 = ∈

∫

+ ^{τ τ} ^for ⁽¹¹⁾

w x t( , )=0 for (x t, )∈∂ ×D0 [ ,t t T0 0+ ], (12)

v x t u x t dx dt Ci i i

D t t T

( , ) ( , )² ( , ).

0 0 0

∫

1 2

∫

^









 = =

+

(13)

(6)

From the assumption that u₁,u₂∈C^2,1(D,ℜ) from the second and fourth part of Assumption (A₂)(i) and from the mean value theorem, there exists θ∈(0,1) such that:

f(x,t,u₁(x,t))–f(x,t,u₂(x,t)) (14)

= ∂ +

∂ ∈

w x t f x t u x t w x t

z x t D

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

By (14), (10), by Assumption (A₁), by (2) and by [5] (Section 17.11),

w f x t u w

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁵⁾

= − ∂

∂













=

∂ =

+

∫ ∑ ∑

∫

^w ^{n x}ⁱ ^a^ij _x^w^d ^dt

j x j

n

i n

D t t T

cos( , ) σ

1

0 1

0 0

− ∂

∂











∫



∫

⁺ ^a^ij _x^w _x^w^{dx dt}

i j

D t t T

0 0 0

− ∂

∂











 + 









∫



∫

⁺ _x^w^{wdx dt}

∫

⁺

∫

^{v u dx dt}

i D t t T

D t t T

0 0 0

1 1 2

+ 









∫



∫

⁺ ^{v u dx dt}

D t t T

2 22

0 0 0

− 









∫



∫

⁺ v u u dx dt

D t t T

1 1 2

0 0 0

− 









∫



∫

⁺ v u u dx dt

D t t T

2 1 2

0 0 0

, where dσ_x is a surface element in ℜⁿ.

From (15), (12), from the last part of Assumption (A₁) and from the inequalities

− 









 ≤ −  +









∫



∫

⁺ v u u dx dti

∫

v u u dx d

D t t T

i D

1 2 1

2 2 2

0 0 0

0

1

2 ( ) tt i

t t T

0 0

1 2

∫

+ ⁽ ⁼ ^{, )}⁽¹⁶⁾

we have

w f x t u w

z dx dt w

t wdx

t T

2 2

0 ∂ +

∂







 ≤ − ∂

∂





∫



∫

⁺ ^{( , ,} ^θ ⁾ ^{t T}⁰

∫

⁺

∫

^_^dt⁽¹⁷⁾

(7)

+ 









 + 









∫



∫

⁺ ^{v u dx dt}

∫

⁺

∫

^{v u dx dt}

D t t T

D t t T 1 12

2 22

0 0 0

−  +









∫



∫

⁺

1

2 ¹ ¹

2 2 2

0 0 0

v u u dx dt

D t t T

( )

−  +









∫



∫

⁺

1

2 ² ¹

2 2 2

0 0 0

v u u dx dt

D t t T

( ) .

Using integration by parts, we obtain:

∂











 = + −

∫

⁺ ^w_t ^{wdx dt}

∫

^{w x t T dx}

∫

^{w x t}

D t t T

D D

0 0 0

0 0

1 2

2

0 2

( , ) ( ,00) .dx

(18) Formulas (17), (18) and (11) imply the inequality:

w f x t u w

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽¹⁹⁾

≤ − + + 









∫ ∫ ∫

⁺ 

1 2

1

2 0

2

0 0 0

0

w x t T dx h x

T w x d dx

D D t

t T

( , ) ( ) ( , )τ τ

+ 









 − 









∫



∫

⁺ ⁺

∫

1 2

1

1 12 2

1 22

0 0 0

v u dx dt v u dx

D t t T

D t

t T

∫∫

^dt

+ 









 − 









∫



∫

⁺ ⁺

∫

1 2

1

2 2 2

2

2 1 2

0 0 0

D t t T

D t

t T

∫∫

^dt.

From (19) and (13), and Assumptions (A₄) and (A₅),we have:

w f x t u w

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽²⁰⁾

≤ −¹₂

∫

² ⁰+ ¹− ²

0

w x t T h x dx

D

( , )[ ( )] .

By (20) and by Assumption (A₃) we obtain:

(8)

w f x t u w

z dx dt

D t t T

2 2

0 0 0

∂ + 0

∂











 ≤

∫

⁺ ^{( , ,} ^θ ⁾ ^.

From the above inequality and from the last part of Assumption (A₂)(i):

w² ≤ 0 in D

and therefore:

w = 0 in D.

The above formula implies that:

u₁ = u₂ in D.

Consequently, by (10):

(v₁ – v₂) u₁ = 0 in D.

Therefore, from Remark 2.1, we have that:

v₁ = v₂ in 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) and the functions f, f₁, f₀ and h are satisfied and that the function k satisfies Assumption (A₂)(ii’). Then, the mixed inverse semilinear problem (4), (5), (8) and (7) admits at most one solution in D in the class of the solutions satisfying Assumptions (A₄) and (A₅).

Proof. Suppose that (u₁,v₁) and (u₂,v₂)are two solutions of problem (4), (5) , (8) and (7) in D and let

w:= v₁ – v₁ in D. (21)

Then, the following formulas hold:

( )( , )Lw x t v x t u x t v x t u x t( , ) ( , ) ( , ) ( , ) w x t( , )

+ − −∂ t

∂

1 1 2 2 (22)

=f x t u x t( , , ( , ))1 −f x t u x t( , , ( , ))2 for ( , )x t D∈ ,

w x t h x

T w x d x D

t t T

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

0

+ 0 = ∈

∫

+ ^{τ τ} ^for ⁽²³⁾

d

d w x t k x t w x t x t D t t T

υx ( , ) ( , ) ( , )+ =0 for ( , )∈∂ ×0 [ ,0 0+ ], (24)

(9)

v x t u x t dx dt Ci i i

D t t T

( , ) ( , )² ( , ).

0 0 0

∫

1 2

∫

^









 = =

+

(25)

Applying a similar argument as in the proof of Theorem 3.1 and using the definition of

∂

∂u

υx (see (3)), we have:

w f x t u w

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽²⁶⁾

= − ∂













∂

+

∫

^w ^d ^wd ^dt

x x D

t t T

υ σ

0 0 0

− ∂

∂













=

+

∫ ∑

∫

^a^ij ^w_x _x^w^{dx dt}

i j n

i j

D t t T

0 , 1

0 0

− ∂

∂











∫



∫

⁺ ^w_x ^{wdx dt}

i D t t T

0 0 0

+ 









 + 









∫



∫

⁺ ^{v u dx dt}

∫

⁺

∫

^{v u dx dt}

D t t T

D t t T 1 1

2

2 2 2

0 0 0

− 









∫



∫

⁺ v u u dx dt

D t t T

1 1 2

0 0 0

− 









∫



∫

⁺ v u u dx dt

D t t T

2 1 2

0 0 0

.

From (26), (24), and as in the proof of Theorem 3.1, we obtain:

w f x t u w

z dx dt

D t t T

2 2

0 0

0 ∂ +

∂











∫



∫

⁺ ^{( , ,} ^θ ⁾ ⁽²⁷⁾

≤ 









 − + −

∂

+

∫

^{kw d} ^x ^dt

∫

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

D t t T

D

2 2

0

2

0 0 0

0

1

2 1

σ ( , )[ ( )]

(10)

+ 









 − 









∫



∫

⁺ ⁺

∫

1 2

1

1 12 2

1 22

0 0 0

D t t T

D t

t T

∫∫

^dt

+ 









 − 









∫



∫

⁺ ⁺

∫

1 2

1

2 22 2

2 12

0 0 0

D t t T

D t

t T

∫∫

^dt.

By (27), by Assumption (A₂)(ii’) and (A₃), and by (25) and Assumption (A₄), we obtain the inequality:

w f x t u w

z dx dt

D t t T

2 2

0 0 0

∂ + 0

∂











 ≤

∫

⁺ ^{( , ,} ^θ ⁾ ^.

From the above inequality and from the last part of Assumption (A₂)(i), w² ≤ 0 in D

and, therefore, the same argument as in the proof of Theorem 3.1 implies that the proof of Theorem 3.2 is complete.

4. Physical interpretation of the nonlocal condition (5)

Theorems 3.1 and 3.2 can be applied to description of physical problems in the 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 (5).

Also, observe that the nonlocal condition (5) considered in Theorem 3.1 and 3.2 is more general than the classical initial condition and the integral periodic condition and the integral antiperiodic condition. Namely, if the function h from condition (5) satisfies the relation:

h(x) = 0 for x∈D₀ then condition (5) is reduced to the initial condition:

u(x,t₀) = f₀(x) for x∈D₀. Instead, if the function h and f in (5) satisfy the conditions:

h(x) = –1 [h(x) = 1] for x∈D₀,

f₀(x) = 0 for x∈D₀

(11)

then condition (5) 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] Beznoshchenko N. J., Prilenko A. I., Inverse problems for parabolic equation, Problems of Mathematical Physics and Computational Mathematics, Nauka, Moscow 1977, 51–63 (in Russian).

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

Math. Prace Matem. 42.1, 2002, 17–30.

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

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

[5] Krzyżański M., Partial Differential Equations of Second Order, Vol. 1, PWN (Polish Scientific Publishers), Warsaw 1971.

[6] Rabczuk R., Elements of Differential Inequalities, PWN (Polish Scientific Publishers), Warsaw 1976 (in Polish).