Recovering of lower order coefficients in forward-backward parabolic equations

Let G be a bounded. The inverse problems is studied in the cylinder Q = G × (0, T ), S = Γ× (0, T ), Γ = ∂ G . The problem is stated as follows: find a pair of functions u ( x , t ) and λ ( x ) satisfying the equation

g(x, t)ut -Lu=λ(x)u+f(x,t),(x,t)∈Q,(1)

and the boundary conditions

Bu|S=ϕ(x,t),(2)

u(x,0)=u(x,T)=u0(x).(3)

Here the operator L is of the form Lu = ⁿ ∂ a ( x ) u - ⁿ a ( x ) u - a ( x ) u and Bu = u or i , j =1 ^xj^{j       x}i         i =1 ^xi

Bu = ∑ in , j =1 a ij u xi ν j + σ ( x ) u , where ν j are the components of the outer unit normal to Γ . We assume that the coefficients of the operator L and the boundary operator B as well as the corresponding function spaces are real. The definitions of the function spaces involved can be found, for instance, in [1]. The operator L is elliptic, i. e., there exists a constant δ ₀ >0 such that n

E a ij ( x)^ j - ^ 0 | ^j i ^{2 v} ^ ^{e 1} n , x ^e G , a y = a ji for all i , j ^.

i , j =1

The inverse problems of the form (1)–(3) in the case of positive function g ( x , t ) are studied in many articles (see [2–5] and the bibliography therein). In our case the function g ( x , t ) can change a sign, i. e., we deal with the forward-backward parabolic equation. Equations of this type often appear in mathematical physics, for example, in fluid dynamics while studying fluid motion with alternating coefficient of viscosity, in transport theory while describing the process of particles motion in some environment. Such equations also occur in geometry, population dynamics, and some other fields. Sufficient number of examples is given in [6]. The boundary value problems for equations of the form (1) are studied in many articles (see, for instance, [7, 8]). The inverse problem of finding the right-hand side in (1) is studied in [9, 10, 12, 13]. We generalize here the results of the article [13]. Our conditions on the coefficients are more general (in particular, the function in front of the derivative in time can depend on t ) and moreover, we prove solvability for an arbitrary n ( n ≤ 3 in [13]).

Математика

2.    Preliminaries

Let E = L ₂ ( G ). The inner product in E is defined by the equality ( u , v ) = ∫ _Gu ( x ) v ( x ) dx . Let

О

D ( L )={ v ∈ W ₂ ² ( G ): Bv | _Γ =0}. The space H ₁ agrees with with W ₂ ¹ ( G ) in the case of Dirichlet boundary conditions and with W ₂ ¹ ( G ) in the case of conditions of the third boundary value problem. The space H ₁ ' is the completion of E in the norm

II v II H 1' = sup ^|( v ^, w )|/ II w II H 1 , w ∈ H ₁, w ≠ 0

i. e., it is a negative space constructed on the pair of H ₁ , E . The operator L extends to an operator of the class L ( H ₁ , H ₁ ') which is the space of linear continuous operator defined on H ₁ with values in H ₁ '. Define the space

W ={ u ∈ L 2 (0, T ; W 2¹ ( G )): u t , u tt ∈ L 2 (0, T ; W 2¹ ( G )), ∂ t^k u ( x ,0) = ∂ t^k u ( x , T ) ( k =0,1) }.

where ∂ _t ^k are generalized derivatives in the Sobolev sense. By W ₀ we mean the subspace of W of functions satisfying the homogeneous Dirichlet conditions in S . Define the norm

II u I k = ^ II ^d t u ¹ I I 2(0, T ; H 1) .

i =0

Next, we present the condition on the data of the problem. We assume that ay e k1( G), ate kp( G) (i, j = 1,^, n) and a о e Ip (G);                    (4)

g , g t , g tt ∈ L ∞ (0, T ; L p ( G )), ∂ t^k g ( x ,0)= ∂ t^k g ( x , T )( k =0,1),                     (5)

where p >  n /2 for n > 2 and p >1 for n ≤ 2 ;

• (i)    f , f t , f tt ∈ L 2 (0, T ; H 1 '), ∂ t^k f ( x ,0)= ∂ t^k f ( x , T )( k =0,1);

• (ii)    ϕ , ϕ _t , ϕ _tt ∈ L ₂ ( S ), ∂ ⁱ _t ϕ ( x , 0) = ∂ ⁱ _t ϕ ( x , T ) ( i = 0,1) in the case of the Robin boundary conditions

and there exists a function Φ ( x , t ) ∈ W such that Φ | _S = ϕ in the case of the Dirichlet boundary conditions (this function Φ exists if, for instance, if ϕ , ϕ _t , ϕ _tt , ϕ _ttt ∈ L ₂ (0, T ; W ₂ ^1/2 ( Γ )) and ∂ ⁱt ϕ ( x ,0)= ∂ ⁱt ϕ ( x , T ) ( i =0,1,2)).

(iii) there exists a constant δ ₁ >0 such that a ₀

( x ) + α g_t ( x , t ) - 12 ∑ _i ⁿ ₌₁ a_ixi

( x )> δ ₁ >0

for all

α ∈ [ - 1/2,3/2] and a.a. ( x , t ) ∈ Q ;

(iv) σ ( x ) ∈ L _∞ ( Γ ) and σ ( x ) + 2 ∑ in =1 a i ν i ≥ 0 for a. a. x ∈ Γ in the case of the Robin boundary conditions.

A pair of functions u ( x , t ), λ ( x ) is called a solution to problem (1)–(3) if λ ( x ) ∈ L_{n /2} ( G ) for n > 2, λ ( x ) ∈ L_p ( G ) with some p >1 for n ≤ 2 , u ∈ W in the case of Robin boundary conditions, u -Φ ∈ W ₀ in the case of the Dirichlet boundary conditions, the conditions (2), (3) holds, and nn

∫gutv +∑aijuxivx+∑aiuxiv + a0uvdQ + ∫σuv -ϕvdΓ=∫λuv + fvdG,(6)

G         i, j=1          ji=1                         Γ

∀ v ∈ H ₁ , where the integral over Γ is absent in the case of the Dirichlet boundary conditions.

Consider an auxiliary problem

Mu=g(x,t)ut -Lu=f(x,t),(x,t)∈Q,(7)

u(x,0)=u(x,T),Bu|S=ϕ(x,t).(8)

We can state that

Theorem 1. Under the conditions (4)–(5), (i)–(iv) there exists a unique solution to the problem (7), (8) such that u ∈ W in the case of the Robin boundary conditions and u -Φ ∈ W ₀ in the case of the Dirichlet boundary conditions. A solution satisfies the estimate

II u ^-ф I k - c о ^ II ^d t ⁽ f ^- M ^{ф )} I L 2(0, t ; h i')                                ⁽⁹⁾

i =0

in the case of the Dirichlet boundary condition and the estimate

¹ u I k ^- c 0 ^ (1 ^d t f ¹ / 2(0, T ; H f) + ^{1 d} t ^{v I} L2 ( S ) )                               ⁽¹⁰⁾

i =0

in the case of the Robin boundary conditions, where the constant c ₀ is some absolute constant c multiplied by the quantity 1/min( δ ₁ , δ ₀ ) .

Proof. We can refer to Theorem 3 in [8], where the corresponding result is stated in an abstract form. We need only to check the conditions of this theorem. In the case of the Dirichlet boundary condition Theorem 1 is reduced to Theorem 3 in [8] after the change of variables u = v +Φ . The corresponding check relies on the embedding theorems and the condition of the theorem.

• 3.    Main results

In this section we consider the inverse problem in question. To justify the corresponding results below, we employ the generalized maximum principle. So we need to impose some additional conditions on the data.

• (v)    g ( x , t ) ∈ L _∞ ( Q ), f , f_t ∈ L _∞ ( Q ); ϕ , ϕ _t ∈ L _∞ ( S ), u ₀ ( x ) ∈ W _∞ ² ( G ) and there exists a constant δ ₂ >0 such that u ₀ ( x ) ≥ δ ₂ ;

• (vi)    there exists a constant δ ₃ >0 such that a ₀ ( x ) + g_t ( x , t ) ≥ δ ₃ >0 for a.a. ( x , t ) ∈ Q ;

• (vii)    in the case of the Robin boundary conditions, we have that σ ( x ) ∈ C 1 ( Γ ), either σ ( x ) ≥ δ 4 >0 for some constant δ 4 and all x ∈ Γ or σ ( x ) ≥ 0 and ϕ ( x , t ) ≡ 0 , ϕ ( x , t ) ∈ W 21/4,1/2 ( S ), and

• ¹    g ⁽ x ^{,0) 1} L ~ ( G ) R 1 ^- vraimin G ^{( Lu} 0 + f ⁽ x ^,0) X R 1 = ^{max( |}l V t ¹ L ~ ( S ) / ^ 4 , H f t ¹ L ~ ( Q ) /М

• (viii)    in the case of the Dirichlet boundary conditions we have that ϕ ( x , t ) ∈ W ₂ ^3/4,3/2 ( S ) and

• ¹    g ^{( x ,0) 1} L „ ( G ) R 2 ^- vraimin G ^{( Lu} 0 + f ⁽ x , ⁰⁾⁾ , R 2 = m ^{ax( |}| V t ¹ L ^ ( S ) , H f t ¹ L _ ( Q ) / ^ 3 ^).

Theorem 2. Under the conditions (4)–(6),   (i)–(viii),   there exists a solution u∈W∩L2(0,T;W22(G)), λ∈L∞(G) to the problem (1)–(3).

Proof. Consider the problem

M ₀ u = g ( x , t ) u_t - Lu - λ ( x ) u = f ( x , t ),( x , t ) ∈ Q ,                       (11)

u ( x ,0)= u ( x , T ), Bu | _S = ϕ ( x , t ),                                 (12)

where we assume that λ ( x ) ≤ 0 a. a. in G . In view of Theorem 1, for a fixed Я е B_R ={ 2 ( x ) e L p ( G ), Я ( x ) - 0 a . e ., || ЛЦ _£ ( _G ) - R }, where p >  n /2 for n >2 and p >1 for n - 2, there exists a unique solution to the problem (11), (12) from the class W . This solution satisfies the estimate

• 1    u -Ф Ik - c0 ^ 1 d t(f - MФ) 1L 2(0, T; H-p + c0 ^ 1 Я( x)d tФ 1L 2(0, T; H^)

i=0

in the case of the Dirichlet boundary condition and the estimate h u Ilk- c0^(I dtf IIL2(0,T;Hf) + h dtVllL2(S))

i =0

in the case of the Robin boundary conditions. In view of the embedding W ₂ ¹ ( G ) ⊂ L _{2 n /( n - 2)} ( G ) (see [1]) we have (let, n > 2, for example)

|( λ ( x ) ∂ ⁱt Φ , v )| ≤ c || λ || L _{n /2} ( G ) || ∂ ⁱt Φ || _{W 2} 1 _{( G )} || v || _{W 2} 1 _{( G )}

Математика

and thus

|| ^λ ( x ) ^∂ it ^Φ || L 2 (0, T ; H 1^' ) ^≤ c || ^λ || L p ( G ) || ^∂ it ^Φ || L 2 (0, T ; W 2¹ ( G )) .                               (15)

Using the conditions on the data and (15), we can rewrite (13) in the form

II u H w - ^c 2 ^ ( 11 ^d t f II L 2(0, T ; H 1') + II ^Ф ll w ) + ^c 3 | R II L p ( G ) ,                            ⁽¹⁶⁾

i =0

where the constants c2 , c3 are independent of λ, u. Differentiate the equality (6) with respect t to and take v=(ut -k)+ , with (ut - k)+ =ut-k if ut ≥ k and (ut -k)+ = 0 otherwise. Take k > 0 and as before assume that λ∈ BR . First, consider the case of the Robin boundary conditions and assume that ϕ≠ 0 . Integrating with respect t to and by parts we infer n11

∫∑ a_ijv_xiv_xj + ( a ₀ + g_t - ∑ a_ixi ) v ² + ( a ₀ + g_t ) kvdQ +

Qi, j=1

1 ⁿ

∫ ( σ + ∑ a i ν i ) v 2 + σ kv - ϕ t vdS = ∫ λ u t v + f t vdQ .

S2 i=1

Here we employ the transformations of the type au_tv = a ( u_t - k ) v + akv = av ² + akv . This equality can be rewritten in the form (see the conditions (iv), (vii) and the ellipticity condition) ∫ δ ₀ | ∇ v | ² + δ ₁ v ² + δ ₃ kvdQ + ∫ δ ₄ kv - ϕ _tvdS ≤ ∫ f_tvdG .

QSQ

Choosing k ≥ || f_t || _{L ∞ ( Q )} / δ ₃ here and k ≥ || ϕ _t || _{L ∞ ( S )} / δ ₄ , we arrive at the inequality

∫ δ ₀ | ∇ v | ² + δ ₁ v ² dQ ≤ 0

G and, therefore, v = 0 a. e. or ut(x,t)≤k=max(||ft||L∞(Q)/δ3,||ϕt||L∞(S)/δ4)=R1. Similar arguments applied to a function - ut yield the estimate

||ut(x, t) ||L∞(G) ≤max(|| ft||L∞(Q)/δ3,||ϕt||L∞(S)/δ4)=R1.(18)

In the case of the Dirichlet boundary condition an analog of the equality (17) is written as ∫∑naijvxivxj+(a0+1gt-1∑naixi)v2+(a0+gt)kvdQ=∫λutv+ftvdQ,(19)

Qi, j=1             22i=1

if we take k ≥ || ϕ _t || _{L ∞ ( S )} . In this case we obtain the estimate

||ut(x,t)||L∞(G)≤max(||ft||L∞(Q)/δ4,||ϕt||L∞(S))=R2.(20)

Consider the mapping A ( λ ) = ( g ( x ,0) u ( x ,0) - Lu ₀ - f ( x ,0))/ u ₀ ( x ), where u is a solution to the problem (11), (12). Let λ ∈ B R , with R = μ ^{1/ p} ( G )(|| g ( x ,0)|| L _∞ ( G ) R i + || Lu 0 || L _∞ ( G ) + || f ( x ,0)|| L _∞ ( G ) ), where i = 1 in the case of the Robin boundary conditions and i = 2 otherwise, and μ ( G ) is the Lebesgue measure of G . Demonstrate that this operator A takes a set B_R into itself and is compact. Let λ ∈ B_R . As we have proven, the inequalities (18), (20) hold and the conditions (vii), (viii) imply that || A ( λ ) || _{Lp ( G )} ≤ 0 a.e. Next, the definition of the quantity and the estimates (14), (20) imply that || A ( λ ) || _{Lp ( G )} ≤ R , i. e., takes the set B_R into itself. The continuity of the mapping A ( λ ) is obvious. Demonstrate that it is compact. Consider a sequence λ _n with λ _n ∈ B_R . The corresponding sequence of solutions satisfies the estimates (13), (14), (18), (20) and thus the sequence is bounded as well as the sequence || u_nt || _{L ∞ ( Q )} . Moreover,

|| A ( λ )|| L _∞ ( G ) ≤ R / μ ^{1/ p} ( G ).                                    (21)

Fix p 0 <  2 n l( n - 2) in the case of n >  2 and is p 0 arbitrary if n <  2. The sequence \\u_nt |I _W 1 (₀ T W 1 ( G )) is bounded and thus so is the sequence || u_nt || _{C ([0, T ]; W 21 ( G ))} . In this case there exists a subsequence u_nk such that u n k ( x ,0) ^ v ( x ) in L_{p 0}( G ) (the embedding theorems). Extracting one more subsequence if necessary we can assume that u n k ( x ,0) ^ v ( x ) a. e. in Lemma 3.2 of Ch.2 in [14] implies that u n k ( x , 0) ^ v ( x ) in any L q ( G ). We have proven that the mapping is compact. By Schauder fixed point theorem, the equation is solvable on the set B R . Consider the equation (11). Since u e W , every summand in this equation belongs C ([0, T ]; H ₁ ^' ) to after a possible change on a set of zero measure. So we can take the trace at t = 0. We obtain that

g ( x , 0) u_t ( x , 0) - Lu ( x , 0) = X ( x ) u ( x , 0) + f ( x , 0).

The equation A ( X ) = X can be rewritten as

g ( x , 0) u_t ( x , 0) - Lu ₀ ( x ) = X (x ) u ( x , 0) + f ( x , 0).

Subtracting these equalities and using the uniqueness theorem for solutions to the problem Lv + X v = 0, Bv | _r = 0, we conclude that u ( x ,0) = u ₀ ( x ). Next, we note that the conditions u_t ( x , t ) e L ^ ( Q ) and u_t ( x , t ) e C ([0, T ]; L ₂ ( G )) (we can state even that u_t ( x , t ) e C ([0, T ]; W 2 (G ))) imply that u_t ( x , t ) e L „ ( G )for every t . Hence, in view of the equality A ( X ) = X we have that Xe L „ ( G ). Rewrite the equation (11) in the form

Lu = gu_t + X ( x ) u - f ( x , t ) e L ^ ( Q ).

In view of the conditions (vii), (viii) and the classical results on solvability of elliptic problem (see [14]), we can conclude that u e L ₂ (0, T ; W ₂ ² ( G )).

In the next theorem we expose the uniqueness conditions. The proof coincides with that in [13, Theorem 6]. So we omit it.

Theorem 3. Let the conditions of Theorem 2 hold and

II g I L . ( Q ) R i / 5 2 <  1, where i = 1 in the case of the Robin boundary conditions and i = 2 otherwise. Then a solution ( u , X ) to the problem (1)–(3) from the class pointed out in the claim of Theorem 2 is unique.

Acknowledgement. The authors were supported by the grant on development of scientific schools with participation of young scientists of the Yugra State University.