Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition
We study the well-posedness of the inverse problem of determination of the coefficient of a minor term of a semilinear parabolic equation in the presence of a nonlinear boundary condition. The additional condition is given in the nonlocal integral form. A uniqueness theorem and a “conditional” stabi...
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| author | Akhundov, A. Ya. Ахундов, А. Я. |
| author_facet | Akhundov, A. Ya. Ахундов, А. Я. |
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| description | We study the well-posedness of the inverse problem of determination of the coefficient of a minor term of a semilinear parabolic equation in the presence of a nonlinear boundary condition. The additional condition is given in the nonlocal integral form. A uniqueness theorem and a “conditional” stability theorem are proved. |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
A.Ya. Akhundov, A. I. Gasanova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
DETERMINATION OF THE COEFFICIENT OF A SEMILINEAR PARABOLIC
EQUATION IN THE CASE OF BOUNDARY-VALUE PROBLEM
WITH NONLINEAR BOUNDARY CONDITION
ВИЗНАЧЕННЯ КОЕФIЦIЄНТА НАПIВЛIНIЙНОГО
ПАРАБОЛIЧНОГО РIВНЯННЯ ДЛЯ ГРАНИЧНОЇ ЗАДАЧI
З НЕЛIНIЙНОЮ ГРАНИЧНОЮ УМОВОЮ
The goal of this paper is to investigate the well-posedness of the inverse problem of determination of the coefficient of
a minor term of a semilinear parabolic equation in the case of nonlinear boundary condition. The additional condition is
given in the nonlocal integral form. A uniqueness theorem and a “conditional” stability theorem are proved.
Дослiджено коректнiсть оберненої задачi про визначення коефiцiєнта молодшого члена напiвлiнiйного параболiч-
ного рiвняння за наявностi нелiнiйної граничної умови. Додаткову умову наведено в нелокальнiй iнтегральнiй
формi. Доведено теорему про єдинiсть та теорему про „умовну” стiйкiсть.
Let Rn be an n-dimensional real Euclidean space, x = (x1, . . . , xn) be an arbitrary point of the
bounded domainD ⊂ Rn with a sufficiently smooth boundary ∂D, Ω = D×(0;T ] , S = ∂D×[0;T ] ,
0 < T be a fixed number.
The spaces C l (·) , C l+α (·) , C l,l/2 (·) , C l+α,(l+α)/2 (·) , l = 0, 1, 2, α ∈ (0, 1) , and the norms
in these spaces are defined as in [1, p. 12 – 20]
‖·‖l = ‖·‖Cl , ‖g (x, t, u)‖0 = sup
Ω
|g (x, t, u(x, t))| ,
ut =
∂u
∂t
, uxi =
∂u
∂xi
, i = 1, n,
∆u =
∑n
i=1
∂2u
∂x2
i
is a Laplacian,
∂u
∂ν
is an internal conormal derivative.
We consider an inverse problem on determining a pair of functions {u (x, t) , c (x)} from the
conditions
ut −∆u+ c (x)u = f (x, t, u) , (x, t) ∈ Ω, (1)
u (x, 0) = ϕ (x) , x ∈ D = D ∪ ∂D, (2)
∂u
∂ν
= ψ (x, t, u) , (x, t) ∈ S, (3)
T∫
0
u (x, t) dt = h (x) , x ∈ D, (4)
c© A.YA. AKHUNDOV, A. I. GASANOVA, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6 847
848 A.YA. AKHUNDOV, A. I. GASANOVA
here f (x, t, p) , ϕ (x) , ψ (x, t, p) , h (x) are the given functions.
The coefficiental inverse problems were studied in the papers [2 – 4] (see also the references
therein).
For the input data of problem (1) – (4), we take the following assumptions:
10) for the function f = f (x, t, p) , we shall assume the following:
the function f (x, t, p) is defined and continuous on the set
A =
{
(x, t, p)|(x, t) ∈ Ω, p ∈ R1
}
,
for each m1 > 0 and for |p| < m1, the function f(x, t, p) is uniformly Hölder continuous in x
and t of orders α and α/2, respectively, for each compact subset of A,
there exists a constant m2 > 0 such that
|f(x, t, p1)− f(x, t, p2)| ≤ m2 |p1 − p2| ,
holds for all p1, p2 and (x, t) ∈ Ω;
20) ϕ (x) ∈ C2+α
(
D
)
;
30) for the function ψ = ψ(x, t, p), we shall assume the following:
the function ψ(x, t, p) is defined and continuous on the set
B =
{
(x, t, p)|(x, t) ∈ S, p ∈ R1
}
,
for each m3 > 0 and for |p| < m3, the function ψ(x, t, p) is uniformly Hölder continuous in x
and t of orders α and α/2, respectively, for each compact subset of B,
there exists a constant m4 > 0 such that
|ψ(x, t, p1)− ψ(x, t, p2)| ≤ m4 |p1 − p2| ,
holds for all p1, p2 and (x, t) ∈ S;
40) h (x) ∈ C2+α
(
D
)
.
Definition 1. The pair of functions {c (x) , u (x, t)} is called the solution of problem (1) – (4) if:
1) c (x) ∈ C
(
D
)
;
2) u (x, t) ∈ C2,1 (Ω) ∩ C1,0
(
Ω
)
;
3) the conditions (1) – (4) hold for these functions, here the condition (3) is defined in the following
sense:
∂u (x, t)
∂ν (x, t)
= lim
y→x
y∈σ
∂u (y, t)
∂ν (x, t)
,
here σ is any closed cone with a vertex x, contained in D ∪ {x} .
The uniqueness theorem and the estimation of stability of the solutions of inverse problems occupy
a central place in investigation of their well-posedness. In the paper, the uniqueness of the solution
of problem (1) – (4) is proved under more general assumptions and the estimation characterizing the
„conditional” stability of the problem is established.
Define the set Kα as
Kα = {(u, c)|u(x, t) ∈ C2+α,1+α/2(Ω), c(x) ∈ Cα(D),
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
DETERMINATION OF THE COEFFICIENT OF A SEMILINEAR PARABOLIC EQUATION . . . 849
|u(x, t)|, |uxi(x, t)|, |uxixj (x, t)| ≤ m5, i, j = 1, n, (x, t) ∈ Ω, |c(x)| ≤ m6, x ∈ D}.
Let {ui (x, t) , ci (x)} be the solutions (1) – (4) corresponding to the given functions.
Definition 2. A solution of problem (1) – (4) is called stable if for any ε > 0 there is a δ (ε) > 0
such that for ‖f1 − f2‖0 < δ, ‖ϕ1 − ϕ2‖2 < δ, ‖ψ1 − ψ2‖0 < δ, ‖h1 − h2‖2 < δ the inequality
‖u1 − u2‖0 + ‖c1 − c2‖0 ≤ ε is fulfilled.
Theorem 1. Let:
1) fi, ϕi, ψi, hi, i = 1, 2, satisfy conditions 10 – 40, respectively;
2) there exist the solutions {ui (x, t) , ci (x)} , i = 1, 2, of problem (1) – (4) in the sense of
Definition 1, and let they belong to the set Kα.
Then there exists a T ∗ > 0 such that for (x, t) ∈ D × [0, T ∗] the solution of problem (1) – (4) is
unique, and the stability estimation
‖u1 − u2‖0 + ‖c1 − c2‖0 ≤
≤ m7 [‖f1 − f2‖0 + ‖ϕ1 − ϕ2‖2 + ‖ψ1 − ψ2‖0 + ‖h1 − h2‖2] (5)
is valid, here m7 > 0 depends on the data of the problem (1) – (4) and on the set Kα.
Proof. First, we prove the validity of the estimation (5). Taking into account (2) and the conditions
of the theorem, from equation (1) for the function c (x) we have
c (x) = −
u (x, T )− ϕ (x)−∆h (x)−
T∫
0
f (x, t, u) dt
(h (x))−1 . (6)
Denote by
z (x, t) = u1 (x, t)− u2 (x, t) , λ (x) = c1 (x)− c2 (x) ,
δ1 (x, t, p) = f1 (x, t, p)− f2 (x, t, p) , δ2 (x) = ϕ1 (x)− ϕ2 (x) ,
δ3 (x, t, p) = ψ1 (x, t, p)− ψ2 (x, t, p) , δ4 (x) = h1 (x)− h2 (x) .
One can verify that the functions λ (x) , w (x, t) = z (x, t)−δ2 (x) satisfy the following conditions:
wt −∆w = F (x, t) , (x, t) ∈ Ω, (7)
w (x, 0) = 0, x ∈ D;
∂w
∂ν
(x, t) = Ψ (x, t) , (x, t) ∈ S, (8)
λ (x) = z (x, T ) (h1 (x))−1 −H (x) , x ∈ D, (9)
where
F (x, t) = δ1 (x, t, u1) + ∆δ2 (x)− c1 (x) z (x, t)−
−λ (x)u2 (x, t) + f2(x, t, u1)− f2(x, t, u2),
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
850 A.YA. AKHUNDOV, A. I. GASANOVA
Ψ (x, t) = δ3 (x, t, u1)− ∂δ2(x)
∂ν
+ ψ2 (x, t, u1)− ψ2 (x, t, u2) ,
H (x) =
δ2 (x) + ∆δ4 (x) +
T∫
0
δ1 (x, t, u1) dt +
+
T∫
0
[f2 (x, t, u1)− f2 (x, t, u2)] dt
h2 (x) +
+
u2 (x, T )− ϕ2 (x)−∆h2 (x)−
T∫
0
f2 (x, t, u2) dt
δ4 (x)
[h1 (x)h2 (x)]−1 .
Under the conditions of the theorem, if follows that there exists a classic solution of problem (7),
(8) on determination of w (x, t) and it may be represented in the following form [5, p. 182]:
w (x, t) =
t∫
0
∫
D
Γ (x, t; ξ, τ)F (ξ, τ) dξdτ +
t∫
0
∫
∂D
Γ (x, t; ξ, τ) ρ (ξ, τ) dξ0dτ, (10)
here Γ (x, t; ξ, τ) is a fundamental solution of the equation wt−∆w = 0, dξ = dξ1 . . . dξn, dξ0 is an
element of the surface ∂D, ρ (x, t)is a continuous bounded solution of the following integral equation
[5, p. 182]:
ρ (x, t) = 2
t∫
0
∫
D
∂Γ (x, t; ξ, τ)
∂ν (x, t)
F (ξ, τ) dξdτ+
+2
t∫
0
∫
∂D
∂Γ (x, t; ξ, τ)
∂ν (x, t)
ρ (ξ, τ) dξ0dτ − 2Ψ (x, t) . (11)
Assume, that
χ = ‖u1 − u2‖0 + ‖c1 − c2‖0 .
Estimate the function |z (x, t)| . Taking into account that z (x, t) = w (x, t) + δ2 (x) , from (10)
we get
|z (x, t)| ≤ |w (x, t)|+ |δ2 (x)| ≤ |δ2 (x)|+
t∫
0
∫
D
Γ (x, t; ξ, τ) |F (ξ, τ)| dξdτ+
+
t∫
0
∫
∂D
Γ (x, t; ξ, τ) |ρ (ξ, τ)| dξ0dτ. (12)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
DETERMINATION OF THE COEFFICIENT OF A SEMILINEAR PARABOLIC EQUATION . . . 851
For the fundamental solutions following estimations are true [1, p. 444]:∫
Rn
Γ(x, t; ξ, τ)dξ ≤ m8, (13)
∫
Rn
∣∣∣Dl
xΓ(x, t; ξ, τ)
∣∣∣ dξ ≤ m9(t− τ)−
l−α
2 , l = 1, 2. (14)
Due to requirements imposed on the input data and on the set Kα, the integrand function F (x, t) in
the second summand of the right-hand side of (12), satisfies the estimation
|F (x, t)| ≤ |δ1 (x, t, u1)|+ |∆δ2 (x)|+ |c1 (x)| |z (x, t)|+
+|λ(x)||u2(x, t)|+ |f2(x, t, u1)− f2(x, t, u2)| ≤
≤ ‖f1 − f2‖0 + ‖ϕ1 − ϕ2‖2 +m10χ, (x, t) ∈ Ω, (15)
here m10 > 0 depends on the data of problem (1) – (4) and the set Kα.
From the Gauss – Ostrogradsky formula and (14) for s = 1, we have∫
∂D
Γ (x, t; ξ, τ) dξ0 ≤ m11(t− τ)−
1−α
2 . (16)
Taking into account expression (11), (14) for s = 1 and s = 2, the conditions of the theorem,
determination of the set Kα for the function ρ (x, t) we get
|ρ (x, t)| ≤ m12 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0 + χ] +m13 ‖ρ‖ tα/2, (x, t) ∈ S,
where m12,m13 > 0 depend on the data of problem (1) – (4) and on the set Kα.
The last inequality is fulfilled for all (x, t) ∈ ∂D × [0, T ] , therefore the following estimation is
true:
‖ρ‖0 ≤ m12 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0 + χ] +m13t
α/2 ‖ρ‖0 .
Let 0 < T1 ≤ T be a number such that m13T
α/2
1 < 1. Then for all (x, t) ∈ ∂D× [0, T1] we have
‖ρ‖0 ≤ m14 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0 + χ] , (17)
where m14 > 0 depends on the data of problem (1) – (4) and on the set Kα.
Taking into account inequalities (13), (15), (16) and (17) from (12) for |z (x, t)| we get
|z (x, t)| ≤ m15 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0] +m16χt
α, (x, t) ∈ Ω, (18)
where m15, m16 > 0 depend on the data of problem (1) – (4) and the set Kα.
Now estimate the function |λ (x)| . From (9) it follows
|λ (x)| ≤ |z (x, t)|
∣∣∣h1 (x)−1
∣∣∣+ |H (x)| .
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
852 A.YA. AKHUNDOV, A. I. GASANOVA
Taking into account the conditions of the theorem, definitions of the set Kα, inequalities (18) and
expressions for H (x) , from the last inequality we get
|λ (x)| ≤ m17 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0 + ‖δ4‖2] +m18t
αχ, x ∈ Ω, (19)
where m17, m18 > 0 depend on the data of the problem (1) – (4) and the set Kα.
Inequalities (18) and (19) are satisfied for any values of (x, t) ∈ D × [0, T ] .
Consequently, combining these inequalities, we obtain
χ ≤ m19 [‖δ1‖0 + ‖δ2‖2 + ‖δ3‖0 + ‖δ4‖2] +m20t
αχ, (20)
where m19,m20 > 0 depend on the data of problem (1) – (4) and the set Kα.
Let T2 (0 < T2 ≤ T ) be a number such that m20T
α
2 < 1. Then from (20) we get that for
(x, t) ∈ D× [0, T ∗] , T ∗ = min (T1, T2) , the stability estimation for the solution of problem (1) – (4)
is true.
Uniqueness of the solution of problem (1) – (4) follows from estimation (5) for f1 (x, t, u) =
= f2 (x, t, u) , ϕ1 (x) = ϕ2 (x) , ψ1 (x, t, u) = ψ2 (x, t, u) , h1 (x) = h2 (x) .
The theorem is proved.
1. Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N. Linear and quasilinear equations of parabolic type. – Moscow:
Nauka, 1967. – 736 p. (in Russian).
2. Ivanchov M. Inverse problems for equationts of parabolic type. – Lviv, 2003. – 238 p.
3. Akhundov A. Y. Some inverse problems for strong parabolic systems // Ukr. Mat. Zh. – 2006. – 58, № 1. – S. 114 – 123
(in Russian).
4. Iskenderov A. D., Akhundov A. Ya. Inverse problem for a linear system of parabolic equations // Dokl. Mat. – 2009.
– 79, № 1. – P. 73 – 75.
5. Friedman A. Partial differential equations of parabolic type. – Moscow: Mir, 1968. – 427 p. (in Russian).
Received 08.01.13,
after revision — 27.01.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
|
| id | umjimathkievua-article-2182 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:16Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/91/a2e227bad232409dc6a839ff13b43891.pdf |
| spelling | umjimathkievua-article-21822019-12-05T10:25:45Z Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition Визначення коефіцієнта напівлінійного параболічного рівняння для граничної задачі з нелінійною граничною умовою Akhundov, A. Ya. Ахундов, А. Я. We study the well-posedness of the inverse problem of determination of the coefficient of a minor term of a semilinear parabolic equation in the presence of a nonlinear boundary condition. The additional condition is given in the nonlocal integral form. A uniqueness theorem and a “conditional” stability theorem are proved. Досліджєно коректність оберненої задачi про визначення коефiцiєнта молодшого члена напівлінійного параболічного рівняння за наявності нелінійної граничної умови. Додаткову умову наведено в нелокальній інтегральній формі. Доведено теорему про єдиність та теорему про „умовну" стійкість. Institute of Mathematics, NAS of Ukraine 2014-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2182 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 6 (2014); 847–852 Український математичний журнал; Том 66 № 6 (2014); 847–852 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2182/1365 https://umj.imath.kiev.ua/index.php/umj/article/view/2182/1366 Copyright (c) 2014 Akhundov A. Ya. |
| spellingShingle | Akhundov, A. Ya. Ахундов, А. Я. Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title | Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title_alt | Визначення коефіцієнта напівлінійного параболічного рівняння для граничної задачі з нелінійною граничною умовою |
| title_full | Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title_fullStr | Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title_full_unstemmed | Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title_short | Determination of the Coefficient of a Semilinear Parabolic Equation for a Boundary-Value Problem with Nonlinear Boundary Condition |
| title_sort | determination of the coefficient of a semilinear parabolic equation for a boundary-value problem with nonlinear boundary condition |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2182 |
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