Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain
In a bounded domain with infinitely increasing quantity of infinitesimal holes whose asymptotic behaviour is described formally, the homogeneous Dirichlet problem for the equation with additional power nonlinearity on the time derivative of the solution is averaged.
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Інститут прикладної математики і механіки НАН України
1999
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| Zitieren: | Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain / N.R. Sidenko // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 92-97. — Бібліогр.: 2 назв. — англ. |
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irk-123456789-1692772020-06-10T01:26:30Z Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain Sidenko, N.R. In a bounded domain with infinitely increasing quantity of infinitesimal holes whose asymptotic behaviour is described formally, the homogeneous Dirichlet problem for the equation with additional power nonlinearity on the time derivative of the solution is averaged. 1999 Article Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain / N.R. Sidenko // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 92-97. — Бібліогр.: 2 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169277 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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English |
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In a bounded domain with infinitely increasing quantity of infinitesimal holes whose asymptotic behaviour is described formally, the homogeneous Dirichlet problem for the equation with additional power nonlinearity on the time derivative of the solution is averaged. |
| format |
Article |
| author |
Sidenko, N.R. |
| spellingShingle |
Sidenko, N.R. Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain Нелинейные граничные задачи |
| author_facet |
Sidenko, N.R. |
| author_sort |
Sidenko, N.R. |
| title |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| title_short |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| title_full |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| title_fullStr |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| title_full_unstemmed |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| title_sort |
averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169277 |
| citation_txt |
Averaging of the periodic by time boundary value problem for the nonlinear wave equation in a perforated domain / N.R. Sidenko // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 92-97. — Бібліогр.: 2 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT sidenkonr averagingoftheperiodicbytimeboundaryvalueproblemforthenonlinearwaveequationinaperforateddomain |
| first_indexed |
2025-07-15T04:02:06Z |
| last_indexed |
2025-07-15T04:02:06Z |
| _version_ |
1837684098832793600 |
| fulltext |
AVERAGING OF THE PERIODIC BY TIME BOUNDARY
VALUE PROBLEM FOR THE NONLINEAR WAVE EQUATION
IN A PERFORATED DOMAIN
c© N.R. SIDENKO
ABSTRACT. In a bounded domain with infinitely increasing quantity of infinitesimal holes
whose asymptotic behaviour is described formally, the homogeneous Dirichlet problem for the
equation with additional power nonlinearity on the time derivative of the solution is averaged.
1. Setting of the problem. Let Ω be a bounded domain in Rn (n ≥ 2). We
denote Ωε a domain obtained by removing from Ω a number N(ε) of closed subsets Si
ε,
i.e. Ωε = Ω \ Sε, Sε = ∪Si
ε (i = 1, N(ε)). Here ε > 0 is a parameter wich tends to zero
on some subsequence, herewith N(ε) → ∞. Let T = const > 0. It being investigated a
behaviour when ε → 0 of a solution uε(t, x) of the problem
u′′ε −∆uε + |u′ε|p−2u′ε = fε(t, x) in Qε,
uε(t, ·)|∂Ωε
= 0, uε(0, x) = uε(T, x), (1)
u′ε(0, x) = u′ε(T, x), x ∈ Ωε,
where u′ε = ∂uε/∂t, u′′ε = ∂2uε/∂t2, Qε = {(t, x) ∈ (0, T ) × Ωε}, fε(t, x) ∈ Lp′(Qε),
p′ = p/(p− 1), 2 < p < ∞. For each value ε it has the unique solution [1]
uε(t, x) = ūε(x) + ũε(t, x), 〈ũε(·, x)〉 ≡ 1
T
T∫
0
ũε(t, x)dt = 0,
ūε ∈
◦
W
1,p′
(Ωε), ũε ∈ L2(0, T ;
◦
H1 (Ωε)) ∩W 1,p(0, T ; Lp(Ωε)).
(2)
The behaviour of a closed set Sε ⊂ Ω̄ when ε → 0 being described by the following
basic hypothesis [2]:
there exists such a sequence of functions wε(x) that
1) wε ∈ H1(Ω) ∩ L∞(Ω), ||wε||L∞(Ω) ≤ M0, 2) wε|Sε
= 0,
3) wε → 1 weakly in H1(Ω) and almost everywhere in Ω,
4) −∆wε = µε − γε where µε, γε ∈ H−1(Ω), µε → µ stron-
gly in H−1(Ω), (γε, v)Ω = 0 for each v ∈
◦
H1 (Ω) if v|Sε
= 0.
(A)
Brackets (·, ·)Ω denote the scalar product in L2(Ω) with respect to the Lebesgue mea-
sure dx and correspondent duality relations. Designations (·, ·)Ωε , (·, ·)Qε , (·, ·)Q are anal-
ogous.
The function ũε(t, x) is defined by the integral identity [1]
T∫
0
[−(ũ′ε, v
′′)Ωε + (5ũε,5v′)Ωε + (F (ũ′ε)− fε, v
′)Ωε ]dt = 0, (3)
where F (u) = |u|p−2u, v(t, x) is an arbitrary function such that
v ∈ W 2,p′(0, T ; Lp′(Ωε)) ∩W 1,p(0, T ; Lp(Ωε))∩
∩H1(0, T ;
◦
H1 (Ωε)), (4)
v(0, x) = v(T, x), v′(0, x) = v′(T, x), 〈v(·, x)〉 = 0, x ∈ Ωε.
2. The estimates of solutions. Periodicity of a function ϕ by t with the period T
is denoted by inclusion ϕ ∈ Tt[0, T ]. Let us choose a sequence of such θn(t) ∈ C∞(R) ∩
Tt[0, T ], N 3 n > T−1, that θn(−t) ≡ θn(t), supp θn = ∪[kT−(2n)−1, kT+(2n)−1] (k ∈ Z),∫ (2n)−1
0 θn(t)dt = 1
2
. We put (ϕ ∗ ψ)(t) =
∫ T
0 ϕ(t − τ)ψ(τ)dτ = (ψ ∗ ϕ)(t) for any ϕ, ψ ∈
Tt[0, T ]∩L1(0, T ). Since ũε ∗θn ∈ C∞(R;
◦
H1 (Ωε)∩Lp(Ωε))∩Tt[0, T ], 〈(ũε ∗θn)(·, x)〉 = 0,
in (3) we can take v = ũε ∗ θn ∗ θn. Herewith (F (ũ′ε)− fε, ũ
′
ε ∗ θn ∗ θn)Qε = 0. As a result
of passing to the limit by n →∞ and in view of (2), we obtain
||ũ′ε||pLp(Qε)
= (fε, ũ
′
ε)Qε , ||ũ′ε||Lp(Qε) ≤ ||fε||1/(p−1)
Lp′ (Qε)
,
||ũε||Lp(Qε) ≤ C0||ũ′ε||Lp(Qε) ≤ C0||fε||1/(p−1)
Lp′ (Qε)
, C0 = const.
(5)
If sup||fε||Lp′ (Qε)
= M < ∞, then the following estimates are valid:
||ũ′ε||Lp(Qε) ≤ M1/(p−1), ||ũε||Lp(Qε) ≤ C0M
1/(p−1) ∀ε. (6)
Putting further
v(t, x) =
t∫
0
ũε(τ, x)dτ − 1
T
T∫
0
(T − τ)ũε(τ, x)dτ,
we have the function satisfying all the conditions (4). Substituting it into (3), we obtain
||5ũε||2L2(Qε)
= ||ũ′ε||L2(Qε)+(fε−F (ũ′ε), ũε)Qε ≤ |Q|(p−2)/p||ũ′ε||2Lp(Qε)
+||ũ′ε||p−1
Lp(Qε)
||ũε||Lp(Qε)+
||fε||Lp′ (Qε)
||ũε||Lp(Qε) ≤ |Q|(p−2)/pM2/(p−1) + 2C0M
p′ = C2
1 , where Q = {(t, x) ∈ (0, T ) ×
Ω}, i.e. the estimate is valid
||ũε||
L2(0,T ;
◦
H1(Ωε))
≤ C1 ∀ε. (7)
3. Passing to the limit. For an arbitrary function v given on Ωε, we denote v̂ its
propagation on Ω by definition by zero on Sε. In view of (2), we have
ˆ̃uε ∈ L2(0, T ;
◦
H1 (Ω)) ∩W 1,p(0, T ; Lp(Ω)) ∩ Tt[0, T ],
〈ˆ̃uε(·, x)〉 = 0,
(8)
and from (6),(7) the estimates follow
||ˆ̃u′ε||Lp(Q) ≤ M1/(p−1), ||ˆ̃uε||Lp(Q) ≤ C0M
1/(p−1),
||ˆ̃uε||
L2(0,T ;
◦
H1(Ω))
≤ C1.
(9)
The last estimates give a possibility to extract a subsequence (again denoted by index ε)
for which the convergences are valid
ˆ̃uε → ũ weakly in L2(0, T ;
◦
H1 (Ω)) ∩ Lp(Q),
ˆ̃u
′
ε → ũ′ weakly in Lp(Q), F (ˆ̃u
′
ε) → χ weakly in Lp′(Q).
(10)
For any function ψ(t, x) such that
ψ ∈ W 1,p′(0, T ; Lp′(Ωε)) ∩ L2(0, T ;
◦
H1 (Ωε))∩
∩Lp(Qε) ∩ Tt[0, T ], 〈ψ(·, x)〉 = 0,
(11)
we define the test function for the identity (3)
v(t, x) =
t∫
0
ψ(τ, x)dτ − 1
T
T∫
0
(T − τ)ψ(τ, x)dτ,
which satisfies the conditions (4). Substituting it into (3), we obtain
−(ˆ̃u
′
ε, ψ̂
′)Q + (5ˆ̃uε,5ψ̂)Q + (F (ˆ̃u
′
ε)− f̂ε, ψ̂)Q = 0, (12)
where ψ̂(t, x) is an arbitrary function defined on R × Ω, whose properties on Q are
analogous to (11), ψ̂(t, ·)|Sε = 0. We put ψ̂(t, x) = wε(x)v(t, x), where wε are the same
functions as in the hypothesis (A) and v(t, x) ∈ C1,2(R × Ω̄) ∩ Tt[0, T ] : v(t, ·)|∂Ω = 0,
〈v(·, x)〉 = 0. After substitution of this function ψ̂ into (12), we have
0 = −(ˆ̃u
′
ε, wεv
′)Q + (ˆ̃uε,−∆(wεv))Q + (F (ˆ̃u
′
ε)− f̂ε, wεv)Q =
= −(ˆ̃u
′
ε, wεv
′)Q + (µε, vˆ̃uε)Q − 2(ˆ̃uε,5wε · 5v)Q−
−(ˆ̃uε, wε∆v)Q + (F (ˆ̃u
′
ε)− f̂ε, wεv)Q.
Here it has been taken into account that (γε, 〈vˆ̃uε〉)Ω = 0. We consider this expression
on the extracted subsequence ε → 0 taking into account (10) and the hypothesis (A) and
supposing that the entire sequence f̂ε → f weakly in Lp′(Q). In the limit we come to the
identity
−(ũ′, v′)Q + (µũ, v)Q + (5ũ,5v)Q + (χ− f, v)Q = 0, (13)
which is valid for all v(t, x) ∈ L2(0, T ;
◦
H1 (Ω)) ∩ Lp(Q) ∩ Tt[0, T ] such that v′ ∈ Lp′(Q),
〈v(·, x)〉 = 0.
Further, we take into consideration that the generalized function µ(x) ∈ H−1(Ω), intro-
duced in the hypothesis (A), generates simultaneously the positive finite Radon measure
dµ(x) = µ(x)dx on Ω [2].
It follows from (8) and (10) that
ũ(0, x) = ũ(T, x)
as a function from C0([0, T ]; Lp(Ω)), 〈ũ(·, x)〉 = 0.
(14)
Substituting v = ũ into (13), we obtain
T∫
0
[||ũ||2L2(Ω,dµ) + || 5 ũ||2L2(Ω)]dt ≤
≤ |Q|(p−2)/p||ũ′||2Lp(Q) + ||f − χ||Lp′ (Q)||ũ||Lp(Q),
i.e. we have additionally to (14)
ũ(t, x) ∈ L2(0, T ; V ) ∩W 1,p(0, T ; Lp(Ω)),
V =
◦
H1 (Ω) ∩ L2(Ω, dµ).
(15)
For the identification of a function χ in (10), we use the equality (5) and consider on
the extracted subsequence the following inequality with an arbitrary v ∈ Lp(Q)
0 ≤ (F (ˆ̃u
′
ε)− F (v), ˆ̃u
′
ε − v)Q =
= (f̂ε, ˆ̃u
′
ε)Q − (F (ˆ̃u
′
ε), v)Q − (F (v), ˆ̃u
′
ε − v)Q.
(16)
In supposition that the entire sequence f̂ε → f strongly in Lp′(Q), after passing to the
limit in (16) we get
0 ≤ (f, ũ′)Q − (χ, v)Q − (F (v), ũ′ − v)Q. (17)
Taking into account (14),(15), we take in (13) the test function v = ũ′∗θn∗θn ∈ C∞(R; V ∩
Lp(Ω)) ∩ Tt[0, t]. As the result we obtain 0 = (f − χ, ũ′ ∗ θn ∗ θn)Q = (f − χ, ũ′)Q. From
this and (17) it follows (χ− F (v), ũ′ − v)Q ≥ 0∀v ∈ Lp(Q) signifying that χ = F (ũ′).
From (13) ∀v ∈ D(Q) such that 〈v(·, x)〉 = 0, we have
(ũ′′ + µũ−∆ũ + F (ũ′)− f, v)Q = 0, (18)
that is
ũ′′ + µũ−∆ũ + F (ũ′)− f = g(x) in D′(Q). (19)
It is easy to see that in (19) g(x) ∈ V ′ + Lp′(Ω) and ũ′′ ∈ L2(0, T ; V ′) + Lp′(Q) ⊂
Lp′(0, T ; V ′ + Lp′(Ω)), so in view of (15)
ũ′ ∈ W 1,p′(0, T ; V ′ + Lp′(Ω)) ⊂ C0([0, T ]; V ′ + Lp′(Ω)). (20)
Then we can write the identity (13) ∀v∈H1(0, T ; V ) ∩W 1,p(0, T ; Lp(Ω))
∩ Tt[0, T ] : 〈v(·, x)〉 = 0 as follows
(ũ′(0, ·)− ũ′(T, ·), v(0, ·))Ω+
(ũ′′ + µũ−∆ũ + F (ũ′)− f, v)Q = 0.
Herewith, for the functions v appointed (18) is valid as before now, therefore (ũ′(0, ·) −
ũ′(T, ·), v(0, ·))Ω = 0 where v(0, x) may be by any function from V ∩ Lp(Ω). In view of
(20), from this it follows that
ũ′(0, x) = ũ′(T, x). (21)
On the base of (14),(20),(21) we make more precise an indefinite function g(x) in (19)
g(x) = 〈ũ′′ + µũ−∆ũ + F (ũ′)− f〉 = 〈F (ũ′)− f〉 ∈ Lp′(Ω). (22)
4. The convergence of the mean by time value. The mean value ūε(x) =
〈uε(·, x)〉 is defined by the following from (1),(2) linear problem
−∆ūε(x) = gε(x) = f̄ε(x)− F (ũ′ε)(x), x ∈ Ωε, ūε ∈
◦
W
1,p′
(Ωε), (23)
in which, if ε → 0 on the extracted subsequence, ĝε → −g weakly in Lp′(Ω). Starting with
this place let us assume that in (1)
2 < p ≤ 2n
n− 2
(n > 2), 2 < p < ∞ (n = 2). (24)
Herewith Lp′(Ω) ⊂ H−1(Ω) and
||ˆ̄uε|| ◦
H1(Ω)
≤ ||ĝε||H−1(Ω) ≤ C2||ĝε||Lp′ (Ω) ≤ C2M1 ∀ε. (25)
Having extracted a subsequence
ˆ̄uε → ū weakly in
◦
H1 (Ω), (26)
we get as in [2]
−∆ū + µū = −g, ū ∈ V, (27)
and ū defined by the problem (27) is unique. Then the convergence (26) is valid for the
entire sequence {ˆ̄uε}.
5. Formulation of the result. Substituting the value of g from (27) into (19) we
obtain for the function u(t, x) = ū(x) + ũ(t, x) the problem
u′′ −∆u + µu + F (u′) = f in Q,
u(0, x) = u(T, x), u′(0, x) = u′(T, x), x ∈ Ω, u(t, ·)|∂Ω = 0,
(28)
in which
u ∈ L2(0, T ; V ) ∩W 1,p(0, T ; Lp(Ω)),
u′′ ∈ Lp′(0, T ; V ′), u′ ∈ C0([0, T ]; V ′).
(29)
It being easily proved that the solution of the problem (28),(29) is unique. Thus the
following result has been established:
Theorem.Let Ω be a bounded domain in Rn (n ≥ 2), and Sε be a sequence of its
closed subsets for which the hypothesis (A) is valid. Let p satisfies inequalities (24), the
sequence f̂ε → f strongly in Lp′(Q). Then for the sequence of solutions of problems (1)
the following convergences take place:
ûε → u weakly in L2(0, T ;
◦
H1 (Ω)) ∩ Lp(Q),
û′ε → u′ weakly in Lp(Q), |û′ε|p−2û′ε → |u′|p−2u′ weakly in Lp′(Q), where u is the unique
solution of the problem (28) satisfying the conditions (29).
References
1. .-. H . –., , 1972. –588 .
2. Cioranescu D.,Donato P.,Murat F.,Zuazua E. Homogenization and corrector for the
wave equation in domains with small holes. –Ann. della scuola norm. super. di Pisa, Sci.
fis. e matem., 1991. ser.4, vol.18, F.2, p.251-293.
Institute of Mathematics,
Ukrainian National Academy of Sciences,
Tereshchenkivska 3, Kyiv, 252602 Ukraine.
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