About bounded properties of smooth solutions of some differential-operator equations
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Інститут прикладної математики і механіки НАН України
1999
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| Назва видання: | Нелинейные граничные задачи |
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| Цитувати: | About bounded properties of smooth solutions of some differential-operator equations / V.V. Gorodetsky, I.I. Drin // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 5-39. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1692692020-06-10T01:26:29Z About bounded properties of smooth solutions of some differential-operator equations Gorodetsky, V.V. Drin, I.I. 1999 Article About bounded properties of smooth solutions of some differential-operator equations / V.V. Gorodetsky, I.I. Drin // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 5-39. — Бібліогр.: 4 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169269 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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English |
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Article |
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Gorodetsky, V.V. Drin, I.I. |
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Gorodetsky, V.V. Drin, I.I. About bounded properties of smooth solutions of some differential-operator equations Нелинейные граничные задачи |
| author_facet |
Gorodetsky, V.V. Drin, I.I. |
| author_sort |
Gorodetsky, V.V. |
| title |
About bounded properties of smooth solutions of some differential-operator equations |
| title_short |
About bounded properties of smooth solutions of some differential-operator equations |
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About bounded properties of smooth solutions of some differential-operator equations |
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About bounded properties of smooth solutions of some differential-operator equations |
| title_full_unstemmed |
About bounded properties of smooth solutions of some differential-operator equations |
| title_sort |
about bounded properties of smooth solutions of some differential-operator equations |
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Інститут прикладної математики і механіки НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169269 |
| citation_txt |
About bounded properties of smooth solutions of some differential-operator equations / V.V. Gorodetsky, I.I. Drin // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 5-39. — Бібліогр.: 4 назв. — англ. |
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Нелинейные граничные задачи |
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2025-07-15T04:01:40Z |
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2025-07-15T04:01:40Z |
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1837684071956742144 |
| fulltext |
ABOUT BOUNDED PROPERTIES OF SMOOTH SOLUTIONS
OF SOME DIFFERENTIAL-OPERATOR EQUATIONS
c© V.V.Gorodetsky, I.I.Drin
1. Fractional differentiation and integration in space D
′
+
Recall that symbol D = D(R) denotes the set of all finite unlimited differentiable on
R functions. Convergence in D is defined as below: sequence {ϕn, n ≥ 1} ⊂ D is called
the converged sequence to function ϕ ∈ D (maps as follows: ϕn → ϕ when n → ∞ in
D) if:
a) exists such R > 0, that suppϕn ⊂ (−R,R), ∀n ∈ N;
b) ϕ
(k)
n ⇒ ϕ(k) when n →∞ on R,∀k ∈ Z+.
Totality of all linear continuous functionals on D with weak convergence is mapped
with symbol D′ ≡ D′(R). Elements D′ are named the generalized functions. Totality of
generalized functions from D′, which are equal zero on the half-axis (−∞, 0), is mapped
by D
′
+. It is known from [1] that D
′
+ creates associative and commutative algebra on
folding operation, and δ ∗ f = f ∗ δ = f,∀f ∈ D
′
+. The δ-function of Dirac is the one in
this algebra.
Let the generalized function fα from D
′
+ depend from parameter α,−∞ < α < +∞,
and be denoted by formula
fα(t) = θ(t)tα−1/Γ(α), α > 0,
fα(t) = f
(m)
α+m(t), α ≤ 0,
where m is the smallest from natural numbers and m+α > 0, θ is the Heaviside function.
The following assertions are valid:
1) ∀{α, β} ⊂ R : fα ∗ fβ = fα+β ;
2) Let I(α)f = f ∗ fα, ∀f ∈ D
′
+. Then
a) ∀f ∈ D
′
+ : I(0)f = f ;
b) ∀f ∈ D
′
+ ∀n ∈ N : I(−n)f = f (n);
c) ∀f ∈ D
′
+ ∀n ∈ N : (I(n)f)(n) = f ;
d) ∀f ∈ D
′
+ ∀{α, β} ⊂ R : I(α)I(β)f = I(α + β)f .
2. Spaces of based and generalized functions
Let H = L2(R),
Φ = lim
m→∞
indΦm, Φm = {ϕ |ϕ =
m∑
k=0
ckhk(x), ck ∈ C},
where
hk(x) = (2kk!)−1/2(−1)kπ−1/4ex2/2(e−x2/2)(k), k ∈ Z+
are Hermite functions which create the ortonormous basis in H. It is evident that Φ
lies densily in H. In space Φ the differentiation operation is defined and continuous.
Let symbol Φ′ map the space of all antilinear continuous functionals on Φ with weak
convergence. Elements of Φ′ also are named the generalized functions. Each element f
from space Φ′ is unlimited differentiable and
< f (n), ϕ >= (−1)n < f, ϕn >, ∀ϕ ∈ Φ, ∀n ∈ N
(here < f, · > maps the action of functional f on the based element).
Series
∞∑
k=0
ckhk, where ck =< f, hk >, k ∈ Z+, f ∈ Φ′, is named Fourier-Hermite
series of generalized function f . For any generalized function f her Fourier-Hermite
series converges in Φ′. Otherwise, any series of type
∞∑
k=0
ckhk converges in Φ′ to some
function f ∈ Φ′ and this series is Fourier-Hermite series for f [2]. Then, Φ′ can be
interpreted as the space of formal series of type
∞∑
k=0
ckhk.
I.M.Gelfand and G.E.Shylov described in [3] the collection of spa- ces, which are called
the spaces of type S. These spaces consist of unlimitly differentiable functions, which
are defined on R and satisfy some decreasing conditions on the infinity and conditions
of increasing of derivatives. Denote some of them.
For arbitrary α, β > 0 let
Sβ
α(R) ≡ Sβ
α := {ϕ ∈ C∞(R)|∃c,B, A,> 0 ∀{k, m} ⊂ Z+
∀x ∈ R : |xkϕ(m)(x)| ≤ cAkBmkkαmmβ}.
Spaces Sβ
α are nontrivial for α + β ≥ 1 and create dense sets in L2(R). If 0 < β < 1
and α > 1 − β then Sβ
α consists only of functions ϕ : R → C which let analytical
continuation into whole complex plane and for which
|ϕ(x + iy)| ≤ c exp{−a|x|1/a + b|y|1/(1−β)}, c, a, b > 0.
Note, that spaces Sβ
α create topological algebras on simple operations of multipli-
cation and folding. In Sβ
α the operations of shear of argument and differentiation are
defined and continuous. This operations translate Sβ
α into itself [3].
Space of all antilinear continuous functionals on Sβ
α with weak convergence is mapped
by symbol (Sβ
α)′. Elements (Sβ
α)′ are called Gevrey ultradistributions of order β.
For abovementioned spaces the following continuous and dense implications are valid:
Φ ⊂ Sβ
α ⊂ L2(R) ⊂ (Sβ
α)′ ⊂ Φ′, α + β ≥ 1.
3. About smooth solutions of parabolic equations with increasing coeffi-
cients
Consider in space Φ′ operator  with such action:
Φ′ 3
∞∑
k=0
ckhk = f 7→ Âf =
∞∑
k=0
(2k + 1)νckhk ∈ Φ′,
where ν > 0 is the fixed parameter. It is evident, that operator  is linear and continuous
in Φ′. The following assertion is valid.
Theorem 1. Let A be the contraction of operator  on L2(R). Then A is nonnegative
selfadjoint operator in L2(R) with dense range of definition D(A) and Φ ⊂ D(A).
Corollary 1. Spectrum of operator A is clear discrete with unique limited point
on infinity. Hermite functions {hk, k ∈ Z+} are eigenfunctios for operator A. These
functions have eigenvalues µk = (2k + 1)ν , k ∈ Z+. Each eigenvalue µk is prime.
Remark 1. If ν = 1 then operator A converges with operator which is created in
L2(R) by the differential expression −d2/dx2 + x2, that is in this case A is harmonic
oscilliator (see [2]).
Consider equation
Dβ
t u(t, x) + (−1)−[β]+1D
{β}
t Aαu(t, x) = 0, (t, x) ∈ (0,∞)× R ≡ Ω, (1)
where β ∈ [−3, 0), α > 0 are fixed numbers, [β] is whole party and {β} is fractional
party of number β, Dβ
t ≡ I(β) is operator of fractional differentiation which acts on
variable t in space D
′
+, Aα is degree of operator A and
D(Aα) = {ϕ ∈ L2(R)|
∞∑
k=0
(2k + 1)2να|ck(ϕ)|2 < ∞,
ck(ϕ) = (ϕ, hk), k ∈ Z+}.
The solution of equation (1) we called the function u which sastisfies the equations:
1) u(·, x) ∈ D
′
+ ∩ C−[β]((0,∞)) for all x ∈ R;
2) u(t, ·) ∈ D(Aα) ⊂ L2(R) for all t > 0; u(t, ·) = 0 for t < 0;
3) u satisfies equation (1).
If β ∈ [−3,−1), we assume that u satisfies also the condition:
4) for arbitrary fixed interval [δ,+∞) ⊂ (0,+∞) constant c = c(δ) > 0 exists such
that
sup
t∈[δ,+∞)
‖D{β}
t u(t, ·)‖L2(R) ≤ c.
Theorem 2. Function u is the solution of equation (1) if and only if it can be
represented in following type
u(t, x) =
∞∑
k=0
(θ(t) exp{−t(2k + 1)να/(−[β])} ∗ f{β}(t))ckhk(x), (2)
t ∈ R\{0}, x ∈ R,
where
f =
∞∑
k=0
ckhk ∈ (Sω
ω )′,
ω = 1/2, if να/(−[β]) ≡ γ ≥ 1 and ω = 1/(2γ), if 0 < γ < 1. And u(t, ·) ∈ Sω
ω for all
t > 0.
Remark 2. From theorem 2 follows that when t > 0 formula (2) describes all
unlimited differentiable on x solutions of equation (1).
Corollary 2. Bounded value D
{β}
t u(t, ·) when t → +0 exists in space (Sω
ω )′, that is
D
{β}
t u(t, ·) → f =
∞∑
k=0
ckhk, t → +0, in (Sω
ω )′.
Then, (Sω
ω )′ is (in some sence) ”maximal” space in which bounded values of function
D
{β}
t u(t, ·) exist for t → +0. Using the representation of function as formal Fourier-
Hermite series we can establish necessary and unique conditions for which the bounded
values D
{β}
t u(t, ·) when t → +0 exist in narrow (intermediate) spaces. These spaces are
situated between L2(R) and (Sω
ω )′. The following assertion are valid.
Theorem 3. In order for bounded value of function D
{β}
t u(t, ·) when t → +0 to
belong to space (Sβ
β )′ (β > 1/2, if γ ≥ 1; β > 1/(2γ), if 0 < γ < 1; γ = να/(−[β])), it
is necessary and enough that
∀µ > 0 ∃c = c(µ) > 0 :
∫
R
|D{β}
t u(t, x)|2dx ≤ ceµt−q
,
q = 1/(2γβ − 1),
for small values t > 0.
Denote, that abovementioned spaces link one to another by following chain:
L2(R) ⊂ (Sβ
β )′ ⊂ (Sω
ω )′ ⊂ Φ′.
Remark 3. If parameter β has one of values of set {−1,−2, −3} then {β} = 0 and
D
{β}
t = D0
t = E (E is identity operator), Dβ
t u(t, ·) ≡ I(β)u(t, ·) = ∂pu(t, ·)/∂tp, p = −β
(see item 1). Then we obtain the equation
∂pu/∂tp + (−1)p+1Aαu = 0, (t, x) ∈ Ω, p ∈ {1, 2, 3}. (3)
f−{β} = f0 = f
′
1 = θ′ = δ
(see item 1), then
θ(t) exp{−t(2k + 1)γ} ∗ f−{β}(t) = θ(t) exp{−t(2k + 1)γ} ∗ δ(t) =
= θ(t) exp{−t(2k + 1)γ}, γ = να/p, p = −[β], p ∈ {1, 2, 3}.
That is why the solutions of these equations are represented in following form (when
t > 0)
u(t, x) =
∞∑
k=0
exp{−t(2k + 1)γ}ckhk(x) =
=< f, Kt,x,γ(·) >, t > 0, x ∈ R,
where
Kt,x,γ(y) =
∞∑
k=0
exp{−t(2k + 1)γ}hk(x)hk(y), t > 0, {x, y} ⊂ R.
Denote that in case γ = 1 the cernel Kt,x,1 can be written explicitly [4]:
Kt,x,1(y) = (2πsh(2t))−1/2 exp{sh−1(2t)xy − 0.5cth(2t)(x2 + y2)}.
Remark 4. If ν = 1, α = m,m ∈ N then (as known from [2]),
Amu(t, x) = (−∂2/∂x2 + x2)mu(t, x) =
=
∑
0≤p+q≤2m
cm
p,qx
p(∂qu(t, x)/∂xq),
where cm
p,q are constant coefficients for which following estimations are valid:
|cm
p,q| ≤ 10mmm−(p+q)/2.
Thus we define equation (3) as the equation of parabolic type with increasing coeffi-
cients.
Corollary 2 from theorem 3 lets us establish Cauchy problem for equation (1) as
described below. For (1) we define initial condition
D
{β}
t u(t, ·)|t=0 = f, (4)
where f ∈ (Sω
ω )′. The solution of Cauchy problem (1), (4) is the solution of equation
(1) which satisfies the initial condition (4) in sence D
{β}
t u(t, ·) → f, t → +0, in space
(Sω
ω )′. The following asssertion is valid.
Theorem 4. Cauchy problem (1),(4) is correctly solved problem in space of initial
data (Sω
ω )′. It’s solution described by formula (2); u(t, ·) ∈ Sω
ω for all t > 0.
References
1. Vladimirov V.S., Equations of mathematics physics, Moscow: Nauka (1976), 528.
2. Gorbachuk V.I., Gorbachuk M.P., Bounded problems for differential-operator equations, Kiev:
Naukova dumka (1984), 284.
3. Gelfand I.M., Shylov G.E., Spaces of based an generalized functions, Moscow: Fizmatgiz (1958),
307.
4. Gorodetsky V.V., Yarmolyk I.I., About the summation of the formal Fourier-Hermite series by the
Abel-Poisson method, Dop. NAN Ukraine (1994), no. 6, 20-26.
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