Lomovtsev, F. E.

English

Электронная библиотека БГУ

ISSN 0012-2661, Differential Equations, 2010, Vol. 46, No. 9, pp. 1374–1378. c© Pleiades Publishing, Ltd., 2010.
Original Russian Text c© F.E. Lomovtsev, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 9, pp. 1364–1368.
SHORT COMMUNICATIONS
Boundary Value Problems for Complete Partial
Differential Equations of Variable Order
F. E. Lomovtsev
Belarus State University, Minsk, Belarus
Received January 26, 2009
Abstract—We prove the well-posedness of new boundary value problems for partially pseu-
dodifferential complete nonclassical equations of variable order in space variables with higher
derivatives of odd order in time.
DOI: 10.1134/S0012266110090144
Boundary value problems for parabolic equations in variable-order partial derivatives (in space
variables) were earlier considered for higher derivatives of the first order [1–3] and odd order [4]
in time, for hyperbolic equations with higher derivatives of the second order [3] and even order [5]
in time, for partially pseudodifferential parabolic equations with higher derivatives of the first
order [6] and higher order [7] in time, and for partially pseudodifferential hyperbolic equations with
even-order derivatives in time [8]. In the present paper, we prove the strong well-posedness of new
boundary value problems for partially pseudodifferential complete nonclassical equations of variable
order with odd-order higher derivatives in time. The variable differentiation order depended on the
time in [1–5] and on space points at which the differentiation is performed in [6, 7].
1. STATEMENT OF THE PROBLEM
In the domain G = ]0, T [× Rn, consider the boundary value problems
(−1)m ∂
2m+1u(t, x)
∂t2m+1
+
2m∑
s=1
∂[(s+1)/2]
∂t[(s+1)/2]
(−1)[s/2](I −Δx)ps(t)−εs ∂
[s/2]u(t, x)
∂t[s/2]
+ λm(I −Δx)p0(t)u(t, x) = f(t, x), t ∈ ]0, T [, x = (x1, . . . , xn) ∈ Rn, (1)
∂iu(0, x)/∂ti = ∂ju(T, x)/∂tj = 0,
x ∈ Rn, i = 0, . . . ,m, j = 0, . . . ,m− 1, m = 0, 1, . . . , (2)
where ps(t) = p(t){1 − (s − 1)/(2m)}, 0 < εs < 1/(2m), [a] is the integer part of the number a,
and the fractional partial derivatives with respect to x are defined with the use of the well-known
Fourier–Plancherel (F–P) transforms in L2(Rn) [9, p. 108]. Here the operators
As(t)u(x) = (−1)[s/2]F−1[(1 + |ξ|2)ps(t)−εsF [u](ξ)](x)
with domains D(As(t)) = {u(x) ∈ L2(Rn) : (1 + |ξ|2)ps(t)−εsF [u](ξ) ∈ L2(Rn)}, t ∈ [0, T ], are
pseudodifferential operators of variable order with respect to x [9, pp. 135–136] corresponding to
the symbols as(t, x, ξ) = (1 + |ξ|2)ps(t)−εs , s ≥ 0, ε0 = 0 [10, pp. 94–98 of the Russian translation].
2. EXISTENCE, UNIQUENESS, AND CONTINUOUS DEPENDENCE
The spaces of strong solutions and right-hand sides are the Hilbert spaces Em(G) defined as the
closures of the sets D(Lm) of all functions C2m+1,p
∗
(G), where p∗ = [4maxt∈[0,T ] p(t)]+1, satisfying
1374
BOUNDARY VALUE PROBLEMS FOR COMPLETE PARTIAL DIFFERENTIAL . . . 1375
conditions (2) and compactly supported in Rn in the Hermitian norms
|||u|||m =
{ T∫
0
∫
Rn
(∣∣∣∣
∂dmu(t, x)
∂dtm
∣∣∣∣
2
+ |(I −Δx)p0(t)/2u(t, x)|2
)
dx dt
}1/2
and the Banach spaces F̂−m(G) defined as the closures of the set L2(G) in the norms
〈||f ||〉−m = sup
v∈Em(G)
{∣∣∣∣∣
T∫
0
(T − t)
∫
Rn
fv dx dt
∣∣∣∣∣(|||v|||m)
−1
}
,
respectively. The boundary value problems (1), (2) correspond to linear unbounded operators
Lm(λm) : Em ⊃ D(Lm) → F̂−m with dense domains D(Lm).
Definition 1. The solutions of the operator equations Lm(λm)u = f , f ∈ F̂−m, m = 0, 1, . . . ,
are referred to as strong solutions of the boundary value problems (1), (2).
Theorem 1. If n/2 < p(t) ∈ Cm+1[0, T ], and the derivative p′(t), t ∈ [0, T ], is nonpositive, then
there exist c0(m) > 0, Λ˜0 = [1,+∞[ for m = 0 and Λ˜m = [λ˜m,+∞[, λ˜m ≥ 1, for m > 0, such that ,
for all f ∈ F̂−m(G) and λm ∈ Λ˜m, the boundary value problems (1) and (2) have a unique strong
solution u ∈ Cm−1([0, T ], L2(Rn)) ∩ Em(G), and
|||u(t, x)|||m ≤ c0(m)〈||f(t, x)||〉−m, m = 0, 1, . . . (3)
Proof. The proof amounts to verifying the assumptions of Theorem 2 in [11] for the operators
A(t) = (I −Δx)p(t) with domains D(A(t)) = {u(x) ∈ L2(Rn) : (1 + |ξ|2)p(t)F [u](ξ) ∈ L2(Rn)} and
for the smoothing operators B−1ε (t) = (I − εB(t))−1, ε > 0, B(t) = −A0(t).
I. Maximal accretivity of the operators A0(t). The proof of the self-adjointness and positive
definiteness of the operators A0(t) in L2(Rn) can be found in [8, p. 533].
II. Existence of self-adjoint operators A(t) subordinate to the operators A0(t). By using two
basic properties of the F–P transforms, the Parseval relation 〈h, g〉 = (2π)−n〈F [h], F [g]〉, h, g ∈
L2(Rn), for the inner product 〈· , ·〉 in L2(Rn), and the inversion formulas F−1F [g] = FF−1[g] = g,
g ∈ L2(Rn), we find that inequalities (4) in [11] hold for all t ∈ [0, T ] and u ∈ D(A0(t)) :
Re 〈A0(t)u, u〉 = Re 〈F−1[(1 + |ξ|2)p0(t)F [u]], u〉 = (2π)−n〈(1 + |ξ|2)p0(t)F [u], F [u]〉
≥ (2π)−n〈(1 + |ξ|2)p(t)F [u], F [u]〉 = 〈F−1[(1 + |ξ|2)p(t)F [u]], u〉 = 〈A(t)u, u〉.
Let us compute the strong derivative dA−1(t)/dt of the operators A−1(t) inverse to the opera-
tors A(t). If p(t) > n/2, then, by using the continuity of F−1 with respect to t in L2(Rn) and the
convolution estimate
‖h ∗ g‖L2 ≤ ‖h‖L1‖g‖L2
[9, p. 62] and by performing the differentiation of the operators A−1(t)g = F−1[(1 + |ξ|2)−p(t)] ∗
g ∈ L2(Rn) with respect to t for all g ∈ L2(Rn) [9, pp. 182–185], we prove the existence of
a strong derivative (dA−1(t)/dt)g = −p′(t)(F−1[(1 + |ξ|2)−p(t) ln(1 + |ξ|2)] ∗ g) in L2(Rn), and the
boundedness of this derivative in L2(Rn) follows from the inequalities [8, p. 533] ‖(dA−1(t)/dt)g‖ ≤
(e)−1|p′(t)|‖g‖, where ‖ · ‖ is the norm in L2(Rn), and  > 0 is an arbitrarily small number for
i = 1 in the inequalities lni z ≤ (i/(e))iz, z ≥ 1,  > 0, i = 1, 2, . . . The verification of the
corresponding assumption of Theorem 2 for the derivative dA−1(t)/dt can be found in [8, p. 537].
III. Differentiability and symmetry of the operators As(t), s > 0. The operators A(t) have
positive fractional powers Aα/(2m)(t)u = F−1[(1 + |ξ|2)p(t)α/(2m)F [u]], 0 ≤ α ≤ 2m. The operators
As(t) belong to B([0, T ],L(W 2m+1−s(t),H)), s > 0, since
‖As(t)u‖2 = ‖F−1[(1 + |ξ|2)ps(t)−εsF [u]]‖2 = (2π)−n‖(1 + |ξ|2)ps(t)−εsF [u]‖2
≤ (2π)−n‖(1 + |ξ|2)ps(t)F [u]‖2 = ‖F−1[(1 + |ξ|2)ps(t)F [u]]‖2 = |u|22m+1−s,t ∀u ∈ W 2m+1−s(t).
DIFFERENTIAL EQUATIONS Vol. 46 No. 9 2010
1376 LOMOVTSEV
The first strong derivative of the linear unbounded operators As(t) with variable domains D(As(t))
in the sense of the definition in [11] was computed in [8, p. 535] and is equal to
(dAs(t)/dt)u(t) = (−1)[s/2]cm(s, p)F−1[(1 + |ξ|2)ps(t)−εs ln(1 + |ξ|2)F [u(t)]],
where cm(s, p) = (1− (s−1)/(2m))p′(t). In a similar way, by using two basic properties of the F–P
transforms, one can show that
‖(dAs(t)/dt)u‖2 ≤ c2m(s, p)(2π)−n‖(1 + |ξ|2)ps(t)−εs+F [u]‖2
≤ c2m(s, p)(2π)−n‖(1 + |ξ|2)ps(t)+p(t)τ/(2m)F [u]‖2 ≤ c2m(s, p)|u|22m+1−s+τ,t ∀u ∈ W 2m+1−s+τ (t)
for an arbitrarily small τ > 0; i.e., dAs(t)/dt ∈ B([0, T ],L(W 2m+1−s+τ (t),H)), s > 0, if 0 <  ≤
εs + nτ/(4m) for i = 1. Since p(t) ∈ Cm+1[0, T ], it follows that the remaining strong derivatives of
the operators As(t) can be evaluated recursively in j, and one can show that if 0 <  ≤ εs+nτ/(4m)
for i = j, then djAs(t)/dtj ∈ B([0, T ],L(W 2m+1−s+τ (t),H)), j = 2, . . . , [(s + 1)/2], s > 0.
The operators As(t) satisfy inequalities (5) in [11] for j = 0, 1; for example, if j = 1, then
|〈(dAs(t)/dt)u, v〉| = |cm(s, p)|(2π)n |〈(1 + |ξ|
2)p(t)(1/2−[s/2]/(2m)+ps/(2m))−εs
× ln(1 + |ξ|2)F [u], (1 + |ξ|2)p(t)(1/2−[(s−1)/2]/(2m)−ps/(2m))F [v]〉|
≤ |cm(s, p)||u|m−[s/2]+ps,t|v|m−[(s−1)/2]−ps,t ∀u ∈ Wm−[s/2]+ps(t), ∀v ∈ Wm−[(s−1)/2]−ps(t)
provided that 0 <  ≤ εs, s > 0, for i = 1. Here we have used two basic properties of F–P
transforms. The proof of the symmetry of the operators As(t), s > 0, can be found in [8, p. 535].
Inequalities (6) in [11] hold for the operators As(t), s > 0, since
(−1)(s+2)/2〈As(t)u, u〉 = −〈F−1[(1 + |ξ|2)p(t)(1−(s−1)/(2m))−εsF [u]], u〉
= −(2π)−n〈(1 + |ξ|2)p(t)(1−(s−1)/(2m))−εsF [u], F [u]〉 ≤ 0 ∀u ∈ D(A(t)). (4)
IV. Properties of smoothing operators B−1ε (t) and the corresponding smoothed operators As(t),
s > 0. 1. In L2(Rn), the dissipativity of the operators B(t) = −A0(t) follows from the positive
definiteness of the operators A0(t), and the proof of the boundedness of the strong derivatives
diB−1(t)/dti = diB∗−1(t)/dti, i = 0, . . . ,m + 1, can be found in [8, p. 534]. For example, for
B−1(t)g = −A−10 (t)g = −(F−1[(1 + |ξ|2)−p0(t)] ∗ g), the strong derivative is
(dB−1(t)/dt)g = (1 + 1/(2m))p′(t)(F−1[(1 + |ξ|2)−p0(t) ln(1 + |ξ|2)] ∗ g),
and ‖(dB−1/dt)g‖ ≤ (1 + 1/(2m))|p′(t)|(e)−1‖g‖ for all g ∈ L2(Rn). The boundedness of the
operators As(t)(dB−1(t)/dt) in L2(Rn) follows from the relations
‖As(t)(dB−1(t)/dt)g‖2 = c2m(0, p)‖F−1[(1 + |ξ|2)−p(t)s/(2m)−εs ln(1 + |ξ|2)F [g]]‖2
≤ c2m(0, p)(2π)−n(e)−2‖(1 + |ξ|2)−εsF [g]‖2 ≤ c2m(0, p)(e)−2‖g‖2,
s = 0, . . . , 2m, ∀g ∈ L2(Rn),
if 0 <  ≤ εs for i = 1. Here we have used the F–P transform of the convolution F [h∗g] = F [h]·F [g],
h ∈ L1(Rn), g ∈ L2(Rn) [9, p. 105]. For the remaining strong derivatives diB−1(t)/dti, the proof
of the boundedness of the operators As(t)(diB−1(t))/dti, i = 2, . . . , [s/2], s = 0, . . . , 2m, in L2(Rn)
can be performed in a similar way with the use of three basic properties of F–P transforms.
2. For m = 0, there exists the first limit in (7) in [11]: if u ∈ Wm+1/2(t), then
|Re 〈(dB−1ε (t)/dt)(B−1ε (t))∗u, v〉| = |Re 〈ε(dB−1(t)/dt)B(t)B−2ε (t)u,B(t)B−1ε (t)u〉|
≤ ‖(dB−1(t)/dt)√ε(−B(t))1/2B−2ε (t)(−B(t))1/2u‖‖
√
ε(−B(t))1/2B−1ε (t)(−B(t))1/2u‖ → 0 (5)
DIFFERENTIAL EQUATIONS Vol. 46 No. 9 2010
BOUNDARY VALUE PROBLEMS FOR COMPLETE PARTIAL DIFFERENTIAL . . . 1377
as ε → 0, since the operators dB−1(t)/dt are bounded in L2(Rn), (−B(t))1/2u ∈ L2(Rn), and
limε→0 ‖
√
ε(−B(t))1/2B−1ε (t)g‖ = 0 for all g ∈ L2(Rn), where (−B(t))1/2 is the square root of the
operators A0(t). The validity of inequalities (7) in [11] follows from (3) in [11].
3(a). For m > 0, the limit (8) in [11] does not exist, but inequality (8′) in [11] takes place:
−Re 〈(dB−1ε (t)/dt)(B−1ε (t))∗v, v〉 = −εRe 〈B−1ε (t)(dB(t)/dt)B−1ε (t)(B−1ε (t))∗v, v〉
≤ ε2cm(0, p)(2π)−n〈(1 + |ξ|2)p0(t) ln(1 + |ξ|2)F [B−2ε (t)v], (1 + |ξ|2)p0(t)F [B−2ε (t)v]〉 ≤ 0
for all v ∈ L2(Rn), since cm(0, p) ≤ 0, t ∈ [0, T ], by the assumption of Theorem 1.
By analogy with the transformations (5), we have
‖(d(B−1ε (t))∗/dt)v‖2 = ‖εB(t)B−1ε (t)B−1(t)(dB(t)/dt)B−1ε (t)v‖2
≤ ‖εB(t)B−1ε (t)‖2‖B−1(t)(dB(t)/dt)B−1ε (t)v‖2 → 0 ∀v ∈ W 1(t), (6)
since limε→0 ‖ε(B(t))B−1ε (t)‖ = 0 and the values
‖B−1(t)(dB(t)/dt)B−1ε (t)v‖2
= ‖F−1[(1 + |ξ|2)−p0(t)] ∗ cm(0, p)F−1[(1 + |ξ|2)p0(t) ln(1 + |ξ|2)F [B−1ε (t)v]]‖2
≤ c2m(0, p)‖B−1ε (t)Aδ/(2m)(t)v‖2 ≤ c2m(0, p)‖Aδ/(2m)(t)v‖2 < c2m(0, p)|v|21,t ∀v ∈ W 1(t) (7)
are bounded uniformly with respect to ε provided that δ = 2m/p(t) < 1 for 0 <  ≤ n/(4m)
if i = 1. Recurrently in i one can show that limε→0 ‖(dj(B−1ε (t))∗/dtj)v‖ = 0 for all v ∈ W j−1(t)
provided that 0 <  ≤ n/(4m) for i = j = 2, . . . ,m + 1, and therefore,
lim
ε→0
‖(diB−1ε (t)/dti)(B−1ε (t))∗v‖ = 0
for all v ∈ W i−1(t) and for i = 2, . . . ,m + 1.
By virtue of the estimates (6), as ε → 0 we have
‖(dB−1ε (t)/dt)(d(B−1ε (t))∗/dt)v‖2
≤ ‖εB(t)B−1ε (t)‖2‖B−1(t)(dB(t)/dt)B−1ε (t)(d(B−1ε (t))∗/dt)v‖2 → 0
for all v ∈ W 1(t), since, by virtue of the estimates (7) and (6) and the commutativity of Aδ/(2m)(t)
with B(t),
‖B−1(t)(dB(t)/dt)B−1ε (t)(d(B−1ε (t))∗/dt)v‖2 ≤ c2m(p)|(d(B−1ε (t))∗/dt)v|2δ,t,
|(d(B−1ε (t))∗/dt)v|2δ,t ≤ ‖εB(t)B−1ε (t)‖2|B−1(t)(dB(t)/dt)B−1ε (t)v|2δ,t → 0, (8)
|B−1(t)(dB(t)/dt)B−1ε (t)v|2δ,t ≤ c2m(0, p)‖B−1ε (t)A2δ/(2m)(t)v‖2 < c2m(0, p)|v|21,t ∀v ∈ W 1(t), (9)
if 2δ = 2m/p(t) < 1 for 0 <  ≤ n/(4m) and i = 1. Recurrently with respect to i′ and j, one can
show that if (i′ + j)δ = 2m/p(t) < i′+ j − 1 for 0 <  ≤ n(i′ + j− 1)/(4m) and for i = max{i′, j},
then limε→0 ‖(di′B−1ε (t)/dti
′
)(djB−1ε (t)/dt
j)v‖ = 0 for all v ∈ W i′+j−1(t), j = 1, . . . ,m+1− i′, and
i′ = 1, . . . ,m + 1.
3(b). For all v˜ ∈ Wm−k(t), there exists a limit (9) in [11] as ε → 0 :
|〈A2k+1(t)(dB−1ε (t)/dt)(B−1ε (t))∗v˜, v˜〉| = |〈F−1[(1 + |ξ|2)p(t)(1/2−k/(2m))−ε2k+1
× F [(dB−1ε (t)/dt)(B−1ε (t))∗v˜]], F−1[(1 + |ξ|2)p(t)(1/2−k/(2m))F [v˜]]〉|
≤ |(dB−1ε (t)/dt)(B−1ε (t))∗v˜|m−k−δ′k,t|v˜|m−k,t → 0,
since, by virtue of the commutativity of the operators A(t) with B(t), the estimates (8) hold for
δ′ = m−k− δ′k, δ′k = 2mε2k+1/maxt∈[0,T ] p(t) ≥ 0, k = 0, . . . ,m−1, v = (B−1ε (t))∗v˜, and, by virtue
of the estimate (9),
|(dB−1ε (t)/dt)(B−1ε (t))∗v˜|2m−k−δ′k,t ≤ c
2
m(0, p)‖B−2ε (t)A(m−k−δ
′
k+
′)/(2m)(t)v˜‖2
≤ c2m(0, p)‖A(m−k−δ
′
k+
′)/(2m)(t)v˜‖2 ≤ c2m(0, p)|v˜|2m−k
DIFFERENTIAL EQUATIONS Vol. 46 No. 9 2010
1378 LOMOVTSEV
provided that 0 < ′ ≤ δ′k, k = 0, . . . ,m − 1, ′ = 2m/maxt∈[0,T ] p(t) ≥ 0, for i = 1. In the same
way, one can show that
lim
ε→0
〈A2k+1(t)B−1ε (t)(d(B−1ε (t))∗/dt)v, v〉 = 0,
lim
ε→0
〈A2k(t)(dB−1ε (t)(B−1ε (t))∗/dt)v,w〉 = 0
for all v ∈ Wm−k(t), for all w ∈ Wm−k+1(t), and for k = 1, . . . ,m−1. If ε → 0 strongly in Wm−k(t),
then
|(d(B−1ε (t))∗/dt)v|2m−k,t ≤ ‖εB(t)B−1ε (t)‖2|B−1(t)(dB(t)/dt)B−1ε (t)v|2m−k,t → 0
by virtue of the estimates (8) and the inequality [which is a consequence of the estimate (9)]
|B−1(t)(dB(t)/dt)B−1ε (t)v|2m−k,t ≤ c2m(0, p)‖B−1ε (t)A(m−k+δ)/(2m)(t)v‖2 < c2m(0, p)|v|2m−k+1,t
for all v ∈ Wm−k+1(t), k = 1, . . . ,m− 1, provided that δ = 2m/p(t) < 1 for 0 <  ≤ n/(4m) and
for i = 1. One can readily show that
lim
ε→0
|(dj(B−1ε (t))∗/dtj)v|m−k,t = 0 ∀v ∈ Wm−k+1(t)
if 0 <  ≤ n/(4m) for i = j = 2, . . . , k and k = 2, . . . ,m− 1;
lim
ε→0
|(djB−1ε (t)(B−1ε (t))∗/dtj)v|m−k,t = 0 ∀v ∈ Wm−k+1(t)
if jδ = 2m/p(t) < m− k + 1 for 0 <  ≤ n(m− k + 1)/(4m) and for j = 1, 2, k = 1, . . . ,m, and
for all v ∈ Wm−k+j−1(t), if jδ = 2m/p(t) < m − k + j − 1 for 0 <  ≤ n(m − k + j − 1)/(4m),
i = j = 2, . . . , k + 1, and k = 2, . . . ,m. The property limε→0 ‖B−1ε (t)w − w‖ = 0, w ∈ L2(Rn), and
the commutativity of the operators A(t) and B(t) imply the existence of the limits (7) and (8).
By virtue of the estimates (4), we have inequalities (12′) from [11]. The limits (13) in [11] exist for
all s = 1, . . . , 2m and are zero for all ε > 0. Inequalities (3) correspond to inequalities (14) in [11].
The proof of Theorem 1 is complete.
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DIFFERENTIAL EQUATIONS Vol. 46 No. 9 2010


Boundary Value Problems for Complete Partial Differential Equations of Variable Order

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Boundary Value Problems for Complete Partial Differential Equations of Variable Order

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