=Paper=
{{Paper
|id=Vol-1662/top2
|storemode=property
|title=The structure of the category of parabolic equations. II
|pdfUrl=https://ceur-ws.org/Vol-1662/top2.pdf
|volume=Vol-1662
|authors=Marina Prokhorova
}}
==The structure of the category of parabolic equations. II==
https://ceur-ws.org/Vol-1662/top2.pdf
The structure of the category of parabolic equations. II
Marina Prokhorova
pmf@imm.uran.ru
Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Russia)
Ural Federal University (Yekaterinburg, Russia)
Abstract
This is the second part of the series consisting of two papers. Here we
investigate the category PE of parabolic equations introduced in the
first paper. The objects of this category are second order parabolic
equations posed on arbitrary manifolds, and the morphisms generalize
the notion of the quotient map by a symmetry group. We introduce a
certain structure in PE formed by the lattice of subcategories. These
subcategories are obtained by the restricting to equations of specific
kind or to morphisms of specific kind or both. We investigate this
structure using a language developed in the first paper. An example
that deals with nonlinear reaction-diffusion equation is discussed in
more detail.
Introduction
This paper is the second part of the series of two papers. In the first part [2] the author defined the category PDE
of partial differential equations and its full subcategory PE that arises from second order parabolic equations on
arbitrary manifolds. This paper is devoted to the investigation of the internal structure of the category PE by
means of the special-purpose language developed in [2, section 4].
Recall the definition of the category of parabolic equations from [2, section 5]. Let us consider the class
P (X, T, Ω) of differential operators on a connected smooth manifold X, which depend additionally on a parameter
t (“time”), locally having the form
X X X
Lu = bij (t, x, u)uij + cij (t, x, u)ui uj + bi (t, x, u)ui + q(t, x, u),
i,j i,j i
x ∈ X, t ∈ T, u ∈ Ω
in some neighborhood of each point, in some (and then arbitrary) local coordinates xi on X. Here subscript i
denotes partial derivative with respect to xi , quadratic form bij = bji is positive definite, and cij = cji . Both T
and Ω may be bounded, semi-bounded or unbounded open intervals of R. The category PE of parabolic equations
is a subcategory of PDE, whose objects are pairs A = (N, E), N = T × X × Ω, where X is a connected smooth
manifold, T and Ω are open intervals, E is an equation of the form ut = Lu, L ∈ P (X, T, Ω). Theorem 1 of [2]
asserts that every morphism in PE has the form
(t, x, u) 7→ (t0 (t), x0 (t, x), u0 (t, x, u)), (1)
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the 47th International Youth School-conference “Modern Problems in
Mathematics and its Applications”, Yekaterinburg, Russia, 02-Feb-2016, published at http://ceur-ws.org
134
with submersive t0 (t), x0 (t, x), and u0 (t, x, u). Isomorphisms in PE are exactly diffeomorphisms of the form (1).
Section 1 of this paper is devoted to the classification of parabolic equations in this framework and to the
description of the internal structure of PE. The proofs of Theorems 1-7 given in the section are postponed to
Section 3.
Section 2 illustrates the using of this structure of PE on the example of the reaction-diffusion equation
ut = a(u) (∆u + η∇u) + q(x, u), x ∈ X, t ∈ R, (2)
posed on a Riemannian manifold X equipped with a vector field η. There are two exceptional cases: a(u) =
eλu H(u) and a(u) = (u − u0 )λ H(ln(u − u0 )), where H(·) is a periodic function; in these cases there are more
morphisms then in a regular case. If only function a(u) does not belong to one of these two exceptional classes,
then Theorems 9-10 assert that every morphism from equation (2) may be transformed by an isomorphism (i.e.
by a bijective global change of variables) of the quotient equation to the “canonical” morphism of very simple
kind so that the “canonical” quotient equation has the same form as (2) with the same function a(u) but is posed
on another Riemannian manifold X 0 , dim X 0 ≤ dim X.
1 The structure of PE and classification of parabolic equations
We formulate here the number of theorems describing the internal structure of PE; the proofs of these theorems
are given in Section 3 below. Certain parts of the structure of PE are depicted schematically on Fig. 1 (the full
picture is not given here in view of its awkwardness).
Let us consider five full subcategories PE k of PE, 1 ≤ k ≤ 5, whose objects are equations that can be written
locally in the following form:
X X
ut = bij (t, x, u) (uij + λ(t, x, u)ui uj ) + bi (t, x, u)ui + q(t, x, u) (PE 1 )
i,j i
X X X
ij ij
ut = a(t, x, u) b̄ (t, x)uij + c (t, x, u)ui uj + bi (t, x, u)ui + q(t, x, u) (PE 2 )
i,j i,j i
X X
ut = a(t, x, u) b̄ij (t, x) (uij + λ(t, x, u)ui uj ) + bi (t, x, u)ui + q(t, x, u) (PE 3 )
i,j i
X X X
ut = bij (t, x)uij + cij (t, x, u)ui uj + bi (t, x, u)ui + q(t, x, u) (PE 4 )
i,j i,j i
X X
ut = bij (t, x) (uij + λ(t, x, u)ui uj ) + bi (t, x, u)ui + q(t, x, u) (PE 5 )
i,j i
Remark 1. Everywhere in the paper we use notation of a category equipped with a subscript and/or primes
for its full subcategory. For example, QPE k , QPE 0 , and QPE 0k defined below are full subcategories of QPE.
Remark 2. In equations of the categories PE 2 and PE 3 , function a(·) is determined up to multiplication by
arbitrary function from T × X to R+ ; moreover, it is determined only locally. Nevertheless we can lead these
equations to the equations of the same form but with globally defined function a : T ×X ×Ω → R+ . For example,
we can require that a(t, x, u0 ) ≡ 1, where u0 is a fixed point of Ω. Everywhere below we will assume that function
a is globally determined on T × X × Ω.
Theorem 1.
1. PE 1 and PE 2 are closed in PE.
2. PE 3 = PE 1 ∩ PE 2 is closed in PE 1 , in PE 2 , and in PE.
3. PE 4 is closed in PE 2 and in PE.
4. PE 5 = PE 3 ∩ PE 4 is closed in PE 3 , in PE 4 , and in PE.
Definition 1. T PE, QPE, SQPE, AQPE, and EPE are wide subcategories of PE, whose morphisms have the
following form:
(t, y(t, x), v (t, x, u)) for T PE
(t, y(t, x), ϕ(t, x)u + ψ(t, x)) for QPE
(t, x, u) → (t, y(x), ϕ(t, x)u + ψ(t, x)) for SQPE
(t, y(x), ϕ(x)u + ψ(x)) for AQPE
(t, y(x), u) for EPE
135
PE TPE QPE
PE 2 PE1 TPE1 QPE
PE 4 PE3 TPE3 QPE ′ QPE ′
PE5 TPE5 QPE1′
QPE ′′ QPE n′′
QPE1′′ QPE a′′
QPE 0′′
QPE SQPE AQPE
QPE1′′q QPE 0′′a
QPE
QPE c′′ QPE ′′ SQPE AQPE
′′
QPE 0c QPE0′′ SQPE0 AQPE0
′′
QPE ca QPE a′′ SQPE a AQPE a
′′
QPE 0ca QPE0′′a SQPE0 a AQPE0 a
SQPE na AQPE na
QPE SQPE AQPE EPE
QPE
QPE ′′ SQPE AQPE EPE
QPE n′′ SQPE n SQPE1
AQPE1 EPE1
QPE0′′ SQPE0
′′
QPE cn
QPE0′′n SQPE0 n SQPEb
SQPE nb AQPE n EPE n
′′
QPE0cn SQPE 0cn
SQPE nba AQPE na EPE na
SQPE0nb
at a ∉ Aexp ∪ Adeg AQPE0na EPE0 na
SQPE 0cnb SQPE0nba
ext ext
at a ∉ Aexp ∪ Adeg SQPE0cnba AQPE0cna EPE0cna
Figure 1: The part of the structure of the category of parabolic equations
136
Denote T PE k = T PE ∩ PE k .
Theorem 2.
1. T PE is wide and plentiful in PE.
2. T PE k is closed in T PE; it is wide and plentiful in PE k , k = 1..5.
Definition 2. The category QPE of quasilinear parabolic equations is the full subcategory of QPE, whose objects
are equations of the form
X X
ut = bij (t, x, u)uij + bi (t, x, u) ui + q(t, x, u), (QPE)
i,j i
(in a local coordinates). In particular, morphisms of QPE are maps of the form
(t, x, u) → (t, y(t, x), ϕ(t, x)u + ψ(t, x)) .
Denote by Anc (M, Ω) the set of continuous positive functions a : M × Ω → R that satisfy the condition
∀m ∈ M ∃u1 , u2 a (m, u1 ) 6= a (m, u2 ) . (Anc )
Define full subcategories of QPE, whose objects are equations of the following form:
X X
ut = a(t, x, u) b̄ij (t, x)uij + bi (t, x, u)ui + q(t, x, u) (QPE 0 )
i,j i
X X
ut = a(t, x, u) ij
b̄ (t, x)uij + bi (t, x, u)ui + q(t, x, u), a ∈ Anc (T × X) (QPE 0n )
i,j i
X X
ut = bij (t, x)uij + bi (t, x, u)ui + q(t, x, u) (QPE 01 )
i,j i
X X X
ut = a(t, x, u) b̄ij (t, x)uij + b̄i (t, x)ui + ξ i (t, x)ui + q(t, x, u) (QPE 00 )
i,j i i
X X
ut = a(t, x, u) b̄ij (t, x)uij + b̄i (t, x)ui + q(t, x, u) (QPE 000 )
i,j i
X X X
ut = a(u) b̄ij (t, x)uij + b̄i (t, x)ui + ξ i (t, x)ui + q(t, x, u) (QPE 00a (a))
i,j i i
X X
ut = bij (t, x)uij + bi (t, x)ui + q(t, x, u), (QPE 001 )
i,j i
X X
ut = ij
b (t, x)uij + bi (t, x)ui + q1 (t, x)u + q0 (t, x), (QPE 001q )
i,j i
where a(·) is a positive function. The family of categories QPE 00a (a) is parameterized by functions a (·), that is
one assigns the category QPE 00a (a) to each continuous positive function a : Ω → R.
We define additionally the full subcategory QPE c of QPE, whose objects are equations from QPE posed on
a compact manifolds X.
Let us introduce the following notation for the intersections of enumerated “basic” subcategories: for a string
σ we set QPE σ = ∩ {QPE α : α ∈ σ}, QPE βσ = QPE σ ∩ QPE β . Particularly, QPE 000n denotes the intersection
QPE 0n ∩ QPE 000 .
In the same manner as in Remark 2, we can obtain a global function a(u) for any equation from QPE 00a (a), for
example, by imposing the condition a(u0 ) = 1. Such function a(u) is independent of the choice of neighborhood
in T × X × Ω and of local coordinates.
Theorem 3.
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1. QPE is closed in QPE and is fully dense in T PE 1 .
2. QPE c is closed in QPE.
3. QPE 0 = QPE ∩ PE 2 = QPE ∩ PE 3 is fully dense in T PE 3 and is closed in QPE.
4. QPE 01 = QPE ∩ PE 5 = QPE 0 ∩ PE 5 is fully dense in T PE 5 and is closed in QPE 0 .
5. QPE 00 is closed in QPE 0 .
6. QPE 001 = QPE 00 ∩ PE 5 = QPE 00 ∩ QPE 01 = QPE 00a (1) is closed in QPE 01 , in QPE 00 , and in QPE 000 .
7. QPE 001q is closed in QPE 001 .
8. QPE 0n is closed in QPE 0 .
9. QPE 000n is fully plentiful in QPE 00n .
10. QPE 000c is fully dense in QPE 00c .
Denote by Aexp the set of functions of the form a(u) = eλu H(u) and by Adeg the set of functions of the form
λ
a(u) = (u − u0 ) H (ln (u − u0 )), where λ, u0 are arbitrary constants and H (·) is arbitrary non-constant periodic
function.
Theorem 4.
/ Aexp ∪ Adeg , then QPE 00a (a) is fully plentiful in QPE 00 .
1. If a ∈
2. QPE 000a (a) is fully plentiful in QPE 00a (a); if a ∈
/ Aexp ∪ Adeg , then QPE 000a (a) is fully plentiful in QPE 000 .
3. QPE 000ca (a) is fully dense in QPE 00ca (a).
4. Suppose A is an object of QPE 00a (a), F : A → B is a morphism in PE such that there is no object of QPE 00a (a)
isomorphic to B in PE (that is a(·) ∈ Aexp ∪ Adeg ). Then there exists an object of QPE 00 isomorphic to B
such that the composition of F : A → B with this isomorphism is of the form
(
(t, y(t, x), u + ψ(t, x)), a ∈ Aexp
(t, x, u) →
(t, y(t, x), v0 + (u − u0 ) exp (ψ(t, x))), a ∈ Adeg
In addition, for each t ∈ T and x1 , x2 ∈ X such that y(t, x1 ) = y(t, x2 ), the difference ψ(t, x2 ) − ψ(t, x1 )
is an integral multiple of Ĥ, where Ĥ is the period of periodic function H. The same assertion holds if we
replace QPE 00a (a) by QPE 000a (a) and QPE 00 by QPE 000 .
Example. The equation
E : ut = (2 + sin u) uxx
is an object of QPE 000a (f ), with X = T = Ω = R, f (u) = 2 + sin u, and f ∈ Aexp . It admits both maps
(t, x, u) 7→ (t, x mod 2π, u) and (t, x, u) 7→ (t, x mod 2π, u + x). In both cases Y = S 1 . In the first case the
quotient equation has the form vt = (2 + sin v) vyy , so it is an object of QPE 000a (f ). In the second case the
quotient equation has the form vt = (2 + sin(v + y)) vyy ; it is an object of QPE 000 , but is not isomorphic to any
object of QPE 000a (f ).
Definition 3. The category of semi-autonomous quasilinear parabolic equations SQPE is the intersection
SQPE ∩ QPE 00 . In other words, SQPE is the full subcategory of SQPE and the wide subcategory of QPE 00 ,
whose objects are equations of the form
X X X
ut = a(t, x, u) b̄ij (t, x)uij + b̄i (t, x)ui + ξ i (t, x)ui + q(t, x, u), (SQPE)
i,j i i
and morphisms are maps of the form (t, x, u) 7→ (t, y(x), ϕ(t, x)u + ψ(t, x)). Define additionally the following full
subcategories of SQPE:
138
SQPE σ = SQPE ∩ QPE 00σ , where σ is one of possible subscripts of QPE 00 ;
SQPE b is the category, whose objects are equations of the form
X X X
ut = a(t, x, u) b̄ij (x)uij + b̄i (t, x)ui + ξ i (t, x)ui + q(t, x, u). (SQPE b )
i,j i i
Theorem 5.
1. SQPE is closed in SQPE.
2. SQPE 0 = SQPE ∩ QPE 000 , SQPE n = SQPE ∩ QPE 00n , and SQPE b are closed in SQPE.
3. SQPE 0n coincides with QPE 000n ; it is closed in SQPE 0 and in SQPE n .
4. SQPE 1 = SQPE ∩ QPE 001 = SQPE a (1) is closed in SQPE 0 .
5. If a ∈
/ Aexp ∪ Adeg , then SQPE a (a) is fully plentiful in SQPE.
Definition 4. The category of autonomous quasilinear parabolic equations AQPE is the full subcategory of
AQPE, whose objects are equations of the form
ut = a(x, u) (∆u + η∇u) + ξ∇u + q(x, u) (AQPE)
posed on a Riemann manifold X equipped with vector fields ξ, η.
Define full subcategories AQPE σ = AQPE ∩ QPE 00σ of AQPE, where σ is one of possible subscripts of QPE 00 .
The objects of these categories are equations of the form
ut = a(x, u) (∆u + η∇u) + ξ∇u + q(x, u), a ∈ Anc (X), (AQPE n )
ut = a(x, u)(∆u + η∇u) + q(x, u), (AQPE 0 )
ut = a(u)(∆u + η∇u) + ξ∇u + q(x, u), (AQPE a (a))
ut = ∆u + ξ∇u + q(x, u). (AQPE 1 )
Theorem 6.
1. AQPE is closed in AQPE.
2. AQPE n is closed in AQPE and full in SQPE bn .
3. AQPE 0 and AQPE 1 are closed in AQPE.
4. If a (·) ∈
/ Aexp ∪ Adeg , then AQPE a (a) is fully plentiful in AQPE.
5. AQPE na (a) is closed in SQPE na (a).
Definition 5. Define the following full subcategories of EPE (its morphisms are maps of the form (t, x, u) 7→
(t, y(x), u)):
EPE = EPE ∩ AQPE,
EPE σ = EPE ∩ AQPE σ ,
EPE a (a) = EPE ∩ AQPE a (a).
Denote by Aext λu ext
exp the set of functions a(u) of the form a(u) = e H(u) and by Adeg the set of functions of the
λ
form a(u) = (u − u0 ) H (ln (u − u0 )), where λ, u0 are arbitrary constants, H(·) is arbitrary periodic function
(that is Aexp ⊂ Aext ext
exp , Adeg ⊂ Adeg ).
Theorem 7.
1. EPE is closed in EPE and wide in AQPE.
2. EPE n , EPE 0 , EPE 1 , and EPE a (a) are closed in EPE.
139
PE TPE QPE QPE ′′ QPE 0′′n SQPE 0 nb AQPE 0 na
TPE1 QPE ′ QPE n′′ SQPE 0 n SQPE 0 nba EPE 0 na
Figure 2: The sequence of arrows from PE to EPE 0na (a)
/ Aext
3. If a ∈ ext
exp ∪ Adeg , then EPE a (a) coincides with AQPE a (a).
Let us consider the sequence depicted on Fig. 2. Selecting the “weakest” arrow in this sequence, we obtain
the following result.
Theorem 8.
1. If a ∈
/ Aexp ∪ Adeg , a 6= const then AQPE 0a (a) is fully plentiful in T PE and plentiful in PE.
/ Aext
2. If a ∈ ext
exp ∪ Adeg , a 6= const then EPE 0a (a) is fully plentiful in T PE and plentiful in PE.
2 Factorization of the reaction-diffusion equation
Let us consider a nonlinear reaction-diffusion equation
ut = a(u) (∆u + η∇u) + q(x, u)
for an unknown function u(t, x), u : T × X → Ω, where T and Ω are open intervals of R and X is a connected
Riemann manifold equipped with a vector field η and a function q : X × T → Ω. This equation defines the object
A of PE.
The following two theorems are the immediate corollaries of Theorem 8.
Theorem 9. Let F : A → B be a morphism of PDE and B be an object of PE. Suppose that a(u) can be written
neither in a form eλu H(u) nor in a form (u − u0 )λ H(ln(u − u0 )) with λ 6= 0, u0 being real constants, H(·) being a
periodic function. Then there exists an isomorphism I : B → B0 of PE (in other words, a bijective global change
of variables of the form (1) in the quotient equation) transforming F to the morphism I ◦ F of the form
(t, x, u) 7→ (t, x0 (x), u)
such that the quotient equation B0 is the reaction-diffusion equation
vt = a(v) (∆v + η 0 ∇v) + q 0 (x0 , v) (3)
for an unknown function v : T × X 0 → Ω, posed on some Riemannian manifold X 0 equipped with a vector field
ξ 0 and a function q 0 : X 0 × T → Ω.
Theorem 10. Let F : A → B be a morphism of PDE and B be an object of PE. Suppose that either a(u) = a0 eλu
or a(u) = a0 (u − u0 )λ for some real constants λ 6= 0, u0 , a0 . Then there exists an isomorphism I : B → B0 of
PE (in other words, a bijective global change of variables of the form (1) in the quotient equation) transforming
F to the morphism I ◦ F of the form
(t, x, u) 7→ (t, x0 (x), ϕ(x)u + ψ(x))
for some smooth functions ϕ : X → R\{0}, ψ : X → R, such that the quotient equation B 0 is the reaction-diffusion
equation (3) for an unknown function v : T × X 0 → Ω0 , posed on some Riemannian manifold X 0 equipped with a
vector field η 0 and a function q 0 : X 0 × T → Ω0 .
140
3 Proofs of Theorems 1-8
Proof of Theorem 1
The map (t, x, u) 7→ (τ (t), y (t, x) , v(t, x, u)) is a morphism in PE if and only if
X
τt B kl = bij yik yjl
i,j
X
kl kl
cij yik yjl
τ C = (ln U ) B + U
t v v
v
i,j
(4)
X X X X
k ij k
τt B = b yij + 2 b (ln Uv )j yik + 2
ij
cij Uj yik + bi yik − ytk
i,j i,j i,j i
X X X
τt Q = U −1
bij Uij + cij Ui Uj + bi Ui + q(t, x, U ) − Ut
v
i,j i,j i
where functionPu = U (t, x, v) P is the inverse of the v (t, x, u). The quotient equation is written as vτ =
kl kl k
P
k,l B v kl + k,l C v k v l + k B v k + Q. Here and below indexes i, j relate to x, indexes k, l relate to
y.
By definition, all PE k are full subcategories of PE.
1. Let us prove that PE 1 is closed in PE. Suppose A ∈ ObPE 1 , F : A → B is a morphism in PE. Then
cij = λ(t, x, u)bij . From the second equation of system (4) we get
C kl (τ, y, v) = B kl (τ, y, v) τt−1 (ln Uv )v + λ (t, x, u) Uv .
The quadratic form B kl is non-degenerated at any point (τ, y, v), so the expression in square brackets is a function
of (τ, y, v): τt−1 (ln Uv )v + λ(t, x, u)Uv = Λ (τ, y, v), and C kl (τ, y, v) = Λ (τ, y, v) B kl (τ, y, v). Thus B ∈ ObPE 1 .
Let us show that PE 2 is closed in PE. Suppose A ∈ ObPE 2 , F : A → B is a morphism in PE. Then
bij = a(t, x, u)b̄ij (t, x). Using the first equation of system (4), we obtain
X
τt B kl = a(t, x, u) b̄ij yik yjl .
i,j
(t,x)
Taking into account that the quadratic form B kl is non-degenerated, we obtain that B 11 6= 0 everywhere. From
the equality
ij k l
P
B kl i,j b̄ yi yj
(τ, y, v) = ij 1 1
(t, x)
B 11
P
i,j b̄ yi yj
we obtain that this fraction is function of (t, y). Thus
B kl (τ, y, v) = A(τ, y, v)B̄ kl (τ, y)
for A (τ, y, v) = B 11 (τ, y, v) and some functions B̄ kl (t, y). Therefore, B ∈ ObPE 2 .
2. PE 3 = PE 1 ∩ PE 2 is closed in PE, in PE 1 , and in PE 2 , because PE 1 and PE 2 are closed in PE.
3. Suppose A ∈ ObPE 4 and F : A → B is a morphism of PE. From the first equation of (4) we obtain that
B kl (τ, y, v) is independent of v. Hence B kl = B kl (τ, y), PE 4 is closed in PE, so it is closed in PE 2 too.
4. Since PE 3 and PE 4 are closed in PE, we obtain that PE 5 = PE 3 ∩ PE 4 is closed in PE, PE 3 and PE 4 .
Proof of Theorem 2
1. By definition, T PE is wide in PE.
Suppose F : A → B is a morphism in PE. By Theorem 1 from [2], the function τ (t) is non-degenerated, so
we can consider the inverse function t (τ ). The map (τ, y, v) → (t (τ ) , y, v) is an isomorphism in PE. Note that
the superposition of F with this isomorphism is a morphism in T PE. Therefore T PE is plentiful in PE.
2. T PE k is closed in PE, while T PE is wide and plentiful in PE. Thus T PE k = PE k ∩ T PE is closed in T PE
and also it is wide and plentiful in PE k .
141
Proof of Theorem 3
Using system (4), we see that the map (t, x, u) → (t, y, ϕu + ψ) is a morphism in QPE if and only if
kl X ij k l
B = b yi yj
i,j
X X X
Bk = bij yij
k
bij (ln ϕ̄)j yik + bi yik − ytk
+2
i,j i,j i , (5)
X X X X
ij i ij i
Qϕ̄ = b ϕ̄ij + b ϕ̄ i − ϕ̄t
v + b ψ̄ ij + b ψ̄ i − ψ̄ t
+ q t, x, ϕ̄v + ψ̄
i,j i i,j i
where ϕ̄ = ϕ−1 , ψ̄ = −ϕ−1 ψ, so U = ϕ̄v + ψ̄. By definition, all subcategories of QPE considered in the Theorem
are full subcategories of QPE.
1a. If cij = 0 and v is linear in u, then C kl = 0. It follows from the second equation of system (4) that QPE
is closed in QPE.
1b. Let F : A → B, (t, x, u) 7→ (t, y (t, x) , v(t, x, u)) be a morphism in T PE 1 , and A, B ∈ ObQPE . Using the
second equation of system (4), we get (ln Uv )v B kl = C kl = 0. It follows that U is linear in v, v is linear in u, F
is a morphism in QPE, and QPE is full in T PE.
1c. Suppose A ∈ ObT PE 1 . Fix u0 ∈ ΩA and consider the map F : (t, x, u) 7→ (t, x, v(t, x, u)), where
ξ
Zu Z
v(t, x, u) = exp λ (t, x, ς) dς dξ.
u0 u0
F defines an isomorphism in T PE 1 from A to B with
C ij = (ln Uv )v bij + Uv λbij = vu−1 (λ − (ln vu )u ) = 0.
Therefore every object of T PE 1 is isomorphic in T PE 1 to some object of QPE, and QPE is full in T PE 1 .
2. The image of a compact under a continuous map is compact. The surjectivity of the map completes the
proof.
3. T PE 3 is closed in PE 1 , QPE is fully dense in PE 1 .
4. T PE 5 is closed in T PE 3 , and QPE 0 is fully dense in T PE 3 . Equality QPE 01 = T PE 5 ∩ QPE 0 completes
the proof.
5. Let A ∈ ObQPE 00 , and suppose F : A → B is a morphism in QPE 0 . From the first equation of system (5)
we obtain
a(t, x, u) = A(t, y, v)ā(t, x), (6)
.P
where ā(t, x) = B 11 (t, y (t, x)) ij 1 1
i,j b (t, x)yi yj (t, x) .
From the second equation of (5) we obtain
B k (t, y, v) = A(t, y, v)ω k (t, x) + µk (t, x), (7)
where
X X X X
ω k (t, x) = ā b̄ij yij
k
+2 b̄ij (ln ϕ̄) yik + j b̄i yik , µk (t, x) = ξ i yik − ytk .
i,j i,j i i
Further we will need the following statement:
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Lemma 1 ((about the extension of a function)). Suppose M , N are C r -manifolds, 1 ≤ r ≤ ∞, F : M → N is
a surjective C r -submersion, µ : M → R is a C s -function, 0 ≤ s ≤ r (if s = 0, then µ is continuous). Take
n o
N0 = n ∈ N : µ|F −1 (n) = const ,
M0 = F −1 (N0 ) = {m ∈ M : ∀m0 ∈ M [F (m0 ) = F (m)] ⇒ [µ (m0 ) = µ (m)]} ,
F0 = F |M0 , µ0 = µ|M0 , and define a function ν0 : N0 → R by the formula ν0 F0 = µ0 (see Fig. 3(a)). Then
ν0 can be extended from N0 to the entire manifold N so that the extended function ν : N → R has class C s of
smoothness (see Fig. 3(b); both diagrams Fig. 3(a, b) are commutative).
M0
F0
/ N0 M0
F0
/ N0
µ0 ν0 µ0 ν0
~ ~
=R =R`
µ µ ν
M
F //N M N
a b
Figure 3: The extension of a function
Proof of Lemma 1
Take an open covering {Vi : i ∈ I} of N such that for every Vi there is a C r -smooth section pi : Vi → M over Vi ,
F ◦ pi = id|Vi (such a covering exists, because F is submersive and surjective). Let {λi } be a C r -partition of
unity subordinated to {Vi } [1, section 2.2]. Let
λi (n) µ (pi (n)) , n ∈ Vi
νi (n) = .
0, n∈/ Vi
P
Then ν (n) = νi (n) is a desired function.
i∈I
Proof of Theorem 3 (continuation)
Fix k. In the notations and assumption of Lemma 1, replace F by the map (t, x) 7→ (t, y(t, x)) and the continuous
function µ by µk (t, x). We obtain that there exists a continuous function ν k (t, y) satisfying the following property
for each (t0 , y0 ): if µk (t, x) is constant on the inverse image of (t0 , y0 ) with respect to the map (t, x) 7→ (t, y(t, x)),
then ν k (t0 , y0 ) coincides with this constant. Let now
B̄ k (t, y, v) = B k (t, y, v) − ν k (t, y) /A(t, y, v) .
(8)
Consider the following two cases for every point (t0 , y0 ):
Case 1: The function A(t0 , y0 , v) is independent of v. Then (7) implies that B k (t0 , y0 , v) is independent of v;
(8) implies that B̄ k is independent of v.
Case 2: For given (t0 , y0 ) the set {A(t0 , y0 , v) : v ∈ Ω} contains more than one element. Then (7) implies that
the restriction of µk (t0 , x) to the inverse image of a point (t0 , y0 ) is constant. Thus µk (t0 , x) = ν k (t0 , y0 ) on this
inverse image, and B̄ k = ω k (t, x) is independent of v in this case too.
In both cases B k (t, y, v) = A(t, y, v)B̄ k (t, y) + ν k (t, y). So, the equation B has the form
X X X
vt = A(t, y, v) B̄ kl (t, y)vkl + B̄ k (t, y)vk + ν k (t, y)vk + Q(t, y, v),
k,l k k
143
and B is an object of QPE 00 .
F is a morphism in QPE 00 if and only if the following system holds; we will use this system in the proof of
the rest of the theorem.
a(t, x, u) = A(t, y, v)ā(t, x)
X
B̄ kl (t, y) = ā b̄ij yik yjl (t, x)
i,j
X X X X
y k + Ξk − ξ i yik = a(t, x, u) b̄ij yij
k
+2 b̄ij (ln ϕ̄)j yik + b̄i yik − B k /ā
t
i i,j i,j i
(9)
X X
ab̄ij ϕ̄ij + ab̄i + ξi ϕ̄i − ϕ̄t v+
Qϕ̄ =
i,j i
X X
ab̄ij ψ̄ij + ab̄i + ξi ψ̄i − ψ̄t + q t, x, ϕ̄v + ψ̄
+
i,j i
6. QPE 001 is closed in QPE 00 and in QPE 01 , because QPE 00 and QPE 01 are closed in QPE 0 . QPE 001 is closed in
QPE 000 , because QPE 000 is the subcategory of QPE 00 .
7. Suppose A ∈ ObQPE 001q , and F : A → B is a morphism in QPE 001 . From the third equation of (5) we get
Q(t, y, v) =
X X X X
bij ϕ̄ij + bi ϕ̄i + q1 (t, x) − ϕ̄t ϕ̄−1 v + bij ψ̄ij + bi ψ̄i + q0 (t, x) − ψ̄t ϕ̄−1 =
i,j i i,j i
Q1 (t, x)v + Q0 (t, x),
so Q1 , Q0 are functions of (t, y), and B ∈ ObQPE 001q . Thus QPE 001q is closed in QPE 001 .
8. Suppose A ∈ ObQPE 0n , F : A → B is a morphism in QPE 0 . For given (t0 , y0 ) let us fix arbitrary x0 such
that y (t0 , x0 ) = y0 . Since a ∈ Anc (T × X), from (6) we get
A (t0 , y0 , v) = a t0 , x0 , ϕ̄ (t0 , x0 ) v + ψ̄ (t0 , x0 ) ā (t0 , x0 ) 6= const.
Finally, we obtain A ∈ Anc (T × Y ), and B ∈ ObQPE 0n , so QPE 0n is closed in QPE 0 .
9. Suppose A ∈ ObQPE 000n , B ∈ ObQPE 00n . Substituting ξi = 0 in the third equation of (9), we get
X X X
ytk + Ξk (t, y) = a(t, x, u) b̄ij yij
k
+2 b̄ij (ln ϕ̄)j yik + b̄i yik − B k /ā (t, x).
i,j i,j i
Since left hand side is independent of u and a ∈ Anc (T × X), both sides of this equality vanish, and we get
ytk = −Ξk (t, y) (10)
The function y(t, x) satisfies the ordinary differential equation (10) with smooth right hand side, so for any
t, t0 the equality y(t, x1 ) = y(t, x2 ) implies that y(t0 , x1 ) = y(t0 , x2 ). Let 1-parameter transformation group
gs : T × Y → T × Y be given by (t, y(t, x)) 7→ (t + s, y(t + s, x)). This group is correctly defined when T = R;
otherwise transformations gs are partially defined, nevertheless reasoning below remains correct after small
refinement.
The composition gs g−s is identity for every s , so gs is bijective. {gs } is the flow map of the smooth vector
field ∂t − k Ξk (t, y)∂yk , so transformations {gs } are smooth by both t and y.
P
Define the map z(t, y) by the equality g−t (t, y) = (0, z(t, y)). Then the map G : T × Y → T × Y , (t, y) 7→
(t, z(t, y)) is an isomorphism in QPE 00 such that z(t, y(t, x)) = z(0, y(0, x)) for every x, t. Therefore G ◦ F ∈
HomQPE 000 .
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10. Suppose APis an object of QPE 00c . Since X is compact, there exists a solution y : T × X → X of the linear
PDE ∂y k /∂t = i ξ i (t, x)∂y k ∂xi . Then the isomorphism (t, x, u) 7→ (t, y(t, x), u) maps A to some object of
QPE 000 . Thus QPE 000c is closed in QPE 00c .
Proof of Theorem 4
If a 6= const, then QPE 000a (a) is fully plentiful in QPE 00a (a) thanks to the part 9 of Theorem 3.
If a = const, then QPE 00a (a) coincides with QPE 001 , which is closed in QPE 00 by Theorem 3. So QPE 00a (a) is
fully plentiful in QPE 00 .
Suppose now that a 6= const, A ∈ ObQPE 00a (a) , and F : A → B is a morphism in QPE 00 . Let us see on equation
(6) as a functional one:
a ϕ̄(t, x)v + ψ̄ (t, x) = A(t, y, v)ā(t, x). (11)
We have three cases:
λu
Case 1. a(u) = He , λ, H = const, and λ 6= 0. Substituting a(u) to (11), we get λϕ̄(t, x)v − ln A(t, y, v) =
ln ā − λψ̄ − ln H . The right hand side of this equality is a function of (t, x), so ϕ̄ = ϕ̄(t, y), and the isomorphism
(t, y, v) 7→ (t, y, ϕ̄(t, y)v) maps B to some object of QPE 00a (a).
λ
Case 2. a(u) = H (u − u0 ) , λ, H, u0 = const, and λ 6= 0. Substituting a(u) to (11), we get
λ
v + ϕ̄−1 (t, x) ψ̄(t, x) − u0 = A(t, y, v)H −1 ϕ̄−λ ā(t, x).
Thus ϕ̄−1 ψ̄ − u0 = q(t, y) for some function q, so the object B maps by the isomorphism (t, y, v) 7→
(t, y, v + q(t, y) + u0 ) to some object of QPE 00a (a).
λ
Case 3. Suppose now that a(u) is neither Heλu nor H (u − u0 ) . Denote x̄ = (t, x), ȳ = (t, y), α = ln a. Fix
arbitrary ȳ0 ∈ T × Y and denote Z = {x̄ : ȳ (x̄) =ȳ0 } ⊂ T × X. Since (11), for any x̄0 , x̄1 ∈ Z and ϕ̄i = ϕ̄ (x̄i ),
ψ̄i = ψ̄ (x̄i )) the value α ϕ̄1 z + ψ̄1 − α ϕ̄0 z + ψ̄0 is independent of v. Let G = G (ȳ0 ) be the additive subgroup
of R generated by the set {ln ϕ̄ (x̄) − ln ϕ̄ (x̄0 ) : x̄ ∈ Z}.
We have the following two subcases.
Case 3.1: G 6= {0}. Put Ĥ1 = ln ϕ̄1 − ln ϕ̄0 ∈ G − {0}, u0 = ψ̄0 − ψ̄1 /(ϕ̄1 − ϕ̄0 ) . Substituting v =
w + u0 − ψ̄0 /ϕ̄0 , for any w we have α eĤ1 w + u0 − α (w + u0 ) = c = const. Consider the function β (x) =
α (ex + u0 ). Since β x + Ĥ1 = β(x) + c, for λ = c/Ĥ1 the function β (x) − λx is Ĥ1 -periodic. Therefore,
λ
a(u) = (u − u0 ) H (ln (u − u0 )) ,
where H is Ĥ1 -periodic, H 6= const, since the case “H = const” have been considered above. Let Ĥ > 0 be
the
n smallest
positive period
o of H. For all x̄ ∈ Z the number ln ϕ̄ (x̄) − ln ϕ̄0 is a multiple of Ĥ, so ϕ̄ (x̄) ∈
ϕ̄0 exp k Ĥ : k ∈ Z for any ȳ0 . Since a(u) is independent of ȳ0 , Ĥ is independent of ȳ0 too.
Case 3.2: G = {0}, that is ϕ̄|Z ≡ ϕ̄0 = const. Here we have two possible sub-subcases:
Case 3.2.a: ψ̄ Z 6= const, that is ∃x̄0 , x̄1 ∈ Z : ψ̄ (x̄1 ) − ψ̄ (x̄0 ) = Ĥ1 6= 0. Then α u + Ĥ1 − α(u) = const.
By the same token as in case 3.1 we get a(u) = H(u)eλu , where λ = const and H is a periodic function with the
smallest period Ĥ > 0. Note that such a representation of a(u) is unique. Substituting this to (11), we obtain
that ∀ȳ ∀x̄0 , x̄1 ∈ Zȳ the number ψ̄ (x̄1 ) − ψ̄ (x̄0 ) is a multiple of Ĥ.
Case 3.2.b: ψ̄ Z = const for given ȳ0 . We already considered the cases a(u) = H(u)eλu and a(u) =
λ
(u − u0 ) H (ln (u − u0 )), so we can assume now without loss of generality that a is not of this form. Then for ev-
ery ȳ0 we have ψ̄ Z = const, ϕ̄ = ϕ̄ (ȳ), and ψ̄ = ψ̄ (ȳ). Thus the isomorphism (t, y, v) → t, y, ϕ̄(t, y)v + ψ̄ (t, y)
maps B to some object of QPE 00a (a).
The proof of the full density of QPE 000ca (a) in QPE 00ca (a) is similar to the proof of part 10 in Theorem 3.
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Proof of Theorem 5
1. QPE 00 is closed in QPE, and SQPE is the subcategory of QPE. Therefore SQPE is closed in SQPE.
2. SQPE n is closed in SQPE for the same reason as in Part 1 of this Theorem. This implies that SQPE n is
closed in SQPE.
Suppose A is an object of SQPE 0 , F : A → B is a morphism in SQPE. Then B k (t, y, v) = A(t, y, v)ω k (t, x),
where ω k is defined as in (7). Hence ω k is a function of (t, y), and B is an object of SQPE 0 .
Suppose A is an object of SQPE b , F : A → B is a morphism in SQPE. From the first equation of (5) we
obtain
ij k l
P
B̄ kl i,j b̄ yi yj
(t, y) = P ij 1 1
(x).
B̄ 11 i,j b̄ yi yj
The right hand side is independent of t, so it is a function of y; denote this function by B̄ 0kl (y). Then AB̄ kl =
A0 (t, y, v)B̄ 0kl (y), where A0 = AB 11 . It follows that B is an object of SQPE b , and SQPE b is closed in SQPE.
3. Let us recall that SQPE 0n is closed in QPE 000n . So it is sufficient to prove that any morphism in QPE 000n is also
a morphism in SQPE 0n . Suppose that F : A → B is a morphism in QPE 000n . Then ytk (t, x) = A(t, y, v)ω k (t, x),
where
X X X
ω k = −B̄ k + ā b̄ij yij
k
+2 b̄ij (ln ϕ̄)j yik + b̄i yik .
i,j i,j i,j
Since the left hand side of this equality is independent of v and A ∈ Anc (Y ), we conclude that ω k = 0. Thus F
is a morphism in SQPE 0n . Finally, SQPE 0n = QPE 000n , is closed in QPE 000 and is fully dense in QPE 00n .
4. QPE 001 is closed in QPE, so SQPE 1 is closed in SQPE and, consequently, is closed in SQPE 0 .
5. The proof is similar to the proof of part 1 of Theorem 4.
Proof of Theorem 6
From (5)-(6) and the fact that SQPE b is closed in SQPE it follows that the map (t, x, u) 7→ (t, y, ϕu + ψ) is a
morphism in SQPE with the source from AQPE if and only if the following conditions are satisfied:
A(t, y, v) =a(x, u)ā(t, x)
kl
=ā(t, x)∇y k ∇y l
B̄ (y)
B k (t, y, v) =A(t, y, v)B̄ k (t, y) + C k (t, y) =
(12)
=a(x, u) ∆y k + (η + 2∇ (ln ϕ̄)) ∇y k + ξ∇y k
Qϕ̄ = (a (∆ϕ̄ + η∇ϕ̄) + ξ∇ϕ̄ − ϕ̄t ) v+
+ a ∆ψ̄ + η∇ψ̄ + ξ∇ψ̄ − ψ̄t + q t, x, ϕ̄v + ψ̄
1. Suppose F : A → B is a morphism in AQPE, A is an object of AQPE. From the second equation of
system (12) it follows that ā = ā(x). Using the first equation of (12) and taking into account that ϕ̄, ψ̄ are
independent of t, we see that A = A (y, v) is independent of t. It follows from the third equation of (12) that
B k is independent of t, B k (y, v) = A(y, v)B̄ k (t, y) + C k (t, y). From this formula, by the same token as in the
proof of part 4 of Theorem 3, we obtain existing of functions Hk (y), Ξk (y) such that B k = A(y, v)Hk (y) + Ξk (y).
Substituting u = ϕ̄(x)v + ψ̄(x) in the last equation of (12), we obtain that Q is independent of t. This implies
that the target B of the morphism F has the form
X X X
vt = A(y, v) B̄ kl (y)vkl + Hk (y)vk + Ξk (y)vk + Q(y, v).
k,l k k
We prove so far that B has such a form only locally. Nevertheless, we can lead it to an equation of the same
form but with globally defined function A(y, v), for example by the way described in Remark 2. Then quadratic
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form B̄ kl is defined on the whole manifold Y , so we can equip Y with a Riemannian metric B̄ kl and finally get
B ∈ ObAQPE .
2. AQPE n = AQPE ∩ SQPE n is closed in AQPE, because SQPE n is closed in SQPE.
Let F : A → B be a morphism in SQPE bn , and both source and target of F are objects of AQPE n . Then ā
is independent of t, and
a x, ϕ̄(t, x)v + ψ̄ (t, x) = A (y(x), v) ā(x). (13)
Let x = x0 . Suppose that the set ϕ̄ (t, x0 ) , ψ̄ (t, x0 ) has more than one element, and consider the intervals
I (v) = ϕ̄ (t, x0 ) v + ψ̄ (t, x0 ) : t ∈ TA ⊆ R.
Then a (x0 , u) is constant on any interval u ∈ I (v), because the right hand side of (13) is independent of t.
Note that I (v) is a continuous function of v in the Hausdorff metric, andS ∀t ϕ̄ (t, x0 ) 6= 0. If at any v the
interval I (v) does not collapses into a point, then a (x0 , u) is constant on I (v). But this contradicts to the
condition a ∈ Anc (X). Therefore I (v0 ) degenerates into a point at some v0 , ϕ̄ (t, x0 ) v0 + ψ̄ (t, x0 ) ≡ u0 , so
ϕ̄v + ψ̄ = ϕ̄ (t, x0 ) (v − v0 ) + u0 . By the assumption, card ϕ̄ (t, x0 ) , ψ̄ (t, x0 ) > 1, so the set {ϕ̄ (t, x0 )} is non-
degenerated interval. Therefore, a (x0 , u) is constant on the sets {u < u0 } and {u > u0 }. But this contradicts to
the condition a ∈ Anc (X) and continuity of a. This contradiction shows that for each x0 the functions ϕ̄, ψ̄ are
independent of t. Consequently F is a morphism in AQPE, and AQPE n is the full subcategory of SQPE bn .
3. Since SQPE 0 and SQPE 1 are closed in SQPE, the subcategories AQPE 0 and AQPE 1 are closed in AQPE.
4. If a ∈
/ Aexp ∪ Adeg , then AQPE a (a) is plentiful in AQPE by the same arguments as used in the proof of
part 1 of Theorem 4, after replacement of x̄, ȳ to x, y respectively.
5. Let F : A → B be a morphism in SQPE na (a), A be an object of AQPE na (a). Then
a ϕ̄(t, x)v + ψ̄(t, x) = A(v)ā(x).
As we proved in part 2, the functions ϕ̄, ψ̄ are independent of t, F is a morphism in AQPE, and
B ∈ ObAQPE ∩ ObSQPE na (a) = ObAQPE na (a) .
Since AQPE na (a) is full in AQPE n , we see that F is a morphism in AQPE na (a).
Proof of Theorem 7
1. EPE is closed in EPE, because AQPE is closed in AQPE.
2. EPE n , EPE 0 , EPE 1 are closed in EPE, because AQPE n , AQPE 0 , AQPE 1 are closed in AQPE.
Suppose F : A → B is a morphism in EPE and A ∈ ObEPE a (a) . Then the first equation of (5) has the
form A(y, u)B̄ kl (y) = a(u)∇y k ∇y l . Hence ∇y k ∇y l = g kl (y) for some functions g kl . For B̄ kl = g kl (y) we have
A(y, u) = a(u). So A is an object of EPE a (a), and EPE a (a) is closed in EPE.
3. The proof is similar to the proof of Theorem 3.
Acknowledgements. This work was partially supported by the RFBR grants 15-01-02352 and 15-51-06001
(Russia).
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[2] M. Prokhorova. The structure of the category of parabolic equations. I. Proceedings of the 47th International
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Russia, 2016.
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