This is the second part of the series consisting of two papers. Here we investigate the category PE of parabolic equations introduced in the rst 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 speci c kind or to morphisms of speci c kind or both. We investigate this structure using a language developed in the rst paper. An example that deals with nonlinear reaction-di usion equation is discussed in more detail.
Introduction
Lu =
X bij (t; x; u)uij +
i;j
x 2 X; t 2 T; u 2
(
Section 1 of this paper is devoted to the classi cation 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-di usion equation
posed on a Riemannian manifold X equipped with a vector eld . There are two exceptional cases: a(u) =
e uH(u) and a(u) = (u u0) 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 (
The structure of P E and classi cation 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 ve full subcategories PE k of PE , 1 k 5, whose objects are equations that can be written locally in the following form: ut =
X bij (t; x; u) (uij + (t; x; u)uiuj ) + i;j
X bi(t; x; u)ui + q(t; x; u)
i ut = a(t; x; u) X bij (t; x)uij + i;j
2. PE 3 = PE 1 \ PE 2 is closed in PE 1, in PE 2, and in PE .
De nition 1. T PE , QPE , SQPE , AQPE , and E PE are wide subcategories of PE , whose morphisms have the following form:
TPE PE PE 2 4
PE1 QPE QPE′ QPE′′ QPE0′′ QPE0′′c QPE0′′ca
QPE QPE QPE′′ QPE0′′
QPE′
QPE0′′a
QPE0′′ QPE0′′a
SQPEn
SQPE0n SQPE0cn at a ∉ Aexp ∪ Adeg at a ∉ Aeexxpt ∪ Adeexgt
QPE QPE QPE′′
SQPE SQPE SQPE0 SQPE0nb SQPE0cnb SQPE0 SQPE0a
AQPE AQPE SQPE1 SQPEb SQPEnba
2. T PEk is closed in T PE; it is wide and plentiful in PEk, k = 1::5.
De nition 2. The category QPE of quasilinear parabolic equations is the full subcategory of QPE, whose objects are equations of the form ut = X bij(t; x; u)uij + X bi (t; x; u) ui + q(t; x; u); i;j i (QPE) (in a local coordinates). In particular, morphisms of QPE are maps of the form
Denote by Anc (M; ) the set of continuous positive functions a : M ! R that satisfy the condition De ne full subcategories of QPE, whose objects are equations of the following form: (t; x; u) ! (t; y(t; x); '(t; x)u + (t; x)) : 8m 2 M
9u1; u2 a (m; u1) 6= a (m; u2) : 0 0 0 ut = a(t; x; u) X bij(t; x)uij + X bi(t; x; u)ui + q(t; x; u)
i;j i ut = a(t; x; u) X bij(t; x)uij + X bi(t; x; u)ui + q(t; x; u); a 2 Anc (T i;j i X) ut = X bij(t; x)uij + X bi(t; x; u)ui + q(t; x; u)
i;j i ut = a(t; x; u) @X bij(t; x)uij + X bi(t; x)uiA + X i(t; x)ui + q(t; x; u)
i;j i i ut = a(t; x; u) @X bij(t; x)uij + X bi(t; x)uiA + q(t; x; u)
i;j i ut = a(u) @X bij(t; x)uij + X bi(t; x)uiA + X i(t; x)ui + q(t; x; u)
i;j i i ut = X bij(t; x)uij + X bi(t; x)ui + q(t; x; u);
i;j i ut = X bij(t; x)uij + X bi(t; x)ui + q1(t; x)u + q0(t; x); i;j i 1 1 1
(Anc) (QPE0) (QPE0n) (QPE01) (QPE00) (QPE000) (QPE0a0(a)) (QPE010) (QPE010q) where a( ) is a positive function. The family of categories QPE0a0(a) is parameterized by functions a ( ), that is one assigns the category QPE0a0(a) to each continuous positive function a : ! R.
We de ne additionally the full subcategory QPEc 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 we set QPE = \ fQPE : 2 g, QPE = QPE \ QPE . Particularly, QPE000n denotes the intersection QPE0n \ QPE000.
In the same manner as in Remark 2, we can obtain a global function a(u) for any equation from QPE0a0(a), for example, by imposing the condition a(u0) = 1. Such function a(u) is independent of the choice of neighborhood in T X and of local coordinates.
Theorem 3. 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.
6. QPE 010 = QPE 00 \ PE 5 = QPE 00 \ QPE 01 = QPE 0a0 (
Denote by Aexp the set of functions of the form a(u) = e uH(u) and by Adeg the set of functions of the form a(u) = (u u0) H (ln (u u0)), where , u0 are arbitrary constants and H ( ) is arbitrary non-constant periodic function.
Theorem 4.
1. If a 2= Aexp [ Adeg, then QPE 0a0(a) is fully plentiful in QPE 00. 2. QPE 000a(a) is fully plentiful in QPE 0a0(a); if a 2= Aexp [ Adeg, then QPE 000a(a) is fully plentiful in QPE 000. 3. QPE 000ca(a) is fully dense in QPE 0c0a(a). 4. Suppose A is an object of QPE 0a0(a), F : A ! B is a morphism in PE such that there is no object of QPE 0a0(a) isomorphic to B in PE (that is a( ) 2 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; x; u) ! ((t; y(t; x); u + (t; x)); (t; y(t; x); v0 + (u
a 2 Aexp u0) exp ( (t; x))); a 2 Adeg In addition, for each t 2 T and x1; x2 2 X such that y(t; x1) = y(t; x2), the di erence (t; x2) (t; x1) is an integral multiple of H^ , where H^ is the period of periodic function H. The same assertion holds if we replace QPE 0a0(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 = = R, f (u) = 2 + sin u, and f 2 Aexp. It admits both maps (t; x; u) 7! (t; x mod 2 ; u) and (t; x; u) 7! (t; x mod 2 ; u + x). In both cases Y = S1. In the rst 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 ).
De nition 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 0
1 ut = a(t; x; u) @X bij (t; x)uij + i;j
X bi(t; x)uiA + X i(t; x)ui + q(t; x; u); i i (SQPE ) and morphisms are maps of the form (t; x; u) 7! (t; y(x); '(t; x)u + (t; x)). De ne additionally the following full subcategories of SQPE : SQPE = SQPE \ QPE 00, where is one of possible subscripts of QPE 00; SQPE b is the category, whose objects are equations of the form 0
1 ut = a(t; x; u) @X bij (x)uij + i;j
X bi(t; x)uiA + X i(t; x)ui + q(t; x; u): i i (SQPE b)
2. SQPE 0 = SQPE \ QPE 000, SQPE n = SQPE \ QPE 0n0, and SQPE b are closed in SQPE .
4. SQPE 1 = SQPE \ QPE 010 = SQPE a (
5. If a 2= Aexp [ Adeg, then SQPE a(a) is fully plentiful in SQPE .
De nition 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 + ru) + ru + q(x; u) (AQPE ) posed on a Riemann manifold X equipped with vector elds , .
De ne full subcategories AQPE = AQPE \ QPE 00 of AQPE , where The objects of these categories are equations of the form is one of possible subscripts of QPE 00. ut = a(x; u) ( u + ut = a(x; u)( u + ut = a(u)( u + ut = u +
ru) + ru + q(x; u): ru) + ru + q(x; u); ru) + q(x; u); ru + q(x; u); a 2 Anc(X); (AQPE n) (AQPE 0) (AQPE a(a)) (AQPE 1)
4. If a ( ) 2= Aexp [ Adeg, then AQPE a(a) is fully plentiful in AQPE .
De nition 5. De ne the following full subcategories of E PE (its morphisms are maps of the form (t; x; u) 7! (t; y(x); u)):
E PE = E PE \ AQPE ; E PE
= E PE \ AQPE ;
E PE a(a) = E PE \ AQPE a(a):
Denote by Aeexxtp the set of functions a(u) of the form a(u) = e uH(u) and by Aedxetg the set of functions of the form a(u) = (u (that is Aexp
u0) H (ln (u Aeexxtp, Adeg
u0)), where , u0 are arbitrary constants, H( ) is arbitrary periodic function
Aedxetg).
TPE
QPE
QPE ′′ TPE1
QPE ′
QPE ′′ n 3. If a 2= Aexp [ Aedxetg, then E PE a(a) coincides with AQPE a(a).
ext
Let us consider the sequence depicted on Fig. 2. Selecting the \weakest" arrow in this sequence, we obtain the following result.
1. If a 2= Aexp [ Adeg, a 6= const then AQPE 0a(a) is fully plentiful in T PE and plentiful in PE . 2. If a 2= Aexp [ Aedxetg, a 6= const then E PE 0a(a) is fully plentiful in T PE and plentiful in PE .
ext 2
Factorization of the reaction-di usion equation Let us consider a nonlinear reaction-di usion equation for an unknown function u(t; x), u : T X ! Riemann manifold equipped with a vector eld A of PE .
The following two theorems are the immediate corollaries of Theorem 8.
, where T and are open intervals of R and X is a connected
and a function q : X T ! . This equation de nes the object
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 e uH(u) nor in a form (u u0) H(ln(u u0)) with 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 (
X0 ! .
Theorem 10. Let F : A ! B be a morphism of PDE and B be an object of PE . Suppose that either a(u) = a0e u
or a(u) = a0(u u0) for some real constants 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 (
(t; x; u) 7! (t; x0(x); '(x)u + (x)) for some smooth functions ' : X ! Rnf0g, : X ! R, such that the quotient equation B0 is the reaction-di usion equation (3) for an unknown function v : T X0 ! 0, posed on some Riemannian manifold X0 equipped with a vector eld 0 and a function q0 : X0 T ! 0.
, posed on some Riemannian manifold X0 equipped with a vector eld (3)
The map (t; x; u) 7! ( (t); y (t; x) ; v(t; x; u)) is a morphism in PE if and only if 8> tBkl = X bij yikyjl >>>> i;j > > > >>> tCkl = (ln Uv)v Bkl + Uv > > > <
X cij yikyjl i;j > > > > > > > > > > > > > > : tBk =
X bij yikj + 2 X bij (ln Uv)j yik + 2 X cij Uj yik + i;j i;j i;j
X biyik i ytk (4) 0
X cij UiUj + i;j i;j
X biUi + q(t; x; U ) i where function u = U (t; x; v) is the inverse of the v (t; x; u). The quotient equation is written as v = Pk;l Bklvkl + Pk;l Cklvkvl + Pk Bkvk + Q. Here and below indexes i, j relate to x, indexes k, l relate to y.
By de nition, all PE k are full subcategories of PE .
1. Let us prove that PE 1 is closed in PE . Suppose A 2 ObPE1 , F : A ! B is a morphism in PE . Then cij = (t; x; u)bij . From the second equation of system (4) we get
Ckl ( ; y; v) = Bkl ( ; y; v) t 1 (ln Uv)v + (t; x; u) Uv : The quadratic form Bkl is non-degenerated at any point ( ; y; v), so the expression in square brackets is a function of ( ; y; v): t 1 (ln Uv)v + (t; x; u)Uv = ( ; y; v), and Ckl ( ; y; v) = ( ; y; v) Bkl( ; y; v). Thus B 2 ObPE1 .
Let us show that PE 2 is closed in PE . Suppose A 2 ObPE2 , F : A ! B is a morphism in PE . Then bij = a(t; x; u)bij (t; x). Using the rst equation of system (4), we obtain Taking into account that the quadratic form Bkl is non-degenerated, we obtain that B11 6= 0 everywhere. From the equality we obtain that this fraction is function of (t; y). Thus for A ( ; y; v) = B11 ( ; y; v) and some functions Bkl(t; y). Therefore, B 2 ObPE2 .
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 2 ObPE4 and F : A ! B is a morphism of PE . From the rst equation of (4) we obtain that Bkl ( ; y; v) is independent of v. Hence Bkl = Bkl ( ; 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.
1. By de nition, T PE is wide in PE .
Suppose F : A ! B is a morphism in PE . By Theorem 1 from [2], the function (t) is non-degenerated, so we can consider the inverse function t ( ). The map ( ; y; v) ! (t ( ) ; 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.
0
1
X bij yikyjl A i;j (t;x)
: Bkl B11 ( ; y; v) = P
P ii;;jj bbiijjyyi1kyy1jl (t; x)
i j Bkl( ; y; v) = A( ; y; v)Bkl ( ; y)
Using system (4), we see that the map (t; x; u) ! (t; y; 'u + ) is a morphism in QPE if and only if 8> Bkl = > > > > > > <>> Bk = X bij yikyjl X bij yikj + 2 X bij (ln ')j yik + i;j
ytk 1 i 0 X bij 'ij +
X bi'i i;j i
X bij ij +
X bi i i;j i where ' = ' 1, = ' 1 , so U = 'v + . By de nition, all subcategories of QPE considered in the Theorem are full subcategories of QPE .
1a. If cij = 0 and v is linear in u, then Ckl = 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 2 ObQPE . Using the second equation of system (4), we get (ln Uv)v Bkl = Ckl = 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 2 ObT PE1 . Fix u0 2
A and consider the map F : (t; x; u) 7! (t; x; v(t; x; u)), where v(t; x; u) = u Z u0 F de nes an isomorphism in T PE 1 from A to B with
Cij = (ln Uv)v bij + Uv bij = vu 1 ( (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 2 ObQPE00 , and suppose F : A ! B is a morphism in QPE 0. From the rst equation of system (5) we obtain
a(t; x; u) = A(t; y; v)a(t; x); where a(t; x) = B11 (t; y (t; x)) . Pi;j bij (t; x)yi1yj1(t; x) .
From the second equation of (5) we obtain where
Bk(t; y; v) = A(t; y; v)!k (t; x) +
k(t; x); 0
1
X bij yikj + 2 X bij (ln ')j yik + i;j i;j
X biyikA ; i k(t; x) = X iyik
ytk: i Further we will need the following statement: (6) (7) Lemma 1 ((about the extension of a function)). Suppose M , N are Cr-manifolds, 1 r 1, F : M ! N is a surjective Cr-submersion, : M ! R is a Cs-function, 0 s r (if s = 0, then is continuous). Take N0 = nn 2 N :
o jF 1(n) = const ;
M0 = F 1 (N0) = fm 2 M : 8m0 2 M [F (m0) = F (m)] ) [ (m0) = (m)]g ; F0 = F jM0 , 0 = jM0 , and de ne a function 0 : N0 ! R by the formula 0F0 = 0 (see Fig. 3(a)). Then 0 can be extended from N0 to the entire manifold N so that the extended function : N ! R has class Cs of smoothness (see Fig. 3(b); both diagrams Fig. 3(a, b) are commutative).
M0 M 0
F0 = R F
~ a 0 / N0 / / N
M0 M 0
F0
~ = R ` b 0 / N0
N Take an open covering fVi : i 2 Ig of N such that for every Vi there is a Cr-smooth section pi : Vi ! M over Vi, F pi = idjVi (such a covering exists, because F is submersive and surjective). Let f ig be a Cr-partition of unity subordinated to fVig [1, section 2.2]. Let i (n) =
i (n) (pi (n)) ; n 2 Vi : 0; n 2= Vi Then (n) = P i (n) is a desired function.
i2I
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 by k(t; x). We obtain that there exists a continuous function k(t; y) satisfying the following property for each (t0; y0): if k(t; x) is constant on the inverse image of (t0; y0) with respect to the map (t; x) 7! (t; y(t; x)), then k (t0; y0) coincides with this constant. Let now
Bk(t; y; v) = Bk(t; y; v) 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 Bk(t0; y0; v) is independent of v; (8) implies that Bk is independent of v.
Case 2: For given (t0; y0) the set fA(t0; y0; v) : v 2 g contains more than one element. Then (7) implies that the restriction of k(t0; x) to the inverse image of a point (t0; y0) is constant. Thus k(t0; x) = k(t0; y0) on this inverse image, and Bk = !k(t; x) is independent of v in this case too.
In both cases Bk(t; y; v) = A(t; y; v)Bk(t; y) + k(t; y). So, the equation B has the form 0
1 vt = A(t; y; v) @X Bkl(t; y)vkl + X Bk(t; y)vkA + X k(t; y)vk + Q(t; y; v); k;l k k and B is an object of QPE00.
F is a morphism in QPE00 if and only if the following system holds; we will use this system in the proof of the rest of the theorem.
0 X iyik = a(t; x; u) @X bijyikj + 2 X bij (ln ')j yik + X biyik i i;j i;j i 1 Bk=aA (9) abi + i 'i
1 'tA v+
1 abi + i i
i 6. QPE010 is closed in QPE00 and in QPE01, because QPE00 and QPE01 are closed in QPE0. QPE010 is closed in QPE000, because QPE000 is the subcategory of QPE00.
7. Suppose A 2 ObQPE010q , and F : A ! B is a morphism in QPE010. From the third equation of (5) we get 0 @X bij'ij + X bi'i + q1(t; x) i;j i 'tA ' 1v + @X bij ij + X bi i + q0(t; x) i;j i 1 tA ' 1 = 1
Q(t; y; v) =
0
Q1(t; x)v + Q0(t; x); so Q1, Q0 are functions of (t; y), and B 2 ObQPE010q . Thus QPE010q is closed in QPE010.
8. Suppose A 2 ObQPE0n , F : A ! B is a morphism in QPE0. For given (t0; y0) let us x arbitrary x0 such that y (t0; x0) = y0. Since a 2 Anc(T X), from (6) we get
A (t0; y0; v) = a t0; x0; ' (t0; x0) v + (t0; x0) a (t0; x0) 6= const: Finally, we obtain A 2 Anc (T
Y ), and B 2 ObQPE0n , so QPE0n is closed in QPE0. 9. Suppose A 2 ObQPE000n , B 2 ObQPE0n0 . Substituting i = 0 in the third equation of (9), we get 0 ytk + k(t; y) = a(t; x; u) @X bijyikj + 2 X bij (ln ')j yik + X biyik i;j i;j i
1 Bk=aA (t; x): Since left hand side is independent of u and a 2 Anc(T
X), both sides of this equality vanish, and we get ytk = k(t; y) (10) The function y(t; x) satis es the ordinary di erential 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 de ned when T = R; otherwise transformations gs are partially de ned, nevertheless reasoning below remains correct after small re nement.
The composition gsg s is identity for every s , so gs is bijective. fgsg is the ow map of the smooth vector eld @t Pk k(t; y)@yk , so transformations fgsg are smooth by both t and y.
De ne 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 QPE00 such that z(t; y(t; x)) = z(0; y(0; x)) for every x, t. Therefore G F 2 HomQPE000 . 10. Suppose A is an object of QPE0c0. Since X is compact, there exists a solution y : T X ! X of the linear PDE @yk /@t = Pi i(t; x)@yk @xi . Then the isomorphism (t; x; u) 7! (t; y(t; x); u) maps A to some object of QPE000. Thus QPE000c is closed in QPE0c0.
If a 6= const, then QPE000a(a) is fully plentiful in QPE0a0(a) thanks to the part 9 of Theorem 3.
If a = const, then QPE0a0(a) coincides with QPE010, which is closed in QPE00 by Theorem 3. So QPE0a0(a) is fully plentiful in QPE00.
Suppose now that a 6= const, A 2 ObQPE0a0(a), and F : A ! B is a morphism in QPE00. Let us see on equation (6) as a functional one: a '(t; x)v + (t; x) = A(t; y; v)a(t; x): (11) We have three cases:
Case 1. a(u) = He u, ; H = const, and 6= 0. Substituting a(u) to (11), we get '(t; x)v ln A(t; y; v) = ln a ln H . The right hand side of this equality is a function of (t; x), so ' = '(t; y), and the isomorphism (t; y; v) 7! (t; y; '(t; y)v) maps B to some object of QPE0a0(a).
Case 2. a(u) = H (u u0) , ; H; u0 = const, and 6= 0. Substituting a(u) to (11), we get v + ' 1(t; x) (t; x) u0 = A(t; y; v)H 1' a(t; x): Thus ' 1 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 QPE0a0(a).
Case 3. Suppose now that a(u) is neither He u nor H (u u0) . Denote x = (t; x), y = (t; y), = ln a. Fix arbitrary y0 2 T Y and denote Z = fx : y (x) = y0g T X. Since (11), for any x0; x1 2 Z and 'i = ' (xi), i = (xi)) the value '1z + 1 '0z + 0 is independent of v. Let G = G (y0) be the additive subgroup of R generated by the set fln ' (x) ln ' (x0) : x 2 Zg.
We have the following two subcases. w + u0 0 /'0 , for any w we have
eH^1 w + u0 Case 3.1: G 6= f0g. Put H^1 = ln '1 ln '0 2 G f0g, u0 = 0 1 /('1
'0) . Substituting v = (w + u0) = c = const. Consider the function (x) = (ex + u0). Since x + H^1 = (x) + c, for = c=H^1 the function (x)
x is H^1-periodic. Therefore, a(u) = (u u0) H (ln (u u0)) ; where H is H^1-periodic, H 6= const, since the case \H = const" have been considered above. Let H^ > 0 be the smallest positive period of H. For all x 2 Z the number ln ' (x) ln '0 is a multiple of H^ , so ' (x) 2 n'0 exp kH^ : k 2 Zo for any y0. Since a(u) is independent of y0, H^ is independent of y0 too.
Case 3.2: G = f0g, that is 'jZ
'0 = const. Here we have two possible sub-subcases: Case 3.2.a:
Z 6= const, that is 9x0; x1 2 Z : (x1) (x0) = H^1 6= 0. Then u + H^1 (u) = const.
By the same token as in case 3.1 we get a(u) = H(u)e u, where = const and H is a periodic function with the smallest period H^ > 0. Note that such a representation of a(u) is unique. Substituting this to (11), we obtain that 8y 8x0; x1 2 Zy the number (x1) (x0) is a multiple of H^ .
Case 3.2.b: Z = const for given y0. We already considered the cases a(u) = H(u)e u and a(u) = (u u0) H (ln (u u0)), so we can assume now without loss of generality that a is not of this form. Then for every y0 we have Z = const, ' = ' (y), and = (y). Thus the isomorphism (t; y; v) ! t; y; '(t; y)v + (t; y) maps B to some object of QPE0a0 (a).
The proof of the full density of QPE000ca(a) in QPE0c0a (a) is similar to the proof of part 10 in Theorem 3.
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 Bk(t; y; v) = A(t; y; v)!k(t; x), where !k is de ned as in (7). Hence !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 rst equation of (5) we obtain
Bkl B11 (t; y) = P
P ii;;jj bbiijjyyi1kyy1jl (x): i j The right hand side is independent of t, so it is a function of y; denote this function by B0kl(y). Then ABkl = A0(t; y; v)B0kl(y), where A0 = AB11. 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 su cient 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)!k(t; x), where 0 1 !k = Since the left hand side of this equality is independent of v and A 2 Anc(Y ), we conclude that !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 0n0. 4. QPE 010 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.
From (5)-(6) and the fact that SQPE b is closed in SQPE it follows that the map (t; x; u) 7! (t; y; 'u + ) is a morphism in SQPE with the source from AQPE if and only if the following conditions are satis ed: 8 A(t; y; v) > > >>> Bkl(y) > > ><> Bk(t; y; v) > > > >>>> Q' > > : =a(x; u)a(t; x) =a(t; x)rykryl =A(t; y; v)Bk(t; y) + Ck(t; y) = =a(x; u) = (a ( ' + + a yk + ( + 2r (ln ')) ryk +
ryk r') + + r r' + r 't) v+ t + q t; x; 'v + (12) 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 a = a(x). Using the rst equation of (12) and taking into account that ', are independent of t, we see that A = A (y; v) is independent of t. It follows from the third equation of (12) that Bk is independent of t, Bk(y; v) = A(y; v)Bk(t; y) + Ck (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), k(y) such that Bk = A(y; v)Hk(y) + k(y). Substituting u = '(x)v + (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 0
k;l
1 X Hk(y)vkA + X k k k(y)vk + Q(y; v): 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 de ned function A(y; v), for example by the way described in Remark 2. Then quadratic B 2 ObAQPE . a x; '(t; x)v + (t; x) = A (y(x); v) a(x): (13) Let x = x0. Suppose that the set
has more than one element, and consider the intervals Then a (x0; u) is constant on any interval u 2 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 Hausdor metric, and 8t ' (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 S I (v). But this contradicts to the condition a 2 Anc (X). Therefore I (v0) degenerates into a point at some v0, ' (t; x0) v0 + (t; x0) u0, so 'v + = ' (t; x0) (v v0) + u0. By the assumption, card ' (t; x0) ; (t; x0) > 1, so the set f' (t; x0)g is nondegenerated interval. Therefore, a (x0; u) is constant on the sets fu < u0g and fu > u0g. But this contradicts to the condition a 2 Anc(X) and continuity of a. This contradiction shows that for each x0 the functions ', are independent of t. Consequently F is a morphism in AQPE , and AQPE n is the full subcategory of SQPE bn.
are independent of t, F is a morphism in AQPE , and a '(t; x)v +
(t; x) = A(v)a(x):
B 2 ObAQPE \ ObSQPEna(a) = ObAQPEna(a) : Since AQPE na(a) is full in AQPE n, we see that F is a morphism in AQPE na(a).
1. E PE is closed in E PE , because AQPE is closed in AQPE .
Acknowledgements. This work was partially supported by the RFBR grants 15-01-02352 and 15-51-06001 (Russia).