Linear and nonlinear heat equations on a $p$ -adic ball
We study the Vladimirov fractional differentiation operator $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, on a $p$-adic ball B$B_N = \{ x \in Q_p : | x|_p \leq p^N\}$. To its known interpretations via the restriction of a similar operator to $Q_p$ and via a certain stochastic process on $B_N$,...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507347513769984 |
|---|---|
| author | Kochubei, A. N. Кочубей, А. Н. |
| author_facet | Kochubei, A. N. Кочубей, А. Н. |
| author_sort | Kochubei, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:18:03Z |
| description | We study the Vladimirov fractional differentiation operator $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, on a $p$-adic ball B$B_N = \{ x \in Q_p : | x|_p \leq p^N\}$. To its known interpretations via the restriction of a similar operator to $Q_p$ and via a certain stochastic process
on $B_N$, we add an interpretation as a pseudodifferential operator in terms of the Pontryagin duality on the additive group
of $B_N$. We investigate the Green function of $D^{\alpha}_N$ and a nonlinear equation on $B_N$, an analog of the classical equation of
porous medium. |
| first_indexed | 2026-03-24T02:07:52Z |
| format | Article |
| fulltext |
UDC 517.9
A. N. Kochubei (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
LINEAR AND NONLINEAR HEAT EQUATIONS ON A \bfitp -ADIC BALL*
ЛIНIЙНЕ ТА НЕЛIНIЙНЕ РIВНЯННЯ ТЕПЛОПРОВIДНОСТI
НА \bfitp -АДИЧНIЙ КУЛI
We study the Vladimirov fractional differentiation operator D\alpha
N , \alpha > 0, N \in \BbbZ , on a p-adic ball BN = \{ x \in \BbbQ p :
| x| p \leq pN\} . To its known interpretations via the restriction of a similar operator to \BbbQ p and via a certain stochastic process
on BN , we add an interpretation as a pseudodifferential operator in terms of the Pontryagin duality on the additive group
of BN . We investigate the Green function of D\alpha
N and a nonlinear equation on BN , an analog of the classical equation of
porous medium.
Вивчається оператор Владимирова диференцiювання дробового порядку D\alpha
N , \alpha > 0, N \in \BbbZ , на p-адичнiй кулi
BN = \{ x \in \BbbQ p : | x| p \leq pN\} . До його вiдомих iнтерпретацiй у термiнах звуження подiбного оператора, визначеного
на \BbbQ p та через деякий випадковий процес на BN , ми додаємо iнтерпретацiю у виглядi псевдодиференцiального
оператора в термiнах дуальностi Понтрягiна на адитивнiй групi BN . Вивчено функцiю Грiна на D\alpha
N та нелiнiйне
рiвняння на BN , що є аналогом класичного рiвняння пористого середовища.
1. Introduction. The theory of linear parabolic equations for real- or complex-valued functions on
the field \BbbQ p of p-adic numbers including the construction of a fundamental solution, investigation
of the Cauchy problem, the parametrix method, is well-developed; see, for example, the monographs
[16, 21]. In such equations, the time variable is real and nonnegative while the spatial variables are p-
adic. There are no differential operators acting on complex-valued functions on \BbbQ p , but there is a lot
of pseudodifferential operators. A typical example is Vladimirov’s fractional differentiation operator
D\alpha , \alpha > 0; see the details below. This operator (as well as its multidimensional generalization,
the so-called Taibleson operator) is a p-adic counterpart of the fractional Laplacian ( - \Delta )\alpha /2 of real
analysis.
Already in real analysis, an interpretation of nonlocal operators on bounded domains is not
straightforward; see [3] for a survey of various possibilities. In the p-adic case, Vladimirov (see
[19]) defined a version D\alpha
N of the fractional differentiation on a ball BN = \{ x \in \BbbQ p : | x| p \leq pN\}
as follows. One takes a test function on BN , extends it onto \BbbQ p by zero, applies D\alpha , and restricts
the resulting function to BN . Then it is possible to consider a closure of the obtained operator, for
example, on L2(BN ).
In [16] (Section 4.6), a probabilistic interpretation of this operator was given. Let \xi \alpha (t) be the
Markov process with the generator D\alpha on \BbbQ p . Suppose that \xi \alpha (0) \in BN and denote by \xi (N)
\alpha (t) the
sum of all jumps of the process \xi \alpha (\tau ), \tau \in [0, t] whose p-adic absolute values exceed pN . Consider
the process \eta \alpha (t) = \xi \alpha (t) - \xi (N)
\alpha (t). Due to the ultrametric inequality, the jumps of \eta \alpha never exceed
pN by absolute value, so that the process remains almost surely in BN . It is proved in [16] that the
generator of the Markov process \eta \alpha on BN equals (on test function) D\alpha
N - \lambda I, where
\lambda =
p - 1
p\alpha +1 - 1
p\alpha (1 - N).
In [16] (Theorem 4.9) the corresponding heat kernel is given explicitly.
* This work was supported in part by Grant 23/16-18 “Statistical dynamics, generalized Fokker – Planck equations, and
their applications in the theory of complex systems” of the Ministry of Education and Science of Ukraine.
c\bigcirc A. N. KOCHUBEI, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2 193
194 A. N. KOCHUBEI
In this paper we find an analytic interpretation of the latter operator using harmonic analysis
on BN as an (additive) compact Abelian group (this group property, just as the above probabilistic
construction, is of purely non-Archimedean nature and has no analogs in the classical theory of partial
differential equations). We give an interpretation of D\alpha
N - \lambda I as a pseudodifferential operator on
BN , then consider it as an operator on L1(BN ) and study its Green function, the integral kernel of
its resolvent. The choice of L1(BN ) as the basic space is motivated by applications to nonlinear
equations.
The first model example of a nonlinear parabolic equation over \BbbQ p is the p-adic analog of the
classical porous medium equation:
\partial u
\partial t
+D\alpha (\Phi (u)) = 0, u = u(t, x), t > 0, x \in \BbbQ p, (1.1)
where \Phi is a strictly monotone increasing continuous real function on \BbbR . Its study was initiated in
[13]. Here we consider this equation on BN , taking the operator D\alpha
N instead of D\alpha :
\partial u
\partial t
+D\alpha
N (\Phi (u)) = 0. (1.2)
As in [13], our study of Eq. (1.2) is based on general results by Crandall – Pierre [10] and
Brézis – Strauss [6] enabling us to consider this equation in the framework of nonlinear semigroups
of operators. Following [3] we consider Eq. (1.2) also in L\gamma (BN ), 1 < \gamma \leq \infty .
An important motivation of the present work is provided by the p-adic model of a poropus
medium introduced in [14, 15].
2. Preliminaries. 2.1. p-Adic numbers [19]. Let p be a prime number. The field of p-adic
numbers is the completion \BbbQ p of the field \BbbQ of rational numbers, with respect to the absolute value
| x| p defined by setting | 0| p = 0,
| x| p = p - \nu if x = p\nu
m
n
,
where \nu ,m, n \in \BbbZ , and m,n are prime to p. \BbbQ p is a locally compact topological field. By
Ostrowski’s theorem there are no absolute values on \BbbQ , which are not equivalent to the “Euclidean”
one, or one of | \cdot | p .
The absolute value | x| p , x \in \BbbQ p , has the following properties:
| x| p = 0 if and only if x = 0,
| xy| p = | x| p \cdot | y| p,
| x+ y| p \leq \mathrm{m}\mathrm{a}\mathrm{x}(| x| p, | y| p).
The latter property called the ultrametric inequality (or the non-Archimedean property) implies
the total disconnectedness of \BbbQ p in the topology determined by the metric | x - y| p , as well as many
unusual geometric properties. Note also the following consequence of the ultrametric inequality:
| x+ y| p = \mathrm{m}\mathrm{a}\mathrm{x}(| x| p, | y| p), if | x| p \not = | y| p .
The absolute value | x| p takes the discrete set of non-zero values pN , N \in \BbbZ . If | x| p = pN , then
x admits a (unique) canonical representation
x = p - N
\bigl(
x0 + x1p+ x2p
2 + . . .
\bigr)
, (2.1)
where x0, x1, x2, . . . \in \{ 0, 1, . . . , p - 1\} , x0 \not = 0. The series converges in the topology of \BbbQ p . For
example,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 195
- 1 = (p - 1) + (p - 1)p+ (p - 1)p2 + . . . , | - 1| p = 1.
We denote \BbbZ p = \{ x \in \BbbQ p : | x| p \leq 1\} . \BbbZ p , as well as all balls in \BbbQ p , is simultaneously open and
closed.
Proceeding from the canonical representation (2.1) of an element x \in \BbbQ p , one can define the
fractional part of x as the rational number
\{ x\} p =
\Biggl\{
0, if N \leq 0 or x = 0,
p - N
\bigl(
x0 + x1p+ . . .+ xN - 1p
N - 1
\bigr)
, if N > 0.
The function \chi (x) = \mathrm{e}\mathrm{x}\mathrm{p}(2\pi i\{ x\} p) is an additive character of the field \BbbQ p, that is a character of its
additive group. It is clear that \chi (x) = 1 if and only if | x| p \leq 1.
Denote by dx the Haar measure on the additive group of \BbbQ p normalized by the equality
\int
\BbbZ p
dx =
= 1.
The additive group of \BbbQ p is self-dual, so that the Fourier transform of a complex-valued function
f \in L1(\BbbQ p) is again a function on \BbbQ p defined as
(\scrF f)(\xi ) =
\int
\BbbQ p
\chi (x\xi )f(x) dx.
If \scrF f \in L1(\BbbQ p), then we have the inversion formula
f(x) =
\int
\BbbQ p
\chi ( - x\xi ) \widetilde f(\xi ) d\xi .
It is possible to extend \scrF from L1(\BbbQ p) \cap L2(\BbbQ p) to a unitary operator on L2(\BbbQ p), so that the
Plancherel identity holds in this case.
In order to define distributions on \BbbQ p , we have to specify a class of test functions. A function f :
\BbbQ p \rightarrow \BbbC is called locally constant if there exists such an integer l \geq 0 that for any x \in \BbbQ p
f(x+ x\prime ) = f(x) if \| x\prime \| \leq p - l.
The smallest number l with this property is called the exponent of local constancy of the function f .
Typical examples of locally constant functions are additive characters, and also cutoff functions
like
\Omega (x) =
\Biggl\{
1, if \| x\| \leq 1,
0, if \| x\| > 1.
In particular, \Omega is continuous, which is an expression of the non-Archimedean properties of \BbbQ p .
Denote by \scrD (\BbbQ p) the vector space of all locally constant functions with compact supports. Note
that \scrD (\BbbQ p) is dense in Lq(\BbbQ p) for each q \in [1,\infty ). In order to furnish \scrD (\BbbQ p) with a topology,
consider first the subspace Dl
N \subset \scrD (\BbbQ p) consisting of functions with supports in a ball
BN = \{ x \in \BbbQ p : | x| p \leq pN\} , N \in \BbbZ ,
and the exponents of local constancy \leq l. This space is finite-dimensional and possesses a natural
direct product topology. Then the topology in \scrD (\BbbQ p) is defined as the double inductive limit topology,
so that
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
196 A. N. KOCHUBEI
\scrD (\BbbQ p) = \mathrm{l}\mathrm{i}\mathrm{m} - \rightarrow
N\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} - \rightarrow
l\rightarrow \infty
Dl
N .
If V \subset \BbbQ p is an open set, the space \scrD (V ) of test functions on V is defined as a subspace of
\scrD (\BbbQ p) consisting of functions with supports in V . For a ball V = BN , we can identify \scrD (BN ) with
the set of all locally constant functions on BN .
The space \scrD \prime (\BbbQ p) of Bruhat – Schwartz distributions on \BbbQ p is defined as a strong conjugate space
to \scrD (\BbbQ p).
In contrast to the classical situation, the Fourier transform is a linear automorphism of the space
\scrD (\BbbQ p). By duality, \scrF is extended to a linear automorphism of \scrD \prime (\BbbQ p). For a detailed theory of
convolutions and direct products of distributions on \BbbQ p closely connected with the theory of their
Fourier transforms see [1, 16, 19].
2.2. Vladimirov’s operator [1, 16, 19]. The Vladimirov operator D\alpha , \alpha > 0, of fractional
differentiation, is defined first as a pseudodifferential operator with the symbol | \xi | \alpha p :
(D\alpha u)(x) = \scrF - 1
\xi \rightarrow x
\bigl[
| \xi | \alpha p\scrF y\rightarrow \xi u
\bigr]
, u \in \scrD (\BbbQ p), (2.2)
where we show arguments of functions and their direct/inverse Fourier transforms. There is also a
hypersingular integral representation giving the same result on \scrD (\BbbQ p) but making sense on much
wider classes of functions (for example, bounded locally constant functions)
(D\alpha u) (x) =
1 - p\alpha
1 - p - \alpha - 1
\int
\BbbQ p
| y| - \alpha - 1
p [u(x - y) - u(x)] dy. (2.3)
The Cauchy problem for the heat-like equation
\partial u
\partial t
+D\alpha u = 0, u(0, x) = \psi (x), x \in \BbbQ p, t > 0,
is a model example for the theory of p-adic parabolic equations. If \psi is regular enough, for example,
\psi \in \scrD (\BbbQ p), then a classical solution is given by the formula
u(t, x) =
\int
\BbbQ p
Z(t, x - \xi )\psi (\xi ) d\xi ,
where Z is, for each t, a probability density and
Z(t1 + t2, x) =
\int
\BbbQ p
Z(t1, x - y)Z(t2, y) dy, t1, t2 > 0, x \in \BbbQ p.
The "heat kernel" Z can be written as the Fourier transform
Z(t, x) =
\int
\BbbQ p
\chi (\xi x)e - t| \xi | \alpha p d\xi . (2.4)
See [16] for various series representations and estimates of the kernel Z .
As it was mentioned in Introduction, the natural stochastic process in BN corresponds to the
Cauchy problem
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 197
\partial u(t, x)
\partial t
+ (D\alpha
Nu) (t, x) - \lambda u(t, x) = 0, x \in BN , t > 0, (2.5)
u(0, x) = \psi (x), x \in BN , (2.6)
where the operator D\alpha
N is defined by restricting D\alpha to functions uN supported in BN and considering
the resulting function D\alpha uN only on BN . Note that D\alpha
N defines a positive definite selfadjoint operator
on L2(BN ), \lambda is its smallest eigenvalue.
Under certain regularity assumptions, for example if \psi \in \scrD (BN ), the problem (2.5), (2.6)
possesses a classical solution
u(t, x) =
\int
BN
ZN (t, x - y)\psi (y) dy, t > 0, x \in BN ,
where
ZN (t, x) = e\lambda tZ(t, x) + c(t), (2.7)
c(t) = p - N - p - N (1 - p - 1)e\lambda t
\infty \sum
n=0
( - 1)n
n!
tn
p - N\alpha n
1 - p - \alpha n - 1
.
Another interpretation of the kernel ZN was given in [8].
It was shown in [13] that the family of operators
(TN (t)u)(x) =
\int
BN
ZN (t, x - y)\psi (y) dy
is a strongly continuous contraction semigroup on L1(BN ). Its generator AN coincides with D\alpha
N - \lambda I
at least on \scrD (BN ). More generally, this is true in the distribution sense on restrictions to BN of
functions from the domain of the generator of the semigroup on L1(\BbbQ p) corresponding to D\alpha .
3. Harmonic analysis on the additive group of a \bfitp -adic ball. Let us consider the p-adic ball
BN as a compact subgroup of \BbbQ p . As we know, any continuous additive character of \BbbQ p has the
form x \mapsto \rightarrow \chi (\xi x), \xi \in \BbbQ p . The annihilator \{ \xi \in \BbbQ p : \chi (\xi x) = 1 for all x \in BN\} coincides with the
ball B - N . By the duality theorem (see, for example, [18], Theorem 27), the dual group \widehat BN to BN
is isomorphic to the discrete group \BbbQ p/B - N consisting of the cosets
pm
\bigl(
r0 + r1p+ . . .+ rN - m - 1p
N - m - 1
\bigr)
+B - N , rj \in \{ 0, 1, . . . , p - 1\} , m \in \BbbZ , m < N.
(3.1)
Analytically, this isomorphism means that any nontrivial continuous character of BN has the form
\chi (\xi x), x \in BN , where | \xi | p > p - N and \xi \in \BbbQ p is considered as a representative of the class
\xi +B - N . Note that | \xi | p does not depend on the choice of a representative of the class.
The normalized Haar measure on BN is p - N dx. The normalization of the Haar measure on
\BbbQ p/B - N can be made in such a way (the normalized measure will be denoted d\mu (x + B - N )) that
the equality
\int
\BbbQ p
f(x) dx =
\int
\BbbQ p/B - N
\left( pN \int
B - N
f(x+ h) dh
\right) d\mu (x+B - N ) (3.2)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
198 A. N. KOCHUBEI
holds for any f \in \scrD (\BbbQ p); see [4], Chapter VII, Proposition 10; [12], (28.54). With this normalization,
the Plancherel identity for the corresponding Fourier transform also holds; see [12], (31.46)(c).
On the other hand, the invariant measure on the discrete group \BbbQ p/B - N equals the sum of
\delta -measures concentrated on its elements multiplied by a coefficient \beta . In order to find \beta , it suffices
to compute both sides of (3.2) for the case where f is the indicator function of the set \{ x \in \BbbQ p :
| x - pN - 1| p \leq p - N\} . Then the left-hand side equals p - N while the right-hand side equals \beta .
Therefore \beta = p - N .
The Fourier transform on BN is given by the formula
(\scrF Nf)(\xi ) = p - N
\int
BN
\chi (x\xi )f(x) dx, \xi \in (\BbbQ p \setminus B - N ) \cup \{ 0\} ,
where the right-hand side, thus also \scrF Nf , can be understood as a function on \BbbQ p/B - N .
The fact that \scrF : \scrD (\BbbQ p) \rightarrow \scrD (\BbbQ p) implies that \scrF maps \scrD (BN ) onto the set of functions on the
discrete set \widehat BN having only a finite number of nonzero values. This set \scrD (\widehat BN ) with a natural locally
convex topology can be seen as the set of test functions on \widehat BN = \BbbQ p/B - N . The conjugate space
\scrD \prime (\widehat BN ) consists of all functions on \widehat BN (see, for example, [11]). Therefore the Fourier transform
is extended, via duality, to the mapping from \scrD \prime (BN ) to \scrD \prime (\widehat BN ). A theory of distributions on
locally compact groups covering the case of BN was developed by Bruhat [7]. To study deeper the
operator D\alpha
N , we need, within harmonic analysis on BN , a construction similar to the well-known
construction of a homogeneous distribution on \BbbQ p [19].
Let us introduce the usual Riesz kernel on \BbbQ p ,
f (N)
\alpha (x) =
1 - p - \alpha
1 - p\alpha - 1
| x| \alpha - 1
p , \mathrm{R}\mathrm{e}\alpha > 0, \alpha \not \equiv 1
\biggl(
\mathrm{m}\mathrm{o}\mathrm{d}
2\pi i
\mathrm{l}\mathrm{o}\mathrm{g} p
\BbbZ
\biggr)
.
Using the formula [19] \int
| x| p\leq pN
| x| \alpha - 1
p dx =
1 - p - 1
1 - p - \alpha
p\alpha N ,
we introduce a distribution from \scrD \prime (BN ) setting\Bigl\langle
f (N)
\alpha , \varphi
\Bigr\rangle
=
1 - p - 1
1 - p\alpha - 1
p\alpha N\varphi (0) +
1 - p - \alpha
1 - p\alpha - 1
\int
BN
[\varphi (x) - \varphi (0)]| x| \alpha - 1
p dx, \varphi \in \scrD (BN ). (3.3)
For \mathrm{R}\mathrm{e}\alpha > 0, this gives\Bigl\langle
f (N)
\alpha , \varphi
\Bigr\rangle
=
1 - p - \alpha
1 - p\alpha - 1
\int
BN
| x| \alpha - 1
p \varphi (x) dx.
On the other hand, the distribution (3.3) is holomorphic in \alpha \not \equiv 1
\biggl(
\mathrm{m}\mathrm{o}\mathrm{d}
2\pi i
\mathrm{l}\mathrm{o}\mathrm{g} p
\BbbZ
\biggr)
. Therefore f (N)
- \alpha
makes sense for any \alpha > 0. Noticing that
1 - p - 1
1 - p - \alpha - 1
p - \alpha N =
p - 1
p\alpha +1 - 1
p - \alpha N+\alpha = \lambda
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 199
(see Introduction), so that\Bigl\langle
f
(N)
- \alpha , \varphi
\Bigr\rangle
= \lambda \varphi (0) +
1 - p\alpha
1 - p - \alpha - 1
\int
BN
[\varphi (x) - \varphi (0)]| x| - \alpha - 1
p dx. (3.4)
The emergence of \lambda in (3.4) “explains” its role in the probabilistic construction of a process on BN
([16], Theorem 4.9).
Theorem 3.1. The operator D\alpha
N , \alpha > 0, acts from \scrD (BN ) to \scrD (BN ) and admits, for each
\varphi \in \scrD (BN ), the representations:
(i) D\alpha
N\varphi = f
(N)
- \alpha \ast \varphi where the convolution is understood in the sense of harmonic analysis on
the additive group of BN ;
(ii) (D\alpha
N\varphi ) (x) = \lambda \varphi (x) +
1 - p\alpha
1 - p - \alpha - 1
\int
BN
| y| - \alpha - 1
p [\varphi (x - y) - \varphi (x)] dy, \alpha > 0;
(iii) on \scrD (BN ), D\alpha
N - \lambda I coincides with the pseudodifferential operator \varphi \mapsto \rightarrow \scrF - 1
N (PN,\alpha \scrF N\varphi ),
where
PN,\alpha (\xi ) =
1 - p\alpha
1 - p - \alpha - 1
\int
BN
| y| - \alpha - 1
p [\chi (y\xi ) - 1] dy. (3.5)
This symbol is extended uniquely from (\BbbQ p \setminus B - N ) \cup \{ 0\} onto \BbbQ p/B - N .
Proof. Denote, for brevity, ap =
1 - p\alpha
1 - p - \alpha - 1
. Let x \in BN . Assuming that \varphi is extended by
zero onto \BbbQ p , we find
(D\alpha
N\varphi ) (x) = ap
\int
\BbbQ p
| y| - \alpha - 1
p [\varphi (x - y) - \varphi (x)] dy = I1 + I2 + I3,
where
I1 = ap
\int
BN
| y| - \alpha - 1
p [\varphi (x - y) - \varphi (x)] dy,
I2 = ap
\int
| y| p>pN
| y| - \alpha - 1
p \varphi (x - y) dy,
I3 = - ap\varphi (x)
\int
| y| p>pN
| y| - \alpha - 1
p dy.
We get using properties of p-adic integrals [19]
I2 = ap
\int
| x - z| p>pN
| x - z| - \alpha - 1
p \varphi (z) dz = ap
\int
| z| p>pN
| z| - \alpha - 1
p \varphi (z) dz = 0,
I3 = - ap\varphi (x)
\infty \sum
j=N+1
\int
| y| p=pj
| y| - \alpha - 1
p dy = - ap\varphi (x)
\biggl(
1 - 1
p
\biggr) \infty \sum
j=N+1
p - \alpha j = \lambda \varphi (x),
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
200 A. N. KOCHUBEI
which implies (ii). Comparing with (3.4) we prove (i).
In order to prove (3.5) we note that
\scrF N (D\alpha
N\varphi - \lambda \varphi ) (\xi ) = app
- N
\int
BN
\chi (x\xi ) dx
\int
BN
| y| - \alpha - 1
p [\varphi (x - y) - \varphi (x)] dy =
= app
- N
\int
BN
| y| - \alpha - 1
p dy
\int
BN
\chi (x\xi )[\varphi (x - y) - \varphi (x)] dx = Pn,\alpha (\xi ) (\scrF N\varphi ) (\xi ),
\xi \in \BbbQ p/B - N .
Theorem 3.1 is proved.
An important consequence of the representations given in Theorem 3.1 is the fact that, in contrast
to operators on \BbbQ p , D\alpha
N : \scrD (BN ) \rightarrow \scrD (BN ), so that we can define in a straightforward way, the
action of this operator on distributions. In particular, the pseudodifferential representation remains
valid on \scrD \prime (BN ). Below (Theorem 4.2) this will be used to describe the domain of the operator AN
on L1(BN ).
4. The Green function. In Section 2 (just as in [13]) we defined the operator AN as the
generator of the semigroup TN on L1(BN ). We can write its resolvent (AN + \mu I) - 1 , \mu > 0, as
\bigl(
(AN + \mu I) - 1u
\bigr)
(x) =
\infty \int
0
e - \mu tdt
\int
BN
ZN (t, x - \xi )u(\xi ) d\xi , u \in L1(BN ), (4.1)
where ZN is given in (2.7).
Theorem 4.1. The resolvent (4.1) admits the representation\bigl(
(AN + \mu I) - 1u
\bigr)
(x) =
\int
BN
K\mu (x - \xi )u(\xi ) d\xi + \mu - 1p - N
\int
BN
u(\xi ) d\xi , u \in L1(BN ), \mu > 0,
(4.2)
where for 0 \not = x \in BN , | x| p = pm,
K\mu (x) =
\int
p - N+1\leq | \eta | p\leq p - m+1
\chi (\eta x)
| \eta | \alpha p - \lambda + \mu
d\eta . (4.3)
If \alpha > 1, then, for any x \in BN ,
K\mu (x) =
\int
| \eta | p\geq p - N+1
\chi (\eta x)
| \eta | \alpha p - \lambda + \mu
d\eta . (4.4)
The kernel K\mu is continuous for x \not = 0 and belongs to L1(BN ).
If \alpha > 1, then K\mu is continuous on BN . If \alpha = 1, then
| K\mu (x)| \leq C| \mathrm{l}\mathrm{o}\mathrm{g} | x| p| , x \in BN . (4.5)
If \alpha < 1, then
| K\mu (x)| \leq C| x| \alpha - 1
p , x \in BN . (4.6)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 201
Proof. Let us use the representation (2.7) substituting it into the equality\int
BN
ZN (t, x) dx = 1
(for the latter see Theorem 4.9 in [16]). We find
c(t) = p - N - e\lambda tp - N
\int
BN
Z(t, y) dy,
so that
ZN (t, x) = e\lambda t
\left[ Z(t, x) - p - N
\int
BN
Z(t, y) dy
\right] + p - N , x \in BN .
Let us consider the expression in brackets proceeding from the definition (2.4) of the kernel Z .
Using the integration formula from Chapter 1, \S 4 of [19] we obtain
Z(t, x) - p - N
\int
BN
Z(t, y) dy = I1(t, x) + I2(t, x),
where
I1(t, x) =
\int
| \xi | p\geq p - N+1
\chi (\xi x)e - t| \xi | \alpha p d\xi ,
I2(t, x) =
\int
| \xi | p\leq p - N
[\chi (\xi x) - 1]e - t| \xi | \alpha p d\xi ,
and I2(t, x) = 0 for x \in BN .
Let | x| p = pm, m \leq N. Then there exists such an element \xi 0 \in \BbbQ p , | \xi 0| p = p - m+1 , that
\chi (\xi 0x) \not = 0. Then making the change of variables \xi = \eta + \xi 0 we find using the ultrametric property\int
| \xi | p\geq p - m+2
\chi (x\xi )e - t| \xi | \alpha p d\xi = \chi (x\xi 0)
\int
| \eta | p\geq p - m+2
\chi (x\eta )e - t| \eta | \alpha p d\eta ,
so that \int
| \xi | p\geq p - m+2
\chi (x\xi )e - t| \xi | \alpha p d\xi = 0.
Therefore
I1(t, x) =
\int
p - N+1\leq | \xi | p\leq p - m+1
\chi (x\xi )e - t| \xi | \alpha p d\xi ,
thus
ZN (t, x) = e\lambda t
\int
p - N+1\leq | \xi | p\leq p - m+1
\chi (x\xi )e - t| \xi | \alpha p d\xi + p - N , | x| p = pm.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
202 A. N. KOCHUBEI
Substituting this in (4.1) and integrating in t we come to (4.2) and (4.3). Note that | \eta | \alpha p > \lambda , as
| \eta | p \geq p - N+1 .
If \alpha > 1, then the integral in (4.4) is convergent. For | x| p = pm we prove repeating the above
argument that \int
| \eta | p\geq p - m+2
\chi (\eta x)
| \eta | \alpha p - \lambda + \mu
d\eta = 0.
Therefore in this case the representation (4.3) can be written in the form (4.4).
Obviously, K\mu (x) is continuous for x \not = 0. If \alpha > 1, then there exists the limit
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0
K\mu (x) =
\int
| \eta | p\geq p - N+1
d\eta
| \eta | \alpha p - \lambda + \mu
<\infty ,
so that in this case K\mu is continuous on BN .
Let \alpha < 1. By (4.3) and an integration formula from [19], Chapter 1, \S 4,
K\mu (x) =
- m+1\sum
l= - N+1
1
p\alpha l - \lambda + \mu
\int
| \xi | p=pl
\chi (\xi x) d\xi =
=
\biggl(
1 - 1
p
\biggr) - m\sum
l= - N+1
pl
p\alpha l - \lambda + \mu
- p - m
p\alpha ( - m+1) - \lambda + \mu
, | x| p = pm.
For some \gamma > 0, p\alpha l - \lambda +\mu \geq \gamma p\alpha l . Computing the sum of a progression we obtain the estimate
(4.6). Similarly, if \alpha = 1, then | K\mu (x)| \leq C( - m+N), which gives, as m \rightarrow - \infty , the inequality
(4.5).
Theorem 4.1 is proved.
If \alpha > 1, we can also give an interpretation of the resolvent (AN + \mu I) - 1 in terms of the
harmonic analysis on BN . We have
(AN + \mu I) - 1u =
\bigl(
K\mu + \mu - 1\bfone
\bigr)
\ast u, u \in L1(BN ), (4.7)
where \bfone (x) \equiv 1, K\mu is given by (4.4), and the convolution is taken in the sense of the additive group
of BN .
Denote by \Pi N the set of all rational numbers of the form
pl
\Bigl(
\nu 0 + \nu 1p+ . . .+ \nu - l+N - 1p
- l+N - 1
\Bigr)
, l < N,
where \nu j \in \{ 0, 1, . . . , p - 1\} , \nu 0 \not = 0. As a set, the quotient group \BbbQ p/B - N coincides with \Pi N\cup \{ 0\} ,
and
\{ \xi \in \BbbQ p : | \xi | p \geq p - N+1\} =
\bigcup
\eta \in \Pi N
(\eta +B - N )
where the sets \eta +B - N with different \eta \in \Pi N are disjoint.
Taking into account the fact that \chi (\rho x) = 1 for x \in BN , \rho \in B - N , we find from (4.4) that
K\mu (x) = p - N
\sum
0\not =\eta \in \BbbQ p/B - N
\chi (\eta x)
| \eta | \alpha p - \lambda + \mu
.
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LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 203
Let us describe the domain DomAN of the generator of our semigroup TN (t) on L1(BN ) in
terms of distributions on BN .
Theorem 4.2. If \alpha > 1, then the set \mathrm{D}\mathrm{o}\mathrm{m}AN consists of those and only those u \in L1(BN ),
for which f
(N)
- \alpha \ast u \in L1(BN ) where the convolution is understood in the sense of the distribution
space \scrD \prime (BN ). If u \in \mathrm{D}\mathrm{o}\mathrm{m}AN , then ANu = f
(N)
- \alpha \ast u - \lambda u where the convolution is understood
in the sense of the distributions from \scrD \prime (BN ).
Proof. Let u = (AN + \mu I) - 1f , f \in L1(BN ), \mu > 0. Representing this resolvent as a
pseudodifferential operator, we prove that f (N)
- \alpha \ast u - \lambda u+ \mu u = f in the sense of \scrD \prime (BN ).
Conversely, let u \in L1(BN ), D\alpha
Nu = f
(N)
- \alpha \ast u \in L1(BN ) where D\alpha
N is understood in the sense
of \scrD \prime (BN ). Set f = (D\alpha
N - \lambda I + \mu I)u, \mu > 0. Denote u\prime = (AN + \mu I) - 1f . Then u\prime \in \mathrm{D}\mathrm{o}\mathrm{m}AN ,
and the above argument shows that
(D\alpha
N - \lambda I + \mu I)(u - u\prime ) = 0.
Applying the pseudodifferential representation we see that
[PN,\alpha (\xi ) + \mu ]
\bigl[
(\scrF Nu)(\xi ) - (\scrF Nu
\prime )(\xi )
\bigr]
= 0, \xi \in \BbbQ p/B - N .
It is seen from (3.5) that the factor PN,\alpha (\xi ) + \mu is real-valued, strictly positive and locally constant
on BN . Therefore the distribution \scrF Nu - \scrF Nu
\prime is zero. Since \scrF N is an isomorphism (see [7]), we
find that u = u\prime , so that u \in \mathrm{D}\mathrm{o}\mathrm{m}AN .
Theorem 4.2 is proved.
5. Nonlinear equations. Let us consider the equation (1.2) where \Phi is a strictly monotone
increasing continuous real function, \Phi (0) = 0, and the linear operator D\alpha
N is understood as the
operator AN + \lambda I on L1(BN ). By the results from [10] and [6], the nonlinear operator D\alpha
N \circ \Phi is
m-accretive, which implies the unique mild solvability of the Cauchy problem for the equation (1.2)
with the initial condition u(0, x) = u0(x), u0 \in L1(BN ); see, e.g., [2] for the definitions. As in the
classical case [3], this mild solution can be interpreted also as a weak solution.
Following [3], we will show that the above construction of the L1-mild solution gives also
L\gamma -solutions for 1 < \gamma \leq \infty .
Theorem 5.1. Let u(t, x), t > 0, x \in BN , be the above mild solution. If 0 < u0 \in L\gamma (BN ),
1 \leq \gamma \leq \infty , then u(t, \cdot ) \in L\gamma (BN ) and
\| u(t, \cdot )\| L\gamma (BN ) \leq \| u0\| L\gamma (BN ). (5.1)
Proof. The case \gamma = 1 has been considered, while the case \gamma = \infty will be implied by the
inequality (5.1) for finite values of \gamma (see Exercise 4.6 in [5]).
Thus, now we assume that 1 < \gamma < \infty . It is sufficient to prove (5.1) for u0 \in \scrD (BN ). Indeed,
if that is proved, we approximate in L\gamma (BN ) an arbitrary function u0 \in L\gamma (BN ) by a sequence
u0,j \in \scrD (BN ). For the corresponding solutions uj(t, x) we have
\| uj(t, \cdot )\| L\gamma (BN ) \leq \| u0,j\| L\gamma (BN ). (5.2)
Since our nonlinear semigroup consists of operators continuous on L1(BN ), we see that, for each
t \geq 0, uj(t, \cdot ) \rightarrow u(t, \cdot ) in L1(BN ). By (5.2), the sequence \{ uj(t, \cdot )\} is bounded in L\gamma (BN ). These
two properties imply the weak convergence uj(t, \cdot )\rightharpoonup u(t, \cdot ) in L\gamma (BN ) (see Exercise 4.16 in [5]).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
204 A. N. KOCHUBEI
Next, we use the weak lower semicontinuity of the L\gamma -norm (see Theorem 2.11 in [17]), that is
the inequality
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
j
\| uj(t, \cdot )\| L\gamma (BN ) \geq \| u(t, \cdot )\| L\gamma (BN ).
Passing to the lower limit in both sides of (5.2), we come to (5.1).
Let us prove (5.1) for u0 \in \scrD (BN ), 1 < \gamma < \infty . By the Crandall – Liggett theorem (see [2] or
[9]), u(t, x) is obtained as a limit in L1(BN ),
u(t, \cdot ) = \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\biggl(
I +
t
k
D\alpha
N \circ \Phi
\biggr) - k
u0,
that is u(t, \cdot ) = \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty uk where uk are found recursively from the relation
t
k + 1
D\alpha
N \circ \Phi (uk+1) + uk+1 = uk. (5.3)
Under our assumptions, u(t, x) > 0 (this follows from Theorem 4 in [10]). The nonlinear
operator
\biggl(
I +
t
k
D\alpha
N \circ \Phi
\biggr) - 1
is also positivity preserving (Proposition 1 in [10]), so that uk > 0 for
all k .
Note that the operator D\alpha
N commutes with shifts while the equation (5.3) for uk+1 has a unique
solution in L1(BN ). As a result, if u0 \in \scrD (BN ), then all the functions uk belong to \scrD (BN ).
Rewriting (5.3) in the form\biggl(
t
k + 1
\biggr) - 1
(uk+1 - uk) = - D\alpha
N \circ \Phi (uk+1), (5.4)
multiplying both sides by u\gamma - 1
k+1 and integrating on BN we find\biggl(
t
k + 1
\biggr) - 1 \int
BN
(uk+1 - uk)u
\gamma - 1
k+1dx = -
\int
BN
u\gamma - 1
k+1D
\alpha
N \circ \Phi (uk+1) dx. (5.5)
Let w = u\gamma - 1
k+1 . Then w \in \scrD (BN ). It follows from (5.4) that D\alpha
N\Phi (uk+1) \in \scrD (BN ). Also we
have \Phi (uk+1) \in \scrD (BN ), so that \Phi (uk+1) belongs to the domain of a selfadjoint realization of the
operator D\alpha
N in L2(BN ). Therefore we can transform the integral in the right-hand side of (5.5) as
follows: \int
BN
u\gamma - 1
k+1D
\alpha
N \circ \Phi (uk+1) dx =
\int
BN
\Phi (w
1
\gamma - 1 )D\alpha
N (w) dx. (5.6)
The right-hand side of (5.6) is nonnegative by Lemma 2 of [6]. Now it follows from (5.5) that\int
BN
u\gamma k+1dx \leq
\int
BN
uku
\gamma - 1
k+1dx.
Applying the Hölder inequality we find
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
LINEAR AND NONLINEAR HEAT EQUATIONS ON A p-ADIC BALL 205
\int
BN
u\gamma k+1dx \leq
\left( \int
BN
u\gamma kdx
\right)
1/\gamma \left( \int
BN
u\gamma k+1dx
\right)
\gamma - 1
\gamma
,
which implies the inequality
\| uk+1\| L\gamma (BN ) \leq \| uk\| L\gamma (BN )
and, by induction, the inequality
\| uk+1\| L\gamma (BN ) \leq \| u0\| L\gamma (BN ).
Passing to the limit, we prove (5.1).
Theorem 5.1 is proved.
References
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Cambridge Univ. Press, 2010.
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Received 09.08.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 2
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| id | umjimathkievua-article-1550 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:52Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/7ce5c86f35931da389734175450abb2a.pdf |
| spelling | umjimathkievua-article-15502019-12-05T09:18:03Z Linear and nonlinear heat equations on a $p$ -adic ball Лiнiйне та нелiнiйне рiвняння теплопровiдностi на $p$ -адичнiй кулi Kochubei, A. N. Кочубей, А. Н. We study the Vladimirov fractional differentiation operator $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, on a $p$-adic ball B$B_N = \{ x \in Q_p : | x|_p \leq p^N\}$. To its known interpretations via the restriction of a similar operator to $Q_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudodifferential operator in terms of the Pontryagin duality on the additive group of $B_N$. We investigate the Green function of $D^{\alpha}_N$ and a nonlinear equation on $B_N$, an analog of the classical equation of porous medium. Вивчається оператор Владимирова диференцiювання дробового порядку $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, на $p$-адичнiй кулi $B_N = \{ x \in Q_p : | x|_p \leq p^N\}$ . До його вiдомих iнтерпретацiй у термiнах звуження подiбного оператора, визначеного на $Q_p$ та через деякий випадковий процес на $B_N$, ми додаємо iнтерпретацiю у виглядi псевдодиференцiального оператора в термiнах дуальностi Понтрягiна на адитивнiй групi BN. Вивчено функцiю Грiна на $D^{\alpha}_N$ та нелiнiйне рiвняння на $B_N$, що є аналогом класичного рiвняння пористого середовища. Institute of Mathematics, NAS of Ukraine 2018-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1550 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 2 (2018); 193-205 Український математичний журнал; Том 70 № 2 (2018); 193-205 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1550/532 Copyright (c) 2018 Kochubei A. N. |
| spellingShingle | Kochubei, A. N. Кочубей, А. Н. Linear and nonlinear heat equations on a $p$ -adic ball |
| title | Linear and nonlinear heat equations on a $p$ -adic ball |
| title_alt | Лiнiйне та нелiнiйне рiвняння теплопровiдностi
на $p$ -адичнiй кулi |
| title_full | Linear and nonlinear heat equations on a $p$ -adic ball |
| title_fullStr | Linear and nonlinear heat equations on a $p$ -adic ball |
| title_full_unstemmed | Linear and nonlinear heat equations on a $p$ -adic ball |
| title_short | Linear and nonlinear heat equations on a $p$ -adic ball |
| title_sort | linear and nonlinear heat equations on a $p$ -adic ball |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1550 |
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