Existence of nonnegative solutions for a fractional parabolic equation in the whole space
UDC 517.9 We study existence of nonnegative solutions for a parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ in $\mathbb{R}^{d}\times (0, T).$ Here $0
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| author | Kenzizi, T. Кензізі, Т. |
| author_facet | Kenzizi, T. Кензізі, Т. |
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| description | UDC 517.9
We study existence of nonnegative solutions for a parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$
in $\mathbb{R}^{d}\times (0, T).$
Here
$0 |
| first_indexed | 2026-03-24T02:06:57Z |
| format | Article |
| fulltext |
UDC 517.9
T. Kenzizi (Univ. Tunis El Manar, Tunisia)
EXISTENCE OF NONNEGATIVE SOLUTIONS
FOR A FRACTIONAL PARABOLIC EQUATION IN THE WHOLE SPACE
IСНУВАННЯ НЕВIД’ЄМНИХ РОЗВ’ЯЗКIВ
ДРОБОВОГО ПАРАБОЛIЧНОГО РIВНЯННЯ В УСЬОМУ ПРОСТОРI
We study existence of nonnegative solutions for a parabolic problem
\partial u
\partial t
= - ( - \bigtriangleup )
\alpha
2 u +
c
| x| \alpha u in \BbbR d \times (0, T ). Here
0 < \alpha < \mathrm{m}\mathrm{i}\mathrm{n}(2, d), ( - \bigtriangleup )
\alpha
2 is the fractional Laplacian on \BbbR d and u0 \in L2(\BbbR d).
Вивчається задача iснування невiд’ємних розв’язкiв параболiчного рiвняння
\partial u
\partial t
= - ( - \bigtriangleup )
\alpha
2 u +
c
| x| \alpha u на \BbbR d \times
\times (0, T ). Тут 0 < \alpha < \mathrm{m}\mathrm{i}\mathrm{n}(2, d), ( - \bigtriangleup )
\alpha
2 — дробовий лапласiан на \BbbR d й u0 \in L2(\BbbR d).
1. Introduction. The purpose of the present paper is to verify that a similar critical behavior of the
Cauchy problem holds when the classical Laplacian is replaced by the fractional Laplacian - ( - \bigtriangleup )
\alpha
2
with 0 < \alpha < \mathrm{m}\mathrm{i}\mathrm{n}(2, d). So we give some existence results of positive solutions for negatively
perturbed Dirichlet fractional Laplacian on \BbbR d.
For every 0 < \alpha < \mathrm{m}\mathrm{i}\mathrm{n}(2, d), we designate by L0 := ( - \bigtriangleup )
\alpha
2 . Let us consider the parabolic
perturbed problem
- \partial u
\partial t
= L0u - V u in \BbbR d \times (0,+\infty ),
u(x, 0) = u0(x) for a.e. x \in \BbbR d,
(1)
where u0 \in L2(\BbbR d), u0 \geq 0 is a Borel measurable function and V is nonnegative potential in
L1
loc(\Omega ).
Although we shall focus on the very special case Vc =
c
| x| \alpha
, 0 < c \leq c\ast :=
2\alpha \Gamma 2
\biggl(
d+ \alpha
4
\biggr)
\Gamma 2
\biggl(
d - \alpha
4
\biggr) .
The present paper addresses several important problems of the potential theory of fractional
Laplacian. One of the results is the existence of nonnegative solution for a parabolic problem per-
turbed by potential. Its main results were motivated by the result of J. A. Goldstein and Q. S. Zhang
[12] for the Laplacian perturbed by a singular potential.
By using the idea in [5, 10, 12], where the problem was addressed and solved for the Dirichlet
Laplacian (i.e., \alpha = 2), Ali Ben Amor and T. Kenzizi [1] established conditions ensuring existence
of nonnegative solutions for a nonlocal case, they proved that for bounded \Omega and for 0 < c \leq
\leq c\ast :=
2\alpha \Gamma 2
\biggl(
d+ \alpha
4
\biggr)
\Gamma 2
\biggl(
d - \alpha
4
\biggr) equation (1) has a nonnegative solution, whereas for c > c\ast and \Omega a
bounded Lipschitz domain then no nonnegative solutions occur.
c\bigcirc T. KENZIZI, 2019
1064 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
EXISTENCE OF NONNEGATIVE SOLUTIONS FOR A FRACTIONAL PARABOLIC EQUATION. . . 1065
The inspiring point for us was the papers of Baras, Goldstein [5, 6] and Goldstein, Zhang [12]
where the problem was addressed and solved for the Lapacian operator (i.e., \alpha = 2). In the latter
cited papers, the authors in [12] generalize the result of existence of nonnegative solutions in [5] to
equations with variable coefficients in the principal part and to a degenerate equations, one of the
most important degenerate equations is the heat equation on the Heisenberg group. However, there
is a substantial difference between the Laplacian and the fractional Laplacian. Whereas it is known
that the first one is local and therefore suitable for describing diffusions, the second one is nonlocal
and commonly used for describing superdiffusions (Lévy flights). These differences are reflected in
the way of computing for both operators (Green formula, integration by part, Leibniz formula, etc.).
The fractional operator appears in numerous fields of mathematical physics, mathematical biology
and mathematical finance and has attracted a lot of attention recently. Nonetheless, we shall show
that the method used in [5, 12] still apply in our setting.
2. Preliminaries and preparing results. To state our main results, it is convenient to introduce
the following notations and definitions. In what follows, \BbbR d denotes the Euclidean space of dimen-
sion d \geq 1, dy is the Lebesgue measure on \BbbR d. We shall write
\int
. . . as a shorthand form
\int
\BbbR d
. . . .
Throughout this paper, letter k, C, c, C \prime , . . . will denote generic positive constants which may
vary in value from line to line.
\bigm| \bigm| A(x, r)
\bigm| \bigm| will denote the volume of the ball A centred at x and of
radius r, (a \wedge b) := \mathrm{m}\mathrm{i}\mathrm{n}(a, b) and (a \vee b) := \mathrm{m}\mathrm{a}\mathrm{x}(a, b). Consider the bilinear symmetric form \scrE \alpha
defined in L2 by
\scrE \alpha (f, g) =
1
2
\scrA (d, \alpha )
\int \int \bigl(
f(x) - f(y)
\bigr) \bigl(
g(x) - g(y)
\bigr)
| x - y| d+\alpha
dx dy,
D(\scrE \alpha ) = W
\alpha
2
,2(\BbbR d) :=
\bigl\{
f \in L2 : \scrE [f ] : \scrE (f, f) < \infty
\bigr\}
,
where
\scrA (d, \alpha ) =
\alpha \Gamma
\biggl(
d+ \alpha
2
\biggr)
21 - \alpha \pi
d
2\Gamma
\Bigl(
1 - \alpha
2
\Bigr) ,
and \Gamma is the Gamma function. It is well known that \scrE \alpha is a transient Dirichlet form and is related (via
Kato representation theorem) to the self-adjoint operator commonly named the fractional Laplacian
on \BbbR d, and which we shall denote by ( - \bigtriangleup )
\alpha
2 . We note that the domain of ( - \bigtriangleup )
\alpha
2 is the fractional
Sobolev space W
\alpha
2
,2(\BbbR d). For smooth compactly supported function \phi \in C\infty
c (\BbbR d), the fractional
Laplacian is defined as the L2(\BbbR d)-closure of the operator
\bigtriangleup
\alpha
2 \phi (x) = \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon - \rightarrow 0
\int
| y| >\epsilon
\bigl[
\phi (x+ y) - \phi (x)
\bigr]
\nu (y) dy, x \in \BbbR d,
where \nu is the Lévy measure given by the density function
\nu (y) =
2\alpha \Gamma
\biggl(
d+ \alpha
2
\biggr)
\pi
d
2
\bigm| \bigm| \bigm| \bigm| \Gamma \biggl(
- \alpha
2
\biggr) \bigm| \bigm| \bigm| \bigm| | y|
- d - \alpha .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1066 T. KENZIZI
This definition is important for applications to probability. Its Fourier symbol is given by
\widehat \bigtriangleup
\alpha
2 \phi (\xi ) = - | \xi | \alpha \widehat \phi (\xi ).
If \phi is regular enough and \alpha \in (0, 2), ( - \bigtriangleup )
\alpha
2 \phi (x) can be computed by the formula
( - \bigtriangleup )
\alpha
2 \phi (x) = cd,\alpha
\int
\phi (x) - \phi (y)
| x - y| d+\alpha
dy,
where cd,\alpha is a constant depending only on d and \alpha . The inverse of the
\alpha
2
power of the Laplacian
is the - \alpha
2
power of the Laplacian ( - \bigtriangleup ) -
\alpha
2 . For 0 < \alpha < \mathrm{m}\mathrm{i}\mathrm{n}(2, d), there is an integral formula
which says that ( - \bigtriangleup ) -
\alpha
2 u is the convolution of the function u with the Riesz potential
( - \bigtriangleup ) -
\alpha
2 \phi (x) = cd,\alpha
\int
\phi (x - y)
| y| d - \alpha
dy,
which holds a long as \phi is integrable enough for the right-hand side to make sense.
Let r > 0 and \phi r(x) = \phi (rx), then we obtain
\bigtriangleup
\alpha
2 \phi r(x) = r\alpha \bigtriangleup
\alpha
2 \phi (rx), x \in \BbbR d.
We let pt the fractional heat kernel which is the fundamental solution to the heat equation
\partial pt(x)
\partial t
+ ( - \bigtriangleup )
\alpha
2 p = 0,
p0(x) = \delta 0(x),
with Fourier transform
\^pt(\xi ) =
\int
pt(x)e
ix\xi dx = e - t| \xi | \alpha , t > 0, x \in \BbbR d, (2)
yield the following identity:
pt(x) = (2\pi ) - d
\int
e - t| \xi | \alpha e - ix\xi d\xi , x \in \BbbR d.
Consequently, we get the scaling property
pt(x) = t -
d
\alpha p1(t
- 1
\alpha x), t > 0, x \in \BbbR d.
It is well-known (see [4]) that p1(x) \approx 1\wedge | x| - d - \alpha , hence the following inequalities holds for some
constant C :
C - 1
\biggl(
t -
d
\alpha \wedge t
| x| d+\alpha
\biggr)
\leq pt(x) \leq C
\biggl(
t -
d
\alpha \wedge t
| x| d+\alpha
\biggr)
, t > 0, x \in \BbbR d.
In particular, the maximum of pt is pt(0) = 21 - \alpha \pi - d
2\alpha - 1
\Gamma
\Biggl(
d
\alpha
\Biggr)
\Gamma
\Biggl(
d
2
\Biggr) t -
d
\alpha .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
EXISTENCE OF NONNEGATIVE SOLUTIONS FOR A FRACTIONAL PARABOLIC EQUATION. . . 1067
The semigroup Pt\phi (x) =
\int
pt(x, y)\phi (y) dy has the fractional Laplacian as generator (see [2, 8,
13]). In particular, for \phi \in C\infty
c (\BbbR d) and x \in \BbbR d, we have
( - \bigtriangleup ) -
\alpha
2 \phi (x) = \mathrm{l}\mathrm{i}\mathrm{m}
t - \rightarrow 0+
1
t
(Pt\phi (x) - \phi (x)) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon - \rightarrow 0+
\int
| y| >\varepsilon
\phi (x+ y) - \phi (x)
| y| d+\alpha
dy.
Using (2) one proves that p is the heat kernel of the fractional Laplacian
\infty \int
s
\int
p(u - s, x, z)
\bigl[
\partial u\phi (u, z) +\bigtriangleup
\alpha
2
z \phi (u, z)
\bigr]
dzdu = - \phi (s, x),
where p(t, x, y) = pt(y - x), s \in \BbbR , x \in \BbbR d and \phi \in C\infty
c (\BbbR \times \BbbR d).
Let D \subseteq \BbbR d be an open set. We denote by pD the heat kernel of the Dirichlet fractional Laplacian
on D. Also pD is jointly continuous when t \not = 0, and we have
0 \leq pD(t, x, y) = pD(t, y, x) \leq p(t, x, y), t > 0, x, y \in \BbbR d.
In particular, \int
pD(t, x, y) \leq 1.
We define the Green function for \bigtriangleup
\alpha
2 on D by
GD(x, y) =
\infty \int
0
pD(t, x, y)dt,
and the scaling property of pD yields the following scaling of GD :
GrD(rx, ry) = r\alpha - dGD(x, y), r > 0, x, y \in \BbbR d.
Definition 2.1. Let 0 < T \leq \infty . A Borel measurable function u : [0, T ) \times \BbbR d - \rightarrow \BbbR is a
solution of problem (1) if :
1) u \in L1
loc
\bigl(
\BbbR d \times (0, T )
\bigr)
,
2) u \in \scrL 2
loc
\bigl(
[0, T ], L2
loc(\BbbR d)
\bigr)
,
3) for every 0 \leq t < T, every \Omega \subset \BbbR d and every \phi \in C\infty
c
\bigl(
[0, T ] \times \Omega
\bigr)
, the following identity
holds true:
\int
\Omega
((u\phi )(t, x) - u0(x)\phi (0, x)) dx+
t\int
0
\int
\Omega
u(s, x)
\Bigl(
- \phi s(s, x) + L0\phi (s, x)
\Bigr)
=
=
t\int
0
\int
\Omega
u(s, x)\phi (s, x)V (x) dx ds.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1068 T. KENZIZI
From now, we set Vk := V \wedge k and denote by (Pk) the heat equation corresponding to the
Dirichlet fractional Laplacian perturbed by - Vk instead of - V :
(Pk) :
\left\{ - \partial u
\partial t
= L0u - Vku in \BbbR d \times (0,+\infty ),
u(x, 0) = u0 for a.e. x \in \BbbR d.
Denote by Hk the self-adjoint operator associated to the quadratic form \scrE Vk := \scrE - Vk defined by
\scrE Vk : D(\scrE Vk) = W
\alpha
2
,2
0 (\BbbR d), \scrE Vk [u] = \scrE -
\int
\BbbR d
u2Vkdx
and, by [11, p. 492] (Remark 1.22), we conclude that
uk(t) = e - tHku0 =
\int
\BbbR d
pt,ku0(y) dy, t \geq 0, (3)
is the solution of (Pk), where pt,k is the nonnegative heat kernel of e - tHk .
Remark 2.1. Let \Omega k be a nondecreasing sequence in \BbbR d such that \Omega \subset
\bigcup
k\geq 0
\Omega k, and let \phi \in
\in C\infty
c
\bigl(
[0, T ] \times \BbbR d
\bigr)
such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\phi \subset \Omega . If V is bounded, the solution of (1) is given by the
integral expression
u(x, t) = e - tL0u0(x) +
t\int
0
\int
\Omega
qt - s(x, y)u(s, y)V (y) dy ds,
where qt is the heat kernel of the operator e - tL0 , t > 0.
By the end of this section, we give some spectral properties of L0 = ( - \bigtriangleup )
\alpha
2 that will be needed
in the proof of the existence part.
Definition 2.2 [9]. The essential spectrum of a bounded self-adjoint operator A on a Hilbert
space, usually denoted \sigma ess(A), is a subset of the spectrum \sigma , and its complement is called the
discrete spectrum, so
\sigma disc(A) = \sigma (A)\setminus \sigma ess(A).
For bounded domain \Omega and for fixed \alpha \in (0,\mathrm{m}\mathrm{i}\mathrm{n}(2, d)], the spectrum of the ( - \bigtriangleup )
\alpha
2 | \Omega is
discrete and consists of a sequence \{ \lambda k(\alpha )\} \infty k=1 of eigenvalues (with finite multiplicity) written in
increasing order according to their multiplicity (see, for example, [7])
0 < \lambda 1(\alpha ) < \lambda 2(\alpha ) \leq . . . \leq \lambda k(\alpha ) . . . \nearrow +\infty .
3. Existence of nonnegative solutions.
Theorem 3.1. Assume that c \leq c\ast . Then the heat equation (1) has at least one nonnegative
solution. Here c\ast is an universal constant ( see, for example, [1, 3]).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
EXISTENCE OF NONNEGATIVE SOLUTIONS FOR A FRACTIONAL PARABOLIC EQUATION. . . 1069
Proof. The idea of studying existence is based on studying the solution (uk) of the approach
problem (Pk), where V is replaced by the trancated potential Vk = V \wedge k. Considering first the
radial function \Phi (x) = | x| - \beta for \beta \in [0, d - \alpha ] (\beta denotes the smaller root of (d - \alpha - \beta )\beta = c).
Let p \in C2(\BbbR ) be a convex function satisfying p(0) = p\prime (0) = 0, such that its derivative is
locally Lipschitz.
First, we will prove that for all uk \in D(\scrE \alpha ) and all \Phi \in W
\alpha
2
,2(\BbbR d) \cap L\infty (\BbbR d), we have
p\prime (uk)\Phi \in D(\scrE \alpha ).
Since \Phi is bounded in D(\scrE \alpha ), we need only to prove that p\prime (uk) \in D(\scrE \alpha ).
Note that p\prime is locally Lipschitz, i.e., there exists M > 0 such that\bigm| \bigm| p\prime (uk(x)) - p\prime (uk(y))
\bigm| \bigm| \leq M
\bigm| \bigm| uk(x) - uk(y)
\bigm| \bigm| .
Thereby we derive
\scrE \alpha
\bigl[
p\prime (uk)
\bigr]
< \infty .
Taking now p\prime (uk)\Phi as a test function, we obtain
t\int
\delta
\int
\partial uk
\partial t
\times p\prime (uk)\Phi +
t\int
\delta
\int
p\prime (uk)\Phi L0uk =
t\int
\delta
\int
Vkukp
\prime (uk)\Phi .
Thus, we get
\int
p(uk(t))\Phi +
t\int
\delta
\int
p\prime (uk(s))\Phi L0uk =
t\int
\delta
\int
Vkukp
\prime (uk)\Phi +
\int
p(uk(\delta ))\Phi .
Since the function p is convex, we have
p\prime (uk(x))
\bigl(
uk(x) - uk(y)
\bigr)
\geq
\bigl(
p(uk)(x) - p(uk)(y)
\bigr)
.
Therefore, \int
p\prime (uk(x))\Phi (x)L0uk(x) dx =
= Cn,sPV
\int \int
p\prime (uk(x))\Phi (x)
\bigl(
uk(x) - uk(y)
\bigr)
| x - y| d+\alpha
dy dx \geq
\geq Cn,sPV
\int \int
\Phi (x)
\bigl(
p(uk)(x) - p(uk)(y)
\bigr)
| x - y| d+\alpha
dy dx =
=
\int
\Phi (x)L0p(uk(x)) dx.
Hence,
\int
p(uk(t))\Phi +
t\int
\delta
\int
\Phi (x)L0p(uk(x, s)) dx ds \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1070 T. KENZIZI
\leq
t\int
\delta
\int
Vkukp
\prime (uk)\Phi +
\int
p(uk(\delta ))\Phi .
Replace p(r) by a sequence pm(r) satisfying the hypotheses, for p and converging to | r| as m - \rightarrow
- \rightarrow \infty . Note that p\prime (r) =
r
| r|
.
First, we see that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
\delta
\int
\Phi (x)L0pm(uk) dx ds -
t\int
\delta
\int
\Phi (x)L0(uk) dx ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \| \Phi \| \infty
t\int
\delta
\int \bigm| \bigm| L0pm(uk) - L0(uk)
\bigm| \bigm| dx ds \leq
\leq | \Phi \| \infty
\bigm\| \bigm\| L0pm(uk) - L0(uk)
\bigm\| \bigm\|
L1
\bigl(
\BbbR d\times [\delta ,t]
\bigr) - \rightarrow 0 as m - \rightarrow +\infty ,
and consequently, we obtain the limiting inequality
\int
uk(t)\Phi dx+
t\int
\delta
\int
\Phi (x).L0(uk) dx ds \leq
\leq
t\int
\delta
\int
Vk(x)uk(s)\Phi dx ds+
\int
uk(\delta )\Phi dx. (4)
We want to let \delta - \rightarrow 0. First we claim that\int
uk(\delta )\Phi dx - \rightarrow
\int
u0(x)\Phi dx.
Since the operator ( - \bigtriangleup )
\alpha
2 is self-adjoint on D
\Bigl(
( - \bigtriangleup )
\alpha
2
\Bigr)
, so, by using the Trotter – Kato theorem
and the spectral theorem, we have
e - \delta (L0 - Vk)u0 = \mathrm{l}\mathrm{i}\mathrm{m}
m - \rightarrow +\infty
\Bigl(
e -
\delta L0
m e(
\delta
m
)Vk
\Bigr) m
u0 \leq e\delta \lambda e - \delta L0u0,
then \| Vk\| \infty \leq \lambda by the positivity preserving property of \{ e - \delta L0\} . It follows that
e - \delta L0u0(x) \leq uk(\delta ) = e - \delta (L0 - Vk)u0(x) = e - \delta L0e\delta Vku0(x).
Thus, \int
(e - \delta L0u0)\Phi \leq
\int
uk(\delta )\Phi =
\int
e - \delta (L0 - Vk)u0\Phi \leq e\delta \lambda
\int
(e - \delta L0u0)\Phi ,
whence \int
(e - \delta L0u0)\Phi =
\int
(e - \delta L0\Phi )u0 - \rightarrow
\int
\Phi u0
as \delta - \rightarrow 0, as asserted.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
EXISTENCE OF NONNEGATIVE SOLUTIONS FOR A FRACTIONAL PARABOLIC EQUATION. . . 1071
Letting \delta - \rightarrow 0 in (4), we deduce
\int
uk(t)\Phi dx+
t\int
0
\int
\Phi (x)L0(uk) dx ds \leq
\leq
t\int
0
\int
Vk(x)uk(s)\Phi dx ds+
\int
u0(x)\Phi dx. (5)
Our aim now is to estimate the LHS of (5). By using [3] (Lemma 2.2), we recall that \Phi (x) = | x| - \beta
is the unique solution for the equation
L0\Phi (x) = c| x| - \alpha | x| - \beta = c| x| - \alpha \Phi (x) in the sense of distributions,
where
c = 2\alpha
\Gamma
\biggl(
\alpha + \beta
2
\biggr)
\Gamma
\biggl(
d - \beta
2
\biggr)
\Gamma
\biggl(
d - (\alpha + \beta )
2
\biggr)
\Gamma
\biggl(
\beta
2
\biggr) ,
which implies
L0\Phi (x) = c| x| - \alpha \Phi (x) \geq Vk(x)\Phi (x).
Therefore, \int
ukL0\Phi dx \geq
\int
Vk(x)uk\Phi dx in the sense of distributions,
and consequently, from (4) we deduce
\int
uk(t)\Phi dx+
t\int
0
\int
Vkuk\Phi dx ds \leq
\int
uk(t)\Phi dx+
t\int
0
\int
\Phi (x)L0(uk) dx ds \leq
\leq
t\int
0
\int
Vkuk\Phi dx ds+
\int
u0(x)\Phi dx.
Thereby we derive that \int
uk(t)\Phi dx \leq
\int
u0(x)\Phi dx.
Let
vk(t) = uk(t)\Phi .
Therefore, if \int
u0(x)\Phi (x) dx < \infty ,
we have
\| vk\| L1(\BbbR d) < M for all k.
Then (vk) is bounded in L1 and by the weak compactness in L1, there exists a subsequence still
called (vk) such that (vk) converge weakly to v \in L1(\Omega ) \forall \Omega \subset \BbbR d and, consequently, a subsequence
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1072 T. KENZIZI
(uk) converging to a function u in L1(\Omega ) and for, every \Omega \subset \BbbR d, we have
\| vkVk - vV \| L1 = \| vkVk - vkV + vkV - vV \| L1 \leq
\leq
\bigm\| \bigm\| vk(Vk - V )
\bigm\| \bigm\|
L1 +
\bigm\| \bigm\| (vk - v)V
\bigm\| \bigm\|
L1 .
Hence, \bigm\| \bigm\| ukVk\Phi - uV \Phi
\bigm\| \bigm\|
L1 - \rightarrow 0 as k - \rightarrow +\infty .
On the other hand, by hypothesis (cf. (3)), (uk) is an increasing sequence, thus (uk) increases to
u(x, t), and u is a solution of (1) in the sense of distributions.
References
1. Ali Ben Amor, Kenzizi T. The heat equation for the Dirichlet fractional Laplacian with negative potentials: Eistence
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Received 09.08.16,
after revision — 18.04.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
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| id | umjimathkievua-article-1498 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:57Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dd/08f3a36e7887f53b4b9e0f27b52e43dd.pdf |
| spelling | umjimathkievua-article-14982019-12-05T08:57:29Z Existence of nonnegative solutions for a fractional parabolic equation in the whole space Існування невiд’ємних розв’язкiв дробового параболiчного рiвняння в усьому просторi Kenzizi, T. Кензізі, Т. UDC 517.9 We study existence of nonnegative solutions for a parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ in $\mathbb{R}^{d}\times (0, T).$ Here $0 УДК 517.9 Вивчається задача існування невід'ємних розв'язків параболічного рівняння $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ на $\mathbb{R}^{d} \times (0, T).$ Тут $0 Institute of Mathematics, NAS of Ukraine 2019-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1498 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 8 (2019); 1064-1072 Український математичний журнал; Том 71 № 8 (2019); 1064-1072 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1498/482 Copyright (c) 2019 Kenzizi T. |
| spellingShingle | Kenzizi, T. Кензізі, Т. Existence of nonnegative solutions for a fractional parabolic equation in the whole space |
| title | Existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| title_alt | Існування невiд’ємних розв’язкiв дробового параболiчного рiвняння в усьому просторi |
| title_full | Existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| title_fullStr | Existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| title_full_unstemmed | Existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| title_short | Existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| title_sort | existence of nonnegative solutions for a fractional parabolic equation in the whole
space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1498 |
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