Existence results for doubly nonlinear parabolic equations with two lower order terms and $L^1$-data
UDC 517.9 We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data.
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2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507218814697472 |
|---|---|
| author | Benkirane, A. El, Hadfi Y. El, Moumni M. Бенкиран, А. Ель, Хадфі Ю. Ель, Мумні М. |
| author_facet | Benkirane, A. El, Hadfi Y. El, Moumni M. Бенкиран, А. Ель, Хадфі Ю. Ель, Мумні М. |
| author_sort | Benkirane, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:56:08Z |
| description | UDC 517.9
We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data. |
| first_indexed | 2026-03-24T02:05:49Z |
| format | Article |
| fulltext |
UDC 517.9
A. Benkirane (Lab. LAMA, Sidi Mohamed Ben Abdellah Univ., Atlas Fez, Morocco),
Y. El Hadfi (Lab. LIPIM, Nat. School Appl. Sci., Sultan Moulay Slimane Univ., Khouribga, Morocco),
M. El Moumni (Dep. Math., Chouaib Doukkali Univ., El Jadida, Morocco)
EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS
WITH TWO LOWER ORDER TERMS AND \bfitL \bfone -DATA
РЕЗУЛЬТАТИ ПРО IСНУВАННЯ РОЗВ’ЯЗКIВ ДВIЧI НЕЛIНIЙНИХ
ПАРАБОЛIЧНИХ РIВНЯНЬ З ДВОМА ЧЛЕНАМИ НИЖЧОГО ПОРЯДКУ
ТА \bfitL \bfone -ДАНИМИ
We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order
terms and L1 -data.
Вивчається проблема iснування перенормованого розв’язку для класу нелiнiйних параболiчних рiвнянь з двома
членами нижчого порядку та L1 -даними.
1. Introduction. We consider the following nonlinear parabolic problem:
\partial b(x, u)
\partial t
- \mathrm{d}\mathrm{i}\mathrm{v}(a(x, t, u,\nabla u)) + g(x, t, u,\nabla u) +H(x, t,\nabla u) = f in QT ,
b(x, u)(t = 0) = b(x, u0) in \Omega ,
u = 0 on \partial \Omega \times (0, T ),
(1.1)
where \Omega is a bounded open subset of \BbbR N , N \geq 1, T > 0, p > 1 and QT is the cylinder \Omega \times (0, T ).
The operator - \mathrm{d}\mathrm{i}\mathrm{v}(a(x, t, u,\nabla u)) is a Leray – Lions operator which is coercive and grows like
| \nabla u| p - 1 with respect to \nabla u, the function b(x, u) is an unbounded on u, and b(x, u0) \in L1(\Omega ). The
functions g and H are two Carathéodory functions with suitable assumptions see below. Finally the
datum f \in L1(QT ).
The problem (1.1) is encountered in a variety of physical phenomena and applications. For
instance, when b(x, u) = u, a(x, t, u,\nabla u) = | \nabla u| p - 2\nabla u, g = f = 0, H(x, t,\nabla u) = \lambda | \nabla u| q,
where q and \lambda are positive parameter, the equation in problem (1.1) can be viewed as the viscosity
approximation of Hamilton – Jacobi-type equation from stochastic control theory [18]. In particular,
when b(x, u) = u, a(x, t, u,\nabla u) = \nabla u, g = f = 0, H(x, t,\nabla u) = \lambda | \nabla u| 2, where \lambda is positive
parameter, the equation in problem (1.1) appears in the physical theory of growth and roughening of
surfaces, where it is known as the Kardar – Parisi – Zhang equation [14]. We introduce the definition
of the renormalized solutions for problem (1.1) as follows. This notion was introduced by P.-L. Lions
and Di Perna [12] for the study of Boltzmann equation (see also P.-L. Lions [17] for a few applications
to fluid mechanics models). This notion was then adapted to an elliptic version of (1.1) by Boccardo
et al. [9] when the right-hand side is in W - 1,p\prime (\Omega ), by Rakotoson [24] when the right-hand side
being a in L1(\Omega ), and by Dal Maso, Murat, Orsina and Prignet [10] for the case of right-hand side
being a general measure data, see also [19, 20].
For b(x, u) = u and H = 0, the existence of a weak solution to problem (1.1) (which belongs to
Lm(0, T ;W 1,m
0 (\Omega )) with p > 2 - 1
N + 1
and m <
p(N + 1) - N
N + 1
was proved in [8] (see also [7])
c\bigcirc A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI, 2019
610 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 611
where g = 0, and in [23] where g = 0, and in [11, 21, 22]. When the function g(x, t, u,\nabla u) \equiv g(u)
is independent on the (x, t,\nabla u) and g is continuous, the existence of a renormalized solution to
problem (1.1) is proved in [5]. Otherwise, recently in [1] is proved the existence of a renormalized
solution to problem (1.1) where the variational case.
The scope of the present paper is to prove an existence result for renormalized solutions to a
class of problems (1.1) with two lower order terms and L1-data. The difficulties connected to our
problem (1.1) are due to the presence of the two terms g and H which induce a lack of coercivity,
noncontrolled growth of the function b(x, s) with respect to s, the functions a(x, t, u,\nabla u) do not
belong to (L1
loc(QT ))
N in general, and the data b(x, u0), f are only integrable.
The rest of this article is organized as follows. In Section 2 we make precise all the assumptions
on b, a, g, H, u0, we also give the concept of a renormalized solution for the problem (1.1). In
Section 3 we establish the existence of our main results.
2. Essential assumptions and different notions of solutions. Throughout the paper, we assume
that the following assumptions hold true. Let \Omega is a bounded open set of \BbbR N , N \geq 1, T > 0 is
given and we set QT = \Omega \times (0, T ), and
b : \Omega \times \BbbR \rightarrow \BbbR is a Carathéodory function,
such that for every x \in \Omega , b(x, .) is a strictly increasing C1-function with b(x, 0) = 0. Next, for any
k > 0, there exists \lambda k > 0 and functions Ak \in L\infty (\Omega ) and Bk \in Lp(\Omega ) such that
\lambda k \leq \partial b(x, s)
\partial s
\leq Ak(x) and
\bigm| \bigm| \bigm| \bigm| \nabla x
\biggl(
\partial b(x, s)
\partial s
\biggr) \bigm| \bigm| \bigm| \bigm| \leq Bk(x), (2.1)
for almost every x \in \Omega , for every s such that | s| \leq k, we denote by \nabla x
\biggl(
\partial b(x, s)
\partial s
\biggr)
the gradient of
\partial b(x, s)
\partial s
defined in the sense of distributions.
Let a : QT \times \BbbR \times \BbbR N \rightarrow \BbbR N be a Carathéodory function, such that\bigm| \bigm| a(x, t, s, \xi )\bigm| \bigm| \leq \beta
\bigl[
k(x, t) + | s| p - 1 + | \xi | p - 1
\bigr]
, (2.2)
for a.e. (x, t) \in QT , all (s, \xi ) \in \BbbR \times \BbbR N , some positive function k(x, t) \in Lp\prime (QT ) and \beta > 0,\bigl[
a(x, t, s, \xi ) - a(x, t, s, \eta )
\bigr]
(\xi - \eta ) > 0 for all (\xi , \eta ) \in \BbbR N \times \BbbR N , with \xi \not = \eta , (2.3)
a(x, t, s, \xi )\xi \geq \alpha | \xi | p, where \alpha is a strictly positive constant. (2.4)
Furthermore, let g(x, t, s, \xi ) : QT \times \BbbR \times \BbbR N \rightarrow \BbbR and H(x, t, \xi ) : QT \times \BbbR N \rightarrow \BbbR are two
Carathéodory functions which satisfy, for almost every (x, t) \in QT and for all s \in \BbbR , \xi \in \BbbR N , the
following conditions: \bigm| \bigm| g(x, t, s, \xi )\bigm| \bigm| \leq L1(| s| )
\bigl(
L2(x, t) + | \xi | p
\bigr)
, (2.5)
g(x, t, s, \xi )s \geq 0, (2.6)
where L1 : \BbbR + \rightarrow \BbbR + is a continuous increasing function, while L2(x, t) is positive and belongs to
L1(QT ),
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
612 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
\exists \delta > 0, \nu > 0 \forall | s| \geq \delta :
\bigm| \bigm| g(x, t, s, \xi )\bigm| \bigm| \geq \nu | \xi | p, (2.7)\bigm| \bigm| H(x, t, \xi )
\bigm| \bigm| \leq h(x, t)| \xi | p - 1, where h(x, t) is positive and belongs to Lp(QT ). (2.8)
We recall that, for k > 1 and s in \BbbR , the truncation is defined as Tk(s) = \mathrm{m}\mathrm{a}\mathrm{x}( - k,\mathrm{m}\mathrm{i}\mathrm{n}(k, s)).
We shall use the following definition of renormalized solution for problem (1.1) in the following
sense.
Definition 1. Let f \in L1(QT ) and b(\cdot , u0(\cdot )) \in L1(\Omega ). A renormalized solution of prob-
lem (1.1) is a function u defined on QT , satisfying the following conditions:
Tk(u) \in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
for all k \geq 0 and b(x, u) \in L\infty \bigl(
0, T ;L1(\Omega )
\bigr)
, (2.9)\int
\{ m\leq | u| \leq m+1\}
a(x, t, u,\nabla u)\nabla u dx dt\rightarrow 0 as m\rightarrow +\infty , (2.10)
\partial BS(x, u)
\partial t
- \mathrm{d}\mathrm{i}\mathrm{v}
\Bigl(
S\prime (u)a(x, t, u,\nabla u)
\Bigr)
+ S\prime \prime (u)a(x, t, u,\nabla u)\nabla u +
+ g(x, t, u,\nabla u)S\prime (u) +H(x, t,\nabla u)S\prime (u) = fS\prime (u) in \scrD \prime (QT ), (2.11)
for all functions S \in W 2,\infty (\BbbR ) which are piecewise \scrC 1(\BbbR ), such that S\prime has a compact support in
\BbbR and
BS(x, u)(t = 0) = BS(x, u0) in \Omega , where BS(x, z) =
z\int
0
\partial b(x, r)
\partial r
S\prime (r) dr. (2.12)
Remark 1. Equation (2.11) is formally obtained through pointwise multiplication of (1.1) by
S\prime (u). However, while a(x, t, u,\nabla u), g(x, t, u,\nabla u), and H(x, t,\nabla u) does not in general make
sense in \scrD \prime (QT ), all the terms in (2.11) have a meaning in \scrD \prime (QT ).
Indeed, if M is such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}S\prime \subset [ - M,M ], the following identifications are made in (2.11):
| BS(x, u)| = | BS(x, TM (u))| \leq M\| S\prime \| L\infty (\BbbR )AM (x) belongs to L\infty (\Omega ) since AM is a bounded
function;
S\prime (u)a(x, t, u,\nabla u) identifies with S\prime (u)a
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
a.e. in QT ; since | TM (u)| \leq
\leq M a.e. in QT and S\prime (u) \in L\infty (QT ), we obtain from (2.2) and (2.9) that
S\prime (u)a
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
\in
\bigl(
Lp\prime (QT )
\bigr) N
;
S\prime \prime (u)a(x, t, u,\nabla u)\nabla u identifies with S\prime \prime (u)a
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
\nabla TM (u) and S\prime \prime (u)a
\bigl(
x,
t, TM (u),\nabla TM (u)
\bigr)
\nabla TM (u) \in L1(QT );
S\prime (u)
\Bigl(
g
\bigl(
x, t, u,\nabla u
\bigr)
+H(x, t,\nabla u)
\Bigr)
identifies with S\prime (u)
\Bigl(
g
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
+H
\bigl(
x, t,
\nabla TM (u)
\bigr) \Bigr)
a.e. in QT ; since | TM (u)| \leq M a.e. in QT and S\prime (u) \in L\infty (QT ), we obtain from (2.2),
(2.5), and (2.8) that
S\prime (u)
\Bigl(
g
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
+H(x, t,\nabla TM (u))
\Bigr)
\in L1(QT );
S\prime (u)f belongs to L1(QT ).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 613
The above considerations show that (2.11) holds in \scrD \prime (QT ) and
\partial BS(x, u)
\partial t
\in Lp\prime
\bigl(
0, T ;W - 1, p\prime (\Omega )
\bigr)
+ L1(QT ). (2.13)
The properties of S, assumptions (2.1) and (2.10) imply that\bigm| \bigm| \nabla BS(x, u)
\bigm| \bigm| \leq \| AM\| L\infty (\Omega )
\bigm| \bigm| \nabla TM (u)
\bigm| \bigm| \| S\prime \| L\infty (\BbbR ) +M\| S\prime \| L\infty (\BbbR )BM (x) (2.14)
and
BS(x, u) belongs to Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
. (2.15)
Then (2.13) and (2.15) imply that BS(x, u) belongs to C0
\bigl(
[0, T ];L1(\Omega )
\bigr)
(for a proof of this trace
result see [21]), so that the initial condition (2.12) makes sense.
Also remark that, for every S \in W 1,\infty (\BbbR ), nondecreasing function such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}S\prime \subset [ - M,M ],
in view of (2.1) we have
\lambda M | S(r) - S(r\prime )| \leq
\bigm| \bigm| \bigm| BS(x, r) - BS(x, r
\prime )
\bigm| \bigm| \bigm| \leq
\leq \| AM\| L\infty (\Omega )| S(r) - S(r\prime )| , a.e. x \in \Omega , \forall r, r\prime \in \BbbR .
3. Statements of results. The main results of this article are stated as follows.
Theorem 1. Let f \in L1(QT ) and u0 is a measurable function such that b(\cdot , u0) \in L1(\Omega ).
Assume that (2.1) – (2.8) hold true. Then there exists a renormalized solution u of problem (1.1) in
the sense of Definition 1.
Proof. The proof of Theorem 1 is done in five steps.
Step 1: Approximate problem and a priori estimates. For n > 0, let us define the following
approximation of b, f and u0.
First, set bn(x, r) = b(x, Tn(r))+
1
n
rbn is a Carathéodory function and satisfies (2.1), there exist
\lambda n > 0 and functions An \in L\infty (\Omega ) and Bn \in Lp(\Omega ) such that \lambda n \leq \partial bn(x, s)
\partial s
\leq An(x) and\bigm| \bigm| \bigm| \bigm| \nabla x
\biggl(
\partial bn(x, s)
\partial s
\biggr) \bigm| \bigm| \bigm| \bigm| \leq Bn(x), a.e. in \Omega , s \in \BbbR .
Next, set
gn(x, t, s, \xi ) =
g(x, t, s, \xi )
1 +
1
n
| g(x, t, s, \xi )|
and Hn(x, t, \xi ) =
H(x, t, \xi )
1 +
1
n
| H(x, t, \xi )|
.
Note that
\bigm| \bigm| gn(x, t, s, \xi )\bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
| g(x, t, s, \xi )| ;n
\bigr\}
and
\bigm| \bigm| Hn(x, t, \xi )
\bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
| H(x, t, \xi )| ;n
\bigr\}
. More-
over, since fn \in Lp\prime (QT ) and fn \rightarrow f a.e. in QT and strongly in L1(QT ) as n\rightarrow \infty ,
u0n \in \scrD (\Omega ), bn(x, u0n) \rightarrow b(x, u0) a.e. in \Omega and strongly in L1(\Omega ) as n\rightarrow \infty .
(3.1)
Let us now consider the approximate problem
\partial bn(x, un)
\partial t
- \mathrm{d}\mathrm{i}\mathrm{v}(a(x, t, un,\nabla un)) + gn(x, t, un,\nabla un) +Hn(x, t,\nabla un) = fn in QT ,
bn(x, un)(t = 0) = bn(x, u0n) in \Omega ,
un = 0 in \partial \Omega \times (0, T ).
(3.2)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
614 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
Since fn \in Lp\prime
\bigl(
0, T ;W - 1, p\prime (\Omega )
\bigr)
, proving existence of a weak solution un \in Lp
\bigl(
0, T ;W 1, p
0 (\Omega )
\bigr)
of (3.2) is an easy task (see, e.g., [16, p. 271]), i.e.,
T\int
0
\biggl\langle
\partial bn(x, un)
\partial t
, v
\biggr\rangle
dt+
\int
QT
a(x, t, un,\nabla un)\nabla v dx dt +
+
\int
QT
gn(x, t, un,\nabla un)v dx dt+
\int
QT
Hn(x, t,\nabla un)v dx dt =
=
\int
QT
fnv dx dt for all v \in Lp
\bigl(
0, T ;W 1,p(\Omega )
\bigr)
\cap L\infty (QT ).
Now, we prove the solution un of problem (3.2) is bounded in Lp
\bigl(
0, T ;W 1, p
0 (\Omega )
\bigr)
.
Lemma 1. Let un \in Lp
\bigl(
0, T ;W 1, p
0 (\Omega )
\bigr)
be a weak solution of (3.2). Then the following
estimates hold:
\| un\| Lp
\bigl(
0,T ;W 1, p
0 (\Omega )
\bigr) \leq D, (3.3)
where D depend only on \Omega , T, N, p, p\prime , f, and \| h\| Lp(QT ).
Proof. To get (3.3), we divide the integral
\int
QT
| \nabla un| p dx dt in two parts and we prove the
following estimates: for all k \geq 0, \int
\{ | un| \leq k\}
\bigm| \bigm| \nabla un\bigm| \bigm| p dx dt \leq M1k, (3.4)
and \int
\{ | un| >k\}
\bigm| \bigm| \nabla un\bigm| \bigm| p dx dt \leq M2, (3.5)
where M1 and M2 are positive constants. In what follows we will denote by Mi, i = 3, 4, . . . , some
generic positive constants. We suppose p < N (the case p \geq N is similar). For \varepsilon > 0 and s \geq 0,
we define
\varphi \varepsilon (r) =
\left\{
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(r) if | r| > s+ \varepsilon ,
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(r)(| r| - s)
\varepsilon
if s < | r| \leq s+ \varepsilon ,
0 otherwise.
We choose v = \varphi \varepsilon (un) as test function in (3.2), we have\left[ \int
\Omega
Bn
\varphi \varepsilon
(x, un) dx
\right] T
0
+
\int
QT
a(x, t, un,\nabla un)\nabla (\varphi \varepsilon (un)) dx dt +
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 615
+
\int
QT
gn(x, t, un,\nabla un)\varphi \varepsilon (un) dx dt+
\int
QT
Hn(x, t,\nabla un)\varphi \varepsilon (un) dx dt =
=
\int
QT
fn\varphi \varepsilon (un) dx dt,
where
Bn
\varphi \varepsilon
(x, r) =
r\int
0
\partial bn(x, s)
\partial s
\varphi \varepsilon (s) ds.
By using Bn
\varphi \varepsilon
(x, r) \geq 0, gn(x, t, un,\nabla un)\varphi \varepsilon (un) \geq 0, (2.4), (2.8), Hölder inequality and letting
\varepsilon go to zero, we obtain
- d
ds
\int
\{ s<| un| \}
\alpha | \nabla un| p dx dt \leq
\leq
\int
\{ s<| un| \}
| fn| dx dt+
+\infty \int
s
\left( - d
d\sigma
\int
\{ \sigma <| un| \}
hp dx dt
\right)
1
p
\left( - d
d\sigma
\int
\{ \sigma <| un| \}
| \nabla un| p dx dt
\right)
1
p\prime
d\sigma ,
where \{ s < | un| \} denotes the set
\bigl\{
(x, t) \in QT , s < | un(x, t)|
\bigr\}
and \mu (s) stands for the distribution
function of un, that is \mu (s) =
\bigm| \bigm| \bigl\{ (x, t) \in QT , | un(x, t)| > s
\bigr\} \bigm| \bigm| for all s \geq 0.
On the other hand, from Fleming – Rishel coarea formula and isoperimetric inequality, we have,
for almost every s > 0,
NC
1
N
N
\bigl(
\mu (s)
\bigr) N - 1
N \leq - d
ds
\int
\{ s<| un| \}
| \nabla un| pdx dt, (3.6)
where CN is the measure of the unit ball in \BbbR N . By using the Hölder’s inequality, we obtain that,
for almost every s > 0,
- d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt \leq
\bigl(
- \mu \prime (s)
\bigr) 1
p\prime
\left( - d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt
\right)
1
p
. (3.7)
Then, combining (3.6) and (3.7), we obtain, for almost every s > 0,
1 \leq
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\bigl( - \mu \prime (s)\bigr) 1
p\prime
\left( - d
ds
\int
\{ s<| un| \}
| \nabla un| pdx dt
\right)
1
p
. (3.8)
By using (3.8), we have
\alpha
\left( - d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt
\right)
1
p\prime
\leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
616 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
\leq
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\bigl( - \mu \prime (s)\bigr) 1
p\prime
\left( \int
\{ s<| un| \}
| fn| dx dt
\right) +
+
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\bigl( - \mu \prime (s)\bigr) 1
p\prime \times
\times
+\infty \int
s
\left( - d
d\sigma
\int
\{ \sigma <| un| \}
hp dx dt
\right)
1
p
\left( - d
d\sigma
\int
\{ \sigma <| un| \}
| \nabla un| p dx dt
\right)
1
p\prime
d\sigma . (3.9)
Now, we consider two functions B and \psi (see Lemma 2.2 of [2]) defined by
\int
\{ s<| un| \}
hp(x, t) dx dt =
\mu (s)\int
0
Bp(\sigma ) d\sigma (3.10)
and
\psi (s) =
\int
\{ s<| un| \}
| fn| dx dt. (3.11)
We have \| B\|
Lp
\bigl(
0,T ;W 1,p
0 (\Omega )
\bigr) \leq \| h\|
Lp
\bigl(
0,T ;W 1,p
0 (\Omega )
\bigr) and | \psi (s)| \leq \| fn\| L1(QT ). From (3.9), (3.10),
and (3.11) we get
\alpha
\left( - d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt
\right)
1
p\prime
\leq
\leq
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\bigl( - \mu \prime (s)\bigr) 1
p\prime \psi (s) +
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\times
\times
\bigl(
- \mu \prime (s)
\bigr) 1
p\prime
+\infty \int
s
B
\bigl(
\mu (\nu )
\bigr) \bigl(
- \mu \prime (\nu )
\bigr) 1
p
\left( - d
d\nu
\int
\{ \nu <| un| \}
| \nabla un| pdxdt
\right)
1
p\prime
d\nu .
From Gronwall’s lemma (see [3]), we obtain
\alpha
\left( - d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt
\right)
1
p\prime
\leq
\leq
\Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (s)
\bigr) 1
N
- 1\bigl( - \mu \prime (s)\bigr) 1
p\prime \psi (s) + (NC
1
N
N ) - 1
\bigl(
\mu (s)
\bigr) 1
N
- 1\times
\times
\bigl(
- \mu \prime (s)
\bigr) 1
p\prime
+\infty \int
s
\Bigl[ \Bigl(
NC
1
N
N
\Bigr) - 1\bigl(
\mu (\sigma )
\bigr) 1
N
- 1
\psi (\sigma )
\Bigr]
B
\bigl(
\mu (\sigma )
\bigr) \bigl(
- \mu \prime (\sigma )
\bigr)
\times
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 617
\times \mathrm{e}\mathrm{x}\mathrm{p}
\left( \sigma \int
s
\Bigl(
NC
1
N
N
\Bigr) - 1
B
\bigl(
\mu (r)
\bigr) \bigl(
\mu (r)
\bigr) 1
N
- 1\bigl( - \mu \prime (r)\bigr) dr
\right) d\sigma . (3.12)
Now, by a variable of change and by Hölder inequality, we estimate the argument of the exponential
function on the right-hand side of (3.12):
\sigma \int
s
B
\bigl(
\mu (r)
\bigr) \bigl(
\mu (r)
\bigr) 1
N
- 1\bigl( - \mu \prime (r)\bigr) dr = \sigma \int
s
B(z)z
1
N
- 1dz \leq
\leq
| \Omega | \int
0
B(z)z
1
N
- 1dz \leq \| B\| Lp
\left( | \Omega | \int
0
z(
1
N
- 1)p\prime
\right)
1
p\prime
.
Raising to the power p\prime in (3.12) and we can write
- d
ds
\int
\{ s<| un| \}
| \nabla un| p dx dt \leq M1,
where M1 depend only on \Omega , N, p, p\prime , f, \alpha , and \| h\| Lp(QT ), integrating between 0 and k, (3.4) is
proved.
We now give the proof of (3.5), using Tk(un) as test function in (3.2), gives\left[ \int
\Omega
Bn
k (x, un) dx
\right] T
0
+
\int
\Omega
a(x, t, un,\nabla un)\nabla Tk(un) dx dt+
+
\int
\Omega
(gn(x, t, un,\nabla un) +Hn(x, t,\nabla un))Tk(un) dx dt =
=
\int
\Omega
fnTk(un) dx dt,
where
Bn
k (x, r) =
r\int
0
\partial bn(x, s)
\partial s
Tk(s) ds.
By using (2.8), we deduce that\left[ \int
\Omega
Bn
k (x, un) dx
\right] T
0
+
\int
\{ | un| \leq k\}
a(x, t, un,\nabla un)\nabla un dx dt+
+
\int
\{ | un| \leq k\}
gn(x, t, un,\nabla un)un dx+
\int
\{ | un| >k\}
gn(x, t, un,\nabla un)Tk(un) dx dt \leq
\leq
\int
\Omega
fnTk(un) dx dt+
\int
\Omega
h(x, t)| \nabla un| p - 1| Tk(un)| dx dt,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
618 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
and by using the fact that Bn
k (x, r) \geq 0, gn(x, t, un,\nabla un)un \geq 0 and (2.4), we have
\alpha
\int
\{ | un| \leq k\}
| \nabla un| p dx dt+
\int
\{ | un| >k\}
g(x, un,\nabla un)Tk(un) dx dt \leq
\leq k\| f\| L1 + k
\int
\{ | un| \leq k\}
h(x, t)| \nabla un| p - 1 dx dt +
+ k
\int
\{ | un| \geq k\}
h(x, t)| \nabla un| p - 1 dx dt.
By Hölder inequality and (3.4), (2.7) and applying Young’s inequality, we get, for all k > \delta ,
\nu k
\int
\{ | un| >k\}
| \nabla un| p dx dt \leq
\leq k\| f\| L1(QT ) + k
1+ 1
p\prime M1\| h\| LpQT ) + k
\int
\{ | un| >k\}
h(x, t)| \nabla un| p - 1 dx dt \leq
\leq k\| f\| L1(QT ) + k
1+ 1
p\prime M1\| h\| LpQT ) +M6k\| h\| pLp +
1
p\prime
\nu k
\int
\{ | un| >k\}
| \nabla un| p dx dt.
Hence,\biggl(
1 - 1
p\prime
\biggr) \int
\{ | un| >k\}
| \nabla un| p dx dt \leq M3\| f\| L1(QT ) + k
1
p\prime M5\| h\| Lp(QT ) +M7\| h\| pLp . (3.13)
Lemma 1 is proved.
Then there exists u \in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
such that, for some subsequence
un \rightharpoonup u weakly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
(3.14)
we conclude that \bigm\| \bigm\| Tk(un)\bigm\| \bigm\| pLp(0,T ;W 1,p
0 (\Omega ))
\leq c2k. (3.15)
We deduce from the above inequalities, (2.1) and (3.15), that\int
\Omega
Bn
k (x, un) dx \leq Ck, (3.16)
where Bn
k (x, z) =
\int z
0
\partial bn(x, s)
\partial s
Tk(s) ds.
Now, we turn to prove the almost every convergence of un and bn(x, un). Consider now a
function nondecreasing \xi k \in C2(\BbbR ) such that \xi k(s) = s for | s| \leq k
2
and \xi k(s) = k for | s| \geq k.
Multiplying the approximate equation by \xi \prime k(un), we obtain
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 619
\partial Bn
\xi (x, un)
\partial t
- \mathrm{d}\mathrm{i}\mathrm{v}
\Bigl(
a(x, t, un,\nabla un)\xi \prime k(un)
\Bigr)
+ a(x, t, un,\nabla un)\xi \prime \prime k(un)\nabla un+
+
\Bigl(
gn(x, t, un,\nabla un) +Hn(x, t,\nabla un)
\Bigr)
\xi \prime k(un) =
= fn\xi
\prime
k(un), (3.17)
in the sense of distributions, where
Bn
\xi (x, z) =
z\int
0
\partial bn(x, s)
\partial s
\xi \prime k(s) ds.
As a consequence of (3.15), we deduce that \xi k(un) is bounded in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
and
\partial Bn
\xi (x, un)
\partial t
is bounded in L1(QT ) + Lp\prime
\bigl(
0, T ;W - 1,p\prime (\Omega )
\bigr)
. Due to the properties of \xi k and (2.1), we conclude
that
\partial \xi k(un)
\partial t
is bounded in L1(QT ) + Lp\prime
\bigl(
0, T ;W - 1,p\prime (\Omega )
\bigr)
, which implies that \xi k(un) strongly
converges in L1(QT ) (see [21]).
Due to the choice of \xi k, we conclude that for each k, the sequence Tk(un) converges almost
everywhere in QT , which implies that un converges almost everywhere to some measurable function
u in QT . Thus, by using the same argument as in [4, 5, 25], we can show
un \rightarrow u a.e. in QT , (3.18)
bn(x, un) \rightarrow b(x, u) a.e. in QT .
We can deduce from (3.15) that
Tk(un)\rightharpoonup Tk(u) weakly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
,
which implies, by using (2.2), for all k > 0, that there exists a function a \in (Lp\prime (QT ))
N , such that
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\rightharpoonup a weakly in
\bigl(
Lp\prime (QT )
\bigr) N
. (3.19)
We now establish that b(., u) belongs to L\infty \bigl(
0, T ;L1(\Omega )
\bigr)
. Using (3.18) and passing to the
limit-inf in (3.16) as n tends to +\infty , we obtain
1
k
\int
\Omega
Bk(x, u)(\tau ) dx \leq C
for almost any \tau in (0, T ). Due to the definition of Bk(x, s) and the fact that
1
k
Bk(x, u) converges
pointwise to b(x, u), as k tends to +\infty , shows that b(x, u) belong to L\infty \bigl(
0, T ;L1(\Omega )
\bigr)
.
Lemma 2. Let un be a solution of the approximate problem (3.2). Then
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\int
\{ m\leq | un| \leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt = 0. (3.20)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
620 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
Proof. We use T1
\bigl(
un - Tm(un)
\bigr) +
= \alpha m(un) \in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
\cap L\infty (QT ) as test function
in (3.2). Then we have
T\int
0
\biggl\langle
\partial bn(x, un)
\partial t
; \alpha m(un)
\biggr\rangle
dt+
\int
\{ m\leq un\leq m+1\}
a(x, t, un,\nabla un)\nabla un\alpha \prime
m(un) dx dt +
+
\int
QT
\bigl(
gn(x, t, un,\nabla un) +Hn(x, t,\nabla un)
\bigr)
\alpha m(un) dx dt \leq
\leq
\int
QT
\bigm| \bigm| fn\alpha m(un)
\bigm| \bigm| dx dt,
which, by setting Bn
m(x, r) =
\int r
0
\partial bn(x, s)
\partial s
\alpha m(s) ds, (2.6) and (2.8) gives
\int
\Omega
Bn
m(x, un)(T ) dx+
\int
\{ m\leq un\leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt \leq
\leq
\int
\{ m\leq un\}
| fn| dx dt+
\int
QT
h(x, t)| \nabla un| p - 1 dx dt.
Now we use Hölder’s inequality and (3.3), in order to deduce\int
\Omega
Bn
m(x, un)(T ) dx+
\int
\{ m\leq un\leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt \leq
\leq
\int
\{ m\leq un\}
| fn| dx dt+ c1
\left( \int
\{ m\leq un\}
| h(x, t)| p dx dt
\right)
1
p\prime
.
Since Bn
m(x, un)(T ) \geq 0 and the strong convergence of fn in L1(QT ), by Lebesgue’s theorem, we
have
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\int
\{ m\leq un\}
| fn| dx dt = 0.
Similarly, since h \in Lp(QT ), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\left( \int
\{ m\leq un\}
| h(x, t)| p dx dt
\right)
1
p\prime
= 0.
We conclude that
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\int
\{ m\leq un\leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt = 0. (3.21)
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 621
On the other hand, using T1
\bigl(
un - Tm(un)
\bigr) -
as test function in (3.2) and reasoning as in the proof
of (3.21) we deduce that
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\int
\{ - (m+1)\leq un\leq - m\}
a(x, t, un,\nabla un)\nabla un dx dt = 0. (3.22)
Thus, (3.20) follows from (3.21) and (3.22).
Step 2: Almost everywhere convergence of the gradients. This step is devoted to introduce
for k \geq 0 fixed a time regularization of the function Tk(u) in order to perform the monotonicity
method (the proof of this steps is similar the Step 4 in [5]). This kind of regularization has been
first introduced by R. Landes (see Lemma 6 and Proposition 3 [15, p. 230] and Proposition 4 in [15,
p. 231]). For k > 0 fixed, and let \varphi (t) = te\gamma t
2
, \gamma > 0. It is well known that when \gamma >
\biggl(
L1(k)
2\alpha
\biggr) 2
,
one has
\varphi \prime (s) -
\biggl(
L1(k)
\alpha
\biggr)
| \varphi (s)| \geq 1
2
for all s \in \BbbR . (3.23)
Let \{ \psi i\} \subset \scrD (\Omega ) be a sequence which converge strongly to u0 in L1(\Omega ). Set wi
\mu = (Tk(u))\mu +
+ e - \mu t Tk(\psi i), where (Tk(u))\mu is the mollification with respect to time of Tk(u). Note that wi
\mu is a
smooth function having the following properties:
\partial wi
\mu
\partial t
= \mu (Tk(u) - wi
\mu ), wi
\mu (0) = Tk(\psi i),
\bigm| \bigm| wi
\mu
\bigm| \bigm| \leq k, (3.24)
wi
\mu \rightarrow Tk(u) strongly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
as \mu \rightarrow \infty . (3.25)
We introduce the following function of one real variable:
hm(s) =
\left\{
1 if | s| \leq m,
0 if | s| \geq m+ 1,
m+ 1 - | s| if m \leq | s| \leq m+ 1,
where m > k. Let \theta \mu ,in = Tk(un) - wi
\mu and z\mu ,in,m = \varphi
\bigl(
\theta \mu ,in
\bigr)
hm(un). By using in (3.2) the test
function z\mu ,in,m, we obtain since gn(x, t, un,\nabla un)\varphi
\bigl(
Tk(un) - wi
\mu
\bigr)
hm(un) \geq 0 on \{ | un| > k\} :
T\int
0
\biggl\langle
\partial bn(x, un)
\partial t
; \varphi (Tk(un) - wi
\mu )hm(un)
\biggr\rangle
dt +
+
\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( \theta \mu ,in
\bigr)
hm(un) dx dt +
+
\int
QT
a(x, t, un,\nabla un)\nabla un\varphi
\bigl(
\theta \mu ,in
\bigr)
h\prime m(un) dx dt +
+
\int
\{ | un| \leq k\}
gn(x, t, un,\nabla un)\varphi
\bigl(
Tk(un) - wi
\mu
\bigr)
hm(un) dx dt \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
622 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
\leq
\int
QT
\bigm| \bigm| fnz\mu ,in,m
\bigm| \bigm| dx dt+ \int
QT
\bigm| \bigm| Hn(x, t,\nabla un)z\mu ,in,m
\bigm| \bigm| dx dt. (3.26)
In the rest of this paper, we will omit for simplicity the denote \varepsilon (n, \mu , i,m) all quantities (possibly
different) such that
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
\mu \rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\varepsilon (n, \mu , i,m) = 0,
and this will be the order in which the parameters we use will tend to infinity, that is, first n, then
\mu , i and finally m. Similarly we will write only \varepsilon (n), or \varepsilon (n, \mu ), . . . to mean that the limits are made
only on the specified parameters.
We will deal with each term of (3.26). First of all, observe that\int
QT
\bigm| \bigm| fnz\mu ,in,m
\bigm| \bigm| dx dt+ \int
QT
\bigm| \bigm| Hn(x, t,\nabla un)z\mu ,in,m
\bigm| \bigm| dx dt = \varepsilon (n, \mu ),
since \varphi
\bigl(
Tk(un) - wi
\mu
\bigr)
hm(un) converges to \varphi
\bigl(
Tk(u) - (Tk(u))\mu + e - \mu tTk(\psi i)
\bigr)
hm(u) strongly in
Lp(QT ) and weakly - \ast in L\infty (QT ) as n\rightarrow \infty and finally \varphi
\bigl(
Tk(u) - (Tk(u))\mu +e
- \mu tTk(\psi i)
\bigr)
hm(u)
converges to 0 strongly in Lp(QT ) and weakly - \ast in L\infty (QT ) as \mu \rightarrow \infty . Thanks to (3.20) the third
and fourth integrals on the right-hand side of (3.26) tend to zero as n and m tend to infinity, and by
Lebesgue’s theorem and F \in (Lp\prime (QT ))
N , we deduce that the right-hand side of (3.26) converges to
zero as n, m and \mu tend to infinity. Since (Tk(un) - wi
\mu )hm(un)\rightharpoonup (Tk(u) - wi
\mu )hm(u) weakly\ast in
L1(QT ) and strongly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
and (Tk(u) - wi
\mu )hm(u) \rightharpoonup 0 weakly\ast in L1(QT ) and
strongly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
as \mu \rightarrow +\infty .
On the one hand, the definition of the sequence wi
\mu makes it possible to establish the following
lemma.
Lemma 3. For k \geq 0 we have
T\int
0
\biggl\langle
\partial bn(x, un)
\partial t
; \varphi (Tk(un) - wi
\mu )hm(un)
\biggr\rangle
dt \geq \varepsilon (n,m, \mu , i). (3.27)
Proof (see Blanchard and Redwane [6]).
On the other hand, the second term of the left-hand side of (3.26) can be written as\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
=
\int
\{ | un| \leq k\}
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt +
+
\int
\{ | un| >k\}
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
=
\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
dx dt +
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 623
+
\int
\{ | un| >k\}
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt,
since m > k and hm(un) = 1 on \{ | un| \leq k\} , we deduce that\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
=
\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
dx dt +
+
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr)
\times
\times \varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt +
+
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\nabla Tk(u)\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt -
-
\int
QT
a(x, t, un,\nabla un)\nabla wi
\mu \varphi
\prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
= K1 +K2 +K3 +K4. (3.28)
By using (2.2), (3.19) and Lebesgue’s theorem, we have a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr)
converges to
a
\bigl(
x, t, Tk(u),\nabla Tk(u)
\bigr)
strongly in
\bigl(
Lp\prime (QT )
\bigr) N
and \nabla Tk(un) converges to \nabla Tk(u) weakly in\bigl(
Lp(QT )
\bigr) N
. Then
K2 = \varepsilon (n). (3.29)
By using (3.19) and (3.25), we have
K3 =
\int
QT
a\nabla Tk(u) dx dt+ \varepsilon (n, \mu ). (3.30)
For what concerns K4 we can write, since hm(un) = 0 on \{ | un| > m+ 1\} :
K4 = -
\int
QT
a
\bigl(
x, t, Tm+1(un),\nabla Tm+1(un)
\bigr)
\nabla wi
\mu \varphi
\prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
= -
\int
\{ | un| \leq k\}
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\nabla wi
\mu \varphi
\prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt -
-
\int
\{ k<| un| \leq m+1\}
a
\bigl(
x, t, Tm+1(un),\nabla Tm+1(un)
\bigr)
\nabla wi
\mu \times
\times \varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt,
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624 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
and, as above, by letting n\rightarrow \infty ,
K4 = -
\int
\{ | u| \leq k\}
a\nabla wi
\mu \varphi
\prime \bigl( Tk(u) - wi
\mu
\bigr)
dx dt -
-
\int
\{ k<| u| \leq m+1\}
a\nabla wi
\mu \varphi
\prime \bigl( Tk(u) - wi
\mu
\bigr)
hm(u) dx dt+ \varepsilon (n),
so that, by letting \mu \rightarrow \infty ,
K4 = -
\int
QT
a\nabla Tk(u) dx dt+ \varepsilon (n, \mu ). (3.31)
In view of (3.28), (3.29), (3.30), and (3.31), we conclude that\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tk(un) - \nabla wi
\mu
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
hm(un) dx dt =
=
\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr)
\varphi \prime \bigl( Tk(un) - wi
\mu
\bigr)
dx dt+ \varepsilon (n, \mu ). (3.32)
To deal with the third term of the left-hand side of (3.26), observe that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
QT
a(x, t, un,\nabla un)\nabla un\varphi
\bigl(
\theta \mu ,in
\bigr)
h\prime m(un) dx dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varphi (2k)
\int
\{ m\leq | un| \leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt.
Thanks to (3.20), we obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
QT
a(x, t, un,\nabla un)\nabla un\varphi
\bigl(
\theta \mu ,in
\bigr)
h\prime m(un) dx dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon (n,m). (3.33)
We now turn to fourth term of the left-hand side of (3.26), we can write\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\{ | un| \leq k\}
gn(x, t, un,\nabla un)\varphi
\bigl(
Tk(un) - wi
\mu
\bigr)
hm(un) dx dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\int
\{ | un| \leq k\}
L1(k)L2(x, t) +
\bigm| \bigm| \nabla Tk(un)\bigm| \bigm| p\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| hm(un) dx dt \leq
\leq L1(k)
\int
QT
L2(x, t)
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt +
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 625
+
L1(k)
\alpha
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\nabla Tk(un)
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt, (3.34)
since L2(x, t) belong to L1(QT ) it is easy to see that
L1(k)
\int
QT
L2(x, t)
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt = \varepsilon (n, \mu ).
On the other hand, the second term of the right-hand side of (3.34), write as
L1(k)
\alpha
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\nabla Tk(un)
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt =
=
L1(k)
\alpha
\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr) \bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt +
+
L1(k)
\alpha
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr) \bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt +
+
L1(k)
\alpha
\int
QT
a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr)
\nabla Tk(u)
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt,
and, as above, by letting first n then finally \mu go to infinity, we can easily seen, that each one of last
two integrals is of the form \varepsilon (n, \mu ). This implies that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\{ | un| \leq k\}
gn(x, t, un,\nabla un)\varphi
\bigl(
Tk(un) - wi
\mu
\bigr)
hm(un) dx dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq L1(k)
\alpha
\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr) \bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| dx dt+ \varepsilon (n, \mu ). (3.35)
Combining (3.26), (3.27), (3.32), (3.33), and (3.35), we get\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr) \biggl(
\varphi \prime (Tk(u) - wi
\mu ) -
L1(k)
\alpha
\bigm| \bigm| \varphi \bigl( Tk(un) - wi
\mu
\bigr) \bigm| \bigm| \biggr) dx dt \leq
\leq \varepsilon (n, \mu , i,m),
and so, thanks to (3.23), we have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
626 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, t, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
\times
\times
\bigl(
\nabla Tk(un) - \nabla Tk(u)
\bigr)
dx dt \leq \varepsilon (n).
Hence by passing to the limit sup over n, we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\int
QT
\Bigl(
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
- a
\bigl(
x, Tk(un),\nabla Tk(u)
\bigr) \Bigr)
(\nabla Tk(un) - \nabla Tk(u)) dx dt = 0.
This implies that
Tk(un) \rightarrow Tk(u) strongly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
for all k. (3.36)
Now, observe that, for every \sigma > 0,
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Bigl\{
(x, t) \in QT : | \nabla un - \nabla u| > \sigma
\Bigr\}
\leq
\leq \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Bigl\{
(x, t) \in QT : | \nabla un| > k
\Bigr\}
+\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Bigl\{
(x, t) \in QT : | u| > k
\Bigr\}
+
+\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Bigl\{
(x, t) \in QT :
\bigm| \bigm| \nabla Tk(un) - \nabla Tk(u)
\bigm| \bigm| > \sigma
\Bigr\}
,
then as a consequence of (3.36) we have that \nabla un converges to \nabla u in measure and, therefore,
always reasoning for a subsequence,
\nabla un \rightarrow \nabla u a.e. in QT ,
which implies
a
\bigl(
x, t, Tk(un),\nabla Tk(un)
\bigr)
\rightharpoonup a
\bigl(
x, t, Tk(u),\nabla Tk(u)
\bigr)
weakly in
\bigl(
Lp\prime (QT )
\bigr) N
. (3.37)
Step 3: Equi-integrability of Hn(x, t,\nabla un) and gn(x, t, un,\nabla un). We shall now prove
that Hn(x, t,\nabla un) converges to H(x, t,\nabla u) and gn(x, t, un,\nabla un) converges to g(x, t, u,\nabla u)
strongly in L1(QT ) by using Vitali’s theorem. Since Hn(x, t,\nabla un) \rightarrow H(x, t,\nabla u) a.e. QT
and gn(x, t, un,\nabla un) \rightarrow g(x, t, u,\nabla u) a.e. QT , thanks to (2.5) and (2.8), it suffices to prove that
Hn(x, t,\nabla un) and gn(x, t, un,\nabla un) are uniformly equi-integrable in QT . We will now prove that
H(x,\nabla un) is uniformly equi-integrable, we use Hölder’s inequality and (3.3), we have, for any
measurable subset E \subset QT ,
\int
E
\bigm| \bigm| H(x,\nabla un)
\bigm| \bigm| dx dt \leq
\left( \int
E
hp(x, t) dx dt
\right) 1
p
\left( \int
QT
| \nabla un| p dx dt
\right)
1
p\prime
\leq
\leq c1
\left( \int
E
hp(x, t) dx dt
\right) 1
p
,
which is small uniformly in n when the measure of E is small.
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 627
To prove the uniform equi-integrability of gn(x, t, un,\nabla un). For any measurable subset E \subset QT
and m \geq 0, \int
E
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt = \int
E\cap \{ | un| \leq m\}
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt +
+
\int
E\cap \{ | un| >m\}
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt \leq
\leq L1(m)
\int
E\cap \{ | un| \leq m\}
\bigl[
L2(x, t) + | \nabla un| p
\bigr]
dx dt +
+
\int
E\cap \{ | un| >m\}
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt =
= K1 +K2. (3.38)
For fixed m, we get
K1 \leq L1(m)
\int
E
\bigl[
L2(x, t) +
\bigm| \bigm| \nabla Tm(un)
\bigm| \bigm| p\bigr] dx dt,
which is thus small uniformly in n for m fixed when the measure of E is small
\bigl(
recall that Tm(un)
tends to Tm(u) strongly in Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr) \bigr)
. We now discuss the behavior of the second integral
of the right-hand side of (3.38), let \psi m be a function such that
\psi m(s) =
\left\{ 0 if | s| \leq m - 1,
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s) if | s| \geq m,
\psi \prime
m(s) = 1 if m - 1 < | s| < m.
We choose for m > 1, \psi m(un) as a test function in (3.2), and we obtain\left[ \int
\Omega
Bn
m(x, un)dx
\right] T
0
+
\int
QT
a(x, t, un,\nabla un)\nabla un\psi \prime
m(un) dx dt +
+
\int
QT
gn(x, t, un,\nabla un)\psi m(un) dx dt+
\int
QT
Hn(x, t,\nabla un)\psi m(un) dx dt =
=
\int
QT
fn\psi m(un) dx dt,
where Bn
m(x, r) =
\int r
0
\partial bn(x, s)
\partial s
\psi m(s) ds, which implies, since Bn
m(x, r) \geq 0 and using (2.4),
Hölder’s inequality\int
\{ m - 1\leq | un| \}
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt \leq \int
E
| Hn(x, t,\nabla un)| dx dt+
\int
\{ m - 1\leq | un| \}
| f | dx dt,
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628 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
and by (3.3), we have
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\int
\{ | un| >m - 1\}
\bigm| \bigm| gn(x, t, un,\nabla un)\bigm| \bigm| dx dt = 0.
Thus we proved that the second term of the right-hand side of (3.38) is also small, uniformly in n
and in E when m is sufficiently large. Which shows that gn(x, t, un,\nabla un) and Hn(x, t,\nabla un) are
uniformly equi-integrable in QT as required, we conclude that
Hn(x, t,\nabla un) \rightarrow H(x, t,\nabla u) strongly in L1(QT ),
gn(x, t, un,\nabla un) \rightarrow g(x, t, u,\nabla u) strongly in L1(QT ).
(3.39)
Step 4: We prove that u satisfies (2.10).
Lemma 4. The limit u of the approximate solution un of (3.2) satisfies
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow +\infty
\int
\{ m\leq | u| \leq m+1\}
a(x, t, u,\nabla u)\nabla u dx dt = 0.
Proof. Note that for any fixed m \geq 0, one has\int
\{ m\leq | un| \leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt =
=
\int
QT
a(x, t, un,\nabla un)
\bigl(
\nabla Tm+1(un) - \nabla Tm(un)
\bigr)
dx dt =
=
\int
QT
a
\bigl(
x, t, Tm+1(un),\nabla Tm+1(un)
\bigr)
\nabla Tm+1(un) dx dt -
-
\int
QT
a
\bigl(
x, t, Tm(un),\nabla Tm(un)
\bigr)
\nabla Tm(un) dx dt.
According to (3.37) and (3.36), one can pass to the limit as n\rightarrow +\infty for fixed m \geq 0, to obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\{ m\leq | un| \leq m+1\}
a(x, t, un,\nabla un)\nabla un dx dt =
=
\int
QT
a
\bigl(
x, t, Tm+1(u),\nabla Tm+1(u)
\bigr)
\nabla Tm+1(u) dx dt -
-
\int
QT
a
\bigl(
x, t, Tm(u),\nabla Tm(u)
\bigr)
\nabla Tm(un) dx dt =
=
\int
\{ m\leq | un| \leq m+1\}
a(x, t, u,\nabla u)\nabla u dx dt. (3.40)
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EXISTENCE RESULTS FOR DOUBLY NONLINEAR PARABOLIC EQUATIONS . . . 629
Taking the limit as m\rightarrow +\infty in (3.40) and using the estimate (3.20) show that u satisfies (2.10) and
the proof is complete.
Step 5: We prove that u satisfies (2.11) and (2.12).
Let S be a function in W 2,\infty (\BbbR ) such that S\prime has a compact support. Let M be a positive real
number such that support of S\prime is a subset of [ - M,M ]. Pointwise multiplication of the approximate
equation (3.2) by S\prime (un) leads to
\partial Bn
S(x, un)
\partial t
- \mathrm{d}\mathrm{i}\mathrm{v}
\Bigl(
S\prime (un)a(x, t, un,\nabla un)
\Bigr)
+ S\prime \prime (un)a(x, t, un,\nabla un)\nabla un +
+ S\prime (un)
\Bigl(
gn(x, t, un,\nabla un) +Hn(x, t,\nabla un)
\Bigr)
= fS\prime (un) in \scrD \prime (QT ), (3.41)
where
Bn
S(x, z) =
z\int
0
\partial bn(x, r)
\partial r
S\prime (r) dr.
In what follows we pass to the limit in (3.41) as n tends to +\infty :
Limit of
\partial Bn
S(x, un)
\partial t
. Since S is bounded and continuous, un \rightarrow u a.e. in QT , implies that
Bn
S(x, un) converges to BS(x, u) a.e. in QT and L\infty (QT )-weak\ast . Then
\partial Bn
S(x, un)
\partial t
converges to
\partial BS(x, u)
\partial t
in \scrD \prime (QT ) as n tends to +\infty .
The limit of - \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
S\prime (un)a(x, t, un,\nabla un)
\bigr)
. Since \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(S\prime ) \subset [ - M,M ], we have for n \geq M :
S\prime (un)an(x, t, un,\nabla un) = S\prime (un)a
\bigl(
x, t, TM (un),\nabla TM (un)
\bigr)
a.e. in QT . The pointwise conver-
gence of un to u, (3.37) and the bounded character of S\prime yield, as n tends to +\infty : S\prime (un)an
\bigl(
x, t, un,
\nabla un
\bigr)
converges to S\prime (u)a
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
in
\bigl(
Lp\prime (QT )
\bigr) N
, and S\prime (u)a
\bigl(
x, t, TM (u),
\nabla TM (u)
\bigr)
has been denoted by S\prime (u)a(x, t, u,\nabla u) in equation (2.11).
The limit of S\prime \prime (un)a(x, t, un,\nabla un)\nabla un. Consider the “energy” term, S\prime \prime (un)a(x, t, un,
\nabla un)\nabla un = S\prime \prime (un)a
\bigl(
x, t, TM (un),\nabla TM (un)
\bigr)
\nabla TM (un) a.e. in QT .
The pointwise convergence of S\prime (un) to S\prime (u) and (3.37) as n tends to +\infty and the bounded
character of S\prime \prime permit us to conclude that S\prime \prime (un)an(x, t, un,\nabla un)\nabla un converges to S\prime \prime (u)a
\bigl(
x, t,
TM (u),\nabla TM (u)
\bigr)
\nabla TM (u) weakly in L1(QT ). Recall that
S\prime \prime (u)a
\bigl(
x, t, TM (u),\nabla TM (u)
\bigr)
\nabla TM (u) = S\prime \prime (u)a(x, t, u,\nabla u)\nabla u a.e. in QT .
The limit of S\prime (un)
\bigl(
gn(x, t, un,\nabla un)+Hn(x, t,\nabla un)
\bigr)
. From \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(S\prime ) \subset [ - M,M ], by (3.39),
we have S\prime (un)gn(x, t, un,\nabla un) converges to S\prime (u)g(x, t, u,\nabla u) strongly in L1(QT ) and
S\prime (un)Hn(x, t,\nabla un) converge to S\prime (u)H(x, t,\nabla u) strongly in L1(QT ), as n tends to +\infty .
The limit of S\prime (un)fn. Since un \rightarrow u a.e. in QT , we have S\prime (un)fn converges to S\prime (u)f
strongly in L1(QT ), as n tends to +\infty .
As a consequence of the above convergence result, we are in a position to pass to the limit as n
tends to +\infty in equation (3.41) and to conclude that u satisfies (2.11).
It remains to show that BS(x, u) satisfies the initial condition (2.12). To this end, firstly re-
mark that, S being bounded and in view of (2.14), (3.15), we have Bn
S(x, un) is bounded in
Lp
\bigl(
0, T ;W 1,p
0 (\Omega )
\bigr)
. Secondly, (3.41) and the above considerations on the behavior of the terms
of this equation show that
\partial Bn
S(x, un)
\partial t
is bounded in L1(QT ) + Lp\prime
\bigl(
0, T ;W - 1,p\prime (\Omega )
\bigr)
. As a conse-
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
630 A. BENKIRANE, Y. EL HADFI, M. EL MOUMNI
quence (see [21]), Bn
S(x, un)(t = 0) = Bn
S(x, u0n) converges to BS(x, u)(t = 0) strongly in L1(\Omega ).
On the other hand, the smoothness of S and in view of (3.1) imply that BS(x, u)(t = 0) = BS(x, u0)
in \Omega . As a conclusion, steps 1 – 5 complete the proof of Theorem 1.
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Received 17.06.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 5
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| id | umjimathkievua-article-1461 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:49Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4e/61b76fb570a607e470e12fbc8621124e.pdf |
| spelling | umjimathkievua-article-14612019-12-05T08:56:08Z Existence results for doubly nonlinear parabolic equations with two lower order terms and $L^1$-data Результати про iснування розв’язкiв двiчi нелiнiйних параболiчних рiвнянь з двома членами нижчого порядку та $L^1$ -даними Benkirane, A. El, Hadfi Y. El, Moumni M. Бенкиран, А. Ель, Хадфі Ю. Ель, Мумні М. UDC 517.9 We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data. УДК 517.9 Вивчається проблема існування перенормованого розв'язку для класу нелінійних параболічних рівнянь з двома членами нижчого порядку та $L^1$-даними. Institute of Mathematics, NAS of Ukraine 2019-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1461 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 5 (2019); 610-630 Український математичний журнал; Том 71 № 5 (2019); 610-630 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1461/445 Copyright (c) 2019 Benkirane A.; El Hadfi Y.; El Moumni M. |
| spellingShingle | Benkirane, A. El, Hadfi Y. El, Moumni M. Бенкиран, А. Ель, Хадфі Ю. Ель, Мумні М. Existence results for doubly nonlinear parabolic equations with two lower order terms and $L^1$-data |
| title | Existence results for doubly nonlinear parabolic
equations with two lower order terms and $L^1$-data |
| title_alt | Результати про iснування розв’язкiв двiчi нелiнiйних
параболiчних рiвнянь з двома членами нижчого порядку
та $L^1$ -даними |
| title_full | Existence results for doubly nonlinear parabolic
equations with two lower order terms and $L^1$-data |
| title_fullStr | Existence results for doubly nonlinear parabolic
equations with two lower order terms and $L^1$-data |
| title_full_unstemmed | Existence results for doubly nonlinear parabolic
equations with two lower order terms and $L^1$-data |
| title_short | Existence results for doubly nonlinear parabolic
equations with two lower order terms and $L^1$-data |
| title_sort | existence results for doubly nonlinear parabolic
equations with two lower order terms and $l^1$-data |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1461 |
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