Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems
We are interested in the existence of distributional solutions for two types of nonlinear evolution problems, whose models are (1.1) and (1.2) below. In the first one the nonlinear reaction term depends on the solution with a slightly superlinear growth. In the second one we consider a first order t...
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Інститут прикладної математики і механіки НАН України
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nasplib_isofts_kiev_ua-123456789-1246292025-02-23T18:01:04Z Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems Dall'Aglio, A. Giachett, D. Segura de Leon, S. We are interested in the existence of distributional solutions for two types of nonlinear evolution problems, whose models are (1.1) and (1.2) below. In the first one the nonlinear reaction term depends on the solution with a slightly superlinear growth. In the second one we consider a first order term depending also on the gradient of the solution in a quadratic way. The two problems are strictly related from the point of view of the a priori estimates we can obtain on their solutions. We point out that no boundedness is assumed on the data of the problems. This implies that the methods involving sub/super-solutions do not apply, and we have to use some convenient test-function to prove the a priori estimates. 2004 Article Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems / A. Dall'Aglio, D. Giachett, S. Segura de Leon // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 518-531. — Бібліогр.: 13 назв. — англ. 1810-3200 2000 MSC. 35B33, 35B45, 35K20, 35K55, 35K57. https://nasplib.isofts.kiev.ua/handle/123456789/124629 en Український математичний вісник application/pdf Інститут прикладної математики і механіки НАН України |
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We are interested in the existence of distributional solutions for two types of nonlinear evolution problems, whose models are (1.1) and (1.2) below. In the first one the nonlinear reaction term depends on the solution with a slightly superlinear growth. In the second one we consider a first order term depending also on the gradient of the solution in a quadratic way. The two problems are strictly related from the point of view of the a priori estimates we can obtain on their solutions. We point out that no boundedness is assumed on the data of the problems. This implies that the methods involving sub/super-solutions do not apply, and we have to use some convenient test-function to prove the a priori estimates. |
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Article |
| author |
Dall'Aglio, A. Giachett, D. Segura de Leon, S. |
| spellingShingle |
Dall'Aglio, A. Giachett, D. Segura de Leon, S. Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems Український математичний вісник |
| author_facet |
Dall'Aglio, A. Giachett, D. Segura de Leon, S. |
| author_sort |
Dall'Aglio, A. |
| title |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| title_short |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| title_full |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| title_fullStr |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| title_full_unstemmed |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| title_sort |
semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2004 |
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https://nasplib.isofts.kiev.ua/handle/123456789/124629 |
| citation_txt |
Semilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems / A. Dall'Aglio, D. Giachett, S. Segura de Leon // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 518-531. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT dallaglioa semilinearparabolicequationswithsuperlinearreactiontermsandapplicationtosomeconvectiondiffusionproblems AT giachettd semilinearparabolicequationswithsuperlinearreactiontermsandapplicationtosomeconvectiondiffusionproblems AT seguradeleons semilinearparabolicequationswithsuperlinearreactiontermsandapplicationtosomeconvectiondiffusionproblems |
| first_indexed |
2025-11-24T06:18:49Z |
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2025-11-24T06:18:49Z |
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| fulltext |
Український математичний вiсник
Том 1 (2004), № 4, 518 – 531
Semilinear Parabolic Equations with
Superlinear Reaction Terms, and Application to
Some Convection-diffusion Problems
Andrea Dall’Aglio, Daniela Giachetti
and Sergio Segura de León
(Presented by A. E. Shishkov)
Abstract. We are interested in the existence of distributional solutions
for two types of nonlinear evolution problems, whose models are (1.1)
and (1.2) below. In the first one the nonlinear reaction term depends
on the solution with a slightly superlinear growth. In the second one we
consider a first order term depending also on the gradient of the solution
in a quadratic way.
The two problems are strictly related from the point of view of the a
priori estimates we can obtain on their solutions. We point out that no
boundedness is assumed on the data of the problems. This implies that
the methods involving sub/super-solutions do not apply, and we have to
use some convenient test-function to prove the a priori estimates.
2000 MSC. 35B33, 35B45, 35K20, 35K55, 35K57.
Key words and phrases. Nonlinear parabolic problems, convection-
diffusion problems, superlinear reaction term, gradient term with quad-
ratic growth, existence, a priori estimates.
1. Introduction
In this paper we are interested in solving nonlinear parabolic problems
of the type
vt − ∆v = f(x, t)
(
1 + |v|
∣∣ log |v|
∣∣α) in Ω×]0, T [;
v(x, t) = 0 on ∂Ω×]0, T [;
v(x, 0) = v0(x) in Ω.
(1.1)
Received 12.11.2003
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
A. Dall’Aglio, D. Giachetti and S. Segura de León 519
Here 0 < α < 1, f(x, t) ∈ Lr(0, T ;Lq(Ω)) for convenient r and q, and v0 ∈
L2(Ω). This type of problems is strictly related to parabolic convection-
diffusion problems whose model is
ut − ∆u = β(u)|∇u|2 + g(x, t) in Ω×]0, T [;
u(x, t) = 0 on ∂Ω×]0, T [;
u(x, 0) = u0(x) in Ω,
(1.2)
where β(s) is a continuous function which grows like an arbitrary power
of s at ±∞.
In these model examples it is easy to see that it is possible to perform
a change of unknown function in (1.2), i.e., v = Ψ(u) (see (2.6) below
for the definition of the function Ψ(s)), and reduce problem (1.2) to a
problem which is similar to (1.1).
Nevertheless, we would like to consider also more general situations
where one has a nonlinear pseudomonotone operator as a principal part
in (1.2) and a general first order term which grows quadratically with
respect to the gradient. In this case one cannot change the unknown
function, but the previous remark suggests the use of convenient test
functions which replace this technique.
We give an existence result of distributional solutions for (1.1) via
test-function method under the hypothesis that f(x, t) ∈ Lr(0, T ;Lq(Ω)),
where the exponents r and q belong to a part of the so-called Aronson-
Serrin region in the (r, q)-plane, part which also depends on the value of
the parameter α in (1.1). As far as the initial datum is concerned, we
assume that v0 ∈ L2(Ω).
The a priori estimates that we will be able to obtain on a sequence
{vn} of approximating solutions of (1.1) will also provide a priori esti-
mates for a sequence of approximate solutions un for problem (1.2), and
therefore an existence result of distributional solutions for this problem.
Besides this application, the result obtained for problem (1.1) seems
to have an autonomous interest.
Let us recall some results concerning the existence of solutions of the
nonlinear heat equation
vt − ∆v = λh(v) in Ω×]0, T [;
v(x, t) = 0 on ∂Ω×]0, T [;
v(x, 0) = v0(x) in Ω,
(1.3)
with 0 ≤ v0 ∈ L∞(Ω), while h(s) is a positive function. In [5] Brezis,
520 Semilinear Parabolic Equations with...
Cazenave, Martel and Ramiandrisoa proved that if
+∞∫
ds
h(s)
< +∞ ,
then there is no solution for large λ. This shows that one cannot hope to
prove global existence for (1.1) if α > 1. On the other hand, if
+∞∫
ds
h(s)
= +∞ ,
and if the data v0(x) and f(x, t) are bounded, it is easy to prove the
existence of a global solution using sub/super-solutions independent on
x. However this method does not work if either v0 or f is unbounded.
One of the aims of this paper is to present some results in this case.
Another interesting remark is the comparison with the case α = 0
and f(x, t) = f(x): it is well known (see, for instance, [11]) that if f(x) ∈
Lq(Ω), with q ≥ N/2, there exists a solution of the following linear heat
equation with singular potential
vt − ∆v = f(x)v in Ω×]0, T [;
v(x, t) = 0 on ∂Ω×]0, T [;
v(x, 0) = v0(x) in Ω,
while, if q < N/2, one can have instantaneous and complete blow-up (for
instance this happens if f(x) = λ/|x|2, with λ large enough (see [2], and
[6] for more general potentials f(x)). In this case our result (see Theorem
2.1 below) states that one can allow α > 0, but has to pay a price by
assuming the stronger condition q > N/[2(1−α)] (see Remark 2.1 below).
As far as the quasi-linear problem (1.2) is concerned, existence results
have been given in [3] under the assumptions that the data f(x, t) and
u0(x) are bounded, via a sub/super-solution method. For more general
data f and u0, the result we obtain for (1.2) (see Theorem 2.2 in next
Section) improves previous results proved in a wider framework in [7]. In
that paper (see also [8] and [10]) a special condition is assumed which
prevents one from considering functions β(s) which tend to +∞ for s→
±∞.
The plan of the paper is as follows: the next section is devoted to
stating the assumptions and the main results of the paper. In Section 3
we will define the approximate problems and prove the related a priori
estimates. In the final section we will study the limiting process.
A. Dall’Aglio, D. Giachetti and S. Segura de León 521
2. Assumptions and main results
Let Ω be a bounded open set in RN , T > 0, QT = Ω×]0, T [, ΣT =
∂Ω×]0, T [. We will denote by Lq(Ω), 1 ≤ q ≤ +∞, the usual Lebesgue
spaces. If X is a Banach space, we will denote by Lq(0, T ;X) the usual
evolution spaces (see, for instance, [4]). We will sometimes write ‖f‖q
instead of ‖f‖Lq(Ω), and ‖f‖r,q instead of ‖f‖Lr(0,T ;Lq(Ω)). Moreover C
will denote a positive constant which only depends on the data of the
problem. Its value may be different from line to line.
We are interested in studying the following two types of nonlinear
evolution problems:
vt − div (a(x, t, v)∇v) = F (x, t, v) in QT ;
v(x, t) = 0 on ΣT ;
v(x, 0) = v0(x) in Ω,
(2.1)
and
ut − div (a(x, t, u)∇u) = B(x, t, u,∇u) in QT ;
u(x, t) = 0 on ΣT ;
u(x, 0) = u0(x) in Ω,
(2.2)
where in both cases the principal part satisfies
• The function a : QT × R → RN2
is a Carathéodory function; that
is, it is measurable with respect to (x, t) for all s ∈ R and
continuous in s for almost all (x, t) ∈ QT ; moreover there exist
two positive constants ν and M such that
[a(x, t, s) · ξ] · ξ ≥ ν|ξ|2 (2.3)
and
|a(x, t, s) · ξ| ≤M |ξ|
hold for almost all (x, t) ∈ QT for all (s, ξ) ∈ R × RN .
We first consider problem (2.1) and state the assumptions and our
results. We will assume that:
• The function F : QT × R → R satisfies the Carathéodory condi-
tions; moreover there exist a constant α, with 0 < α < 1, and a
positive measurable function f(x, t) ∈ Lr(0, T ;Lq(Ω)), with
q, r > 1 , q >
N
2
max
{
1
1 − α
,
r
r − 1
}
, (2.4)
522 Semilinear Parabolic Equations with...
such that
|F (x, t, s)| ≤
(
1 + |s|
∣∣ log |s|
∣∣α
)
f(x, t) ;
• v0(x) ∈ L2(Ω).
Remark 2.1. If f ∈ L∞(0, T ;Lq(Ω)), with q > N/[2(1 − α)], and in
particular if f(x, t) = f(x) ∈ Lq(Ω), then obviously there exists r < ∞
such that (2.4) holds.
We can now state the first of our main existence results:
Theorem 2.1. Under the above hypotheses, problem (2.1) admits at least
one distributional solution v ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0 (Ω)).
Remark 2.2. The condition
q >
Nr
2(r − 1)
is the same one which ensures the local boundedness of the solutions of
the equation
vt − ∆v = f(x, t) , f ∈ Lr(0, T ;Lq(Ω)) ,
as shown by Aronson and Serrin in [1].
We now turn our attention to the second quasi-linear problem (2.2).
We will assume that:
• The function B : QT × R × RN → R satisfies the Carathéodory
conditions; moreover there exist two positive constants λ and C1,
and a positive measurable function g(x, t) ∈ Lr(0, T ;Lq(Ω)), with
q, r > 1 , q >
N
2
max
{
λ+ 1 ,
r
r − 1
}
,
such that
|B(x, t, s, ξ)| ≤ C1(1 + |s|λ)|ξ|2 + g(x, t) ; (2.5)
In order to state the assumption on the initial datum, we define two
auxiliary functions by
γ(s) =
C1
ν
s∫
0
(1 + |σ|λ) dσ =
C1
ν
(
s+
|s|λ+1
λ+ 1
sign s
)
,
A. Dall’Aglio, D. Giachetti and S. Segura de León 523
and
Ψ(s) =
s∫
0
e|γ(σ)| dσ . (2.6)
The assumption on u0 reads as follows
• Ψ(u0(x)) ∈ L2(Ω).
Theorem 2.2. Under the above hypotheses, problem (2.2) admits at least
one distributional solution u such that
Ψ(u) ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0 (Ω)) . (2.7)
Remark 2.3. Recalling that Ψ′(s) ≥ 1, the estimate (2.7) implies
u ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0 (Ω)) .
Remark 2.4. It is easy to show that for every δ > λ + 1 there exists
Cδ > 0 such that
|Ψ(s)| ≤ Cδ(1 + e|s|
δ
)
for every s ∈ R. It follows that the assumption on the initial datum of
problem (2.1), i. e., Ψ(u0) ∈ L2(Ω), is satisfied if
∫
Ω
e2|u0|δ dx <∞
for some δ > λ+ 1, and, a fortiori, if u0 ∈ L∞(Ω).
3. A priori estimates
We first consider problem (2.1). For n ∈ N, we define the following
approximate problems
(vn)t − div (a(x, t, vn)∇vn) = Tn(F (x, t, vn)) in QT ;
vn(x, t) = 0 on ΣT ;
vn(x, 0) = Tn(v0(x)) in Ω,
(3.1)
where Tn(s) = min
{
n,max{s,−n}
}
is the usual truncation at levels ±n.
It is well known that problem (3.1) admits at least one weak solution
vn ∈ C0([0, T ];L2(Ω)) ∩ L2(0, T ;H1
0 (Ω)).
Proposition 3.1. Under the assumptions of Theorem 2.1, there exists
a constant C, depending only on the data of the problem, such that
‖vn‖L∞(0,T ;L2(Ω)) + ‖vn‖L2(0,T ;H1
0 (Ω)) ≤ C .
524 Semilinear Parabolic Equations with...
Proof. We first observe that, under the hypothesis (2.4), we can assume
that the exponent r satisfies rα ≤ 1. Indeed, if this is not the case, we
can replace r by a smaller value which still satisfies (2.4), and apply the
usual inclusions between Lebesgue spaces.
We multiply the equation (3.1) by vn and integrate on Ω, for t fixed.
Using the assumptions (2.3) and (2.4) we obtain
1
2
d
dt
∫
Ω
v2
n dx+ ν
∫
Ω
|∇vn|2 dx
≤
∫
Ω
(
1 + |vn|
∣∣ log |vn|
∣∣α
)
|vn| f dx
≤ C
[∫
Ω
f +
∫
Ω
v2
n
∣∣ log |vn|
∣∣αf dx
]
.
The last integral can be estimated as follows. For δ ∈]0, 1[, to be chosen
hereafter, we can write
∫
Ω
f v2
n
∣∣ log |vn|
∣∣α dx =
∫
Ω
f |vn|2δ|vn|2(1−δ)
∣∣ log |vn|
∣∣α dx
≤ ‖f(t)‖q
[∫
Ω
|vn|2
∗
dx
] 2δ
2∗
[∫
Ω
|vn|2ρ(1−δ)
∣∣ log |vn|
∣∣αρ dx
] 1
ρ
,
where ρ > 1 is defined by
1
ρ
+
2δ
2∗
+
1
q
= 1 .
Using Young’s and Sobolev’s inequalities, one obtains, for every ε > 0,
∫
Ω
f v2
n
∣∣ log |vn|
∣∣α dx
≤ ε
[∫
Ω
|vn|2
∗
dx
] 2
2∗
+ C(ε) ‖f(t)‖
1
1−δ
q
[∫
Ω
|vn|2ρ(1−δ)
∣∣ log |vn|
∣∣αρdx
] 1
ρ(1−δ)
≤ εC
∫
Ω
|∇vn|2 dx+C(ε) ‖f(t)‖
1
1−δ
q
[∫
Ω
|vn|2ρ(1−δ)
∣∣ log |vn|
∣∣αρ dx
] 1
ρ(1−δ)
.
A. Dall’Aglio, D. Giachetti and S. Segura de León 525
Choosing ε = ν/(2C), and δ such that 1
1−δ = r, that is, δ = r−1
r , we
obtain:
1
2
d
dt
∫
Ω
v2
n dx+ ν
∫
Ω
|∇vn|2 dx
≤ C
(
‖f(t)‖1 + ‖f(t)‖rq
[∫
Ω
|vn|
2ρ
r
∣∣ log |vn|
∣∣αρ dx
] r
ρ
)
. (3.2)
Since q > Nr
2(r−1) , from the definition of δ and ρ one can check that
ρ < r. Therefore there exists an increasing and convex function η(s) :
[0,+∞) → [0,+∞) such that η(0) = 0 and
η
(
|Ω| s 2ρ
r (log s)αρ
)
∼ Cs2 for s→ +∞.
By Jensen’s inequality
η
(∫
Ω
|vn|
2ρ
r
∣∣ log |vn|
∣∣αρ dx
)
≤ 1
|Ω|
∫
Ω
η(|Ω| |vn|
2ρ
r
∣∣ log |vn|
∣∣αρ) dx ≤ C
(∫
Ω
v2
n dx+ 1
)
.
This implies
∫
Ω
|vn|
2ρ
r
∣∣ log |vn|
∣∣αρ dx ≤ η−1
C
(∫
Ω
v2
n dx+ 1
)
. (3.3)
It is easy to see that the function η satisfies
η−1(s) ∼ s
ρ
r (log s)αρ for s→ +∞.
Therefore, (3.3) implies
[∫
Ω
|vn|
2ρ
r
∣∣ log |vn|
∣∣αρ dx
] r
ρ
≤ CH
(∫
Ω
v2
n dx
)
, (3.4)
where
H(s) = 1 + s| log s|αr . (3.5)
If we define
ξn(t) =
∫
Ω
[vn(t)]
2 dx ,
526 Semilinear Parabolic Equations with...
we have proved that
ξ′n(t) ≤ C
[
‖f(t)‖1 + ‖f(t)‖rqH(ξn(t))
]
≤ C
(
‖f(t)‖1 + ‖f(t)‖rq
)
[1 +H(ξn(t))] .
Since ‖f(t)‖1 + ‖f(t)‖rq is an integrable function on ]0, T [, the last in-
equality yields
G(ξn(t)) −G(ξn(0)) ≤ C , (3.6)
where
G(s) =
s∫
0
dσ
1 +H(σ)
.
Recall than we are assuming that αr ≤ 1, therefore the function 1/(1 +
H(s)) is not integrable on [0,+∞). Since G(ξn(0)) is bounded, the esti-
mate (3.6) provides a uniform estimate of vn in L∞(0, T ;L2(Ω)). Using
(3.2) and (3.4), after integration with respect to time, one obtains an
estimate in L2(0, T ;H1
0 (Ω)).
We can now turn our attention to proving an estimate for problem
(2.2). Once again we have to define a sequence of approximate problems:
(un)t − div (a(x, t, un)∇un) = Tn(B(x, t, un,∇un)) in QT ;
un(x, t) = 0 on ΣT ;
un(x, 0) = u0,n(x) in Ω.
(3.7)
Notice that the prescribed datum at time t = 0 is not simply the trun-
cation of u0 as in the previous proposition, but is a function u0,n ∈
L∞(Ω) ∩H1
0 (Ω) such that
1
n
‖u0,n‖H1
0 (Ω) → 0 as n→ ∞ ,
Ψ(u0,n) → Ψ(u0) a.e. and strongly in L2(Ω) .
The existence of such a sequence may be proved by truncation and con-
volution. These assumptions are required in order to prove the strong
convergence of the gradients ∇un (see [8] and [7]).
Proposition 3.2. Under the assumptions above, there exists a constant
C, depending only on the data of the problem, such that
‖Ψ(un)‖L∞(0,T ;L2(Ω)) + ‖Ψ(un)‖L2(0,T ;H1
0 (Ω)) ≤ C .
A. Dall’Aglio, D. Giachetti and S. Segura de León 527
To prove Proposition 3.2, we need the following elementary lemma:
Lemma 3.1. For every α > λ
λ+1 , there exists a constant Cα such that
e|γ(s)| ≤ Cα
(
1 + |Ψ(s)|
∣∣ log |Ψ(s)|
∣∣α)
for every s ∈ R.
Proof. It suffices to show that
lim
s→+∞
eγ(s)
Ψ(s) (log Ψ(s))α
< +∞ ,
and this follows easily from a repeated application of De L’Hôpital’s
rule.
Proof of Proposition 3.2. Let us take e|γ(un)|Ψ(un) as test function in
(3.7). Recalling the assumptions on the terms of the equation and inte-
grating on Ω, we obtain for every fixed t:
1
2
d
dt
∫
Ω
Ψ(un)
2 dx+ ν
∫
Ω
|∇Ψ(un)|2 dx
+ C1
∫
Ω
|∇un|2e|γ(un)|Ψ(un)(1 + |un|λ)sign un dx
≤ C1
∫
Ω
|∇un|2e|γ(un)||Ψ(un)|(1 + |un|λ) dx
+
∫
Ω
ge|γ(un)||Ψ(un)| dx .
Under our hypotheses on g, we can find α such that λ
λ+1 < α < 1 and
such that (2.4) holds. Then, using Lemma 3.1, after cancellation we can
estimate the last integral in the previous formula as follows:
∫
Ω
ge|γ(un)||Ψ(un)| dx ≤ Cα
∫
Ω
g|Ψ(un)| (1 + |Ψ(un)|)
∣∣ log |Ψ(un)|
∣∣α dx
≤ C
(∫
Ω
g dx+
∫
Ω
g|Ψ(un)|2
∣∣ log |Ψ(un)|
∣∣α dx
)
.
From here, reasoning exactly as in the proof of Proposition 3.1, we obtain
the differential inequality
ξ′n(t) ≤ C
(
‖g(t)‖1 + ‖g(t)‖rqH(ξn(t))
)
,
528 Semilinear Parabolic Equations with...
where
ξn(t) =
∫
Ω
Ψ(un(t))
2 dx ,
and H(s) is defined by (3.5). In view of the assumptions on the initial
datum, the desired estimate follows immediately.
Remark 3.1. From Proposition 3.2 and Gagliardo-Nirenberg’s interpo-
lation result (see, for instance, [9], Chapter I, Proposition 3.1), it follows
that Ψ(un) is bounded (uniformly with respect to n) in Lρ(0, T ;Lσ(Ω))
for every ρ and σ such that
2 ≤ σ ≤ 2N
N − 2
, 2 ≤ ρ ≤ ∞
and
N
σ
+
2
ρ
=
N
2
.
Proposition 3.3. There exists a positive constant C, depending only on
the data of the problem, such that
∫
{|un|>k}
∣∣TnB(x, t, un,∇un)
∣∣
≤ C‖gχ
{|un|>k}
‖r,q + C
∫
Ω∩{|u0,n|>k}
|Ψ(u0,n)| (3.8)
for every n∈N and k≥0. In particular the sequence {TnB(x, t, un,∇un)}
is bounded and equi-intergable in L1(QT ).
Proof. The estimate (3.8) follows easily from the following inequality:
∫
{|un|>k}
(1 + |un|λ)|∇un|2 ≤ C‖gχ
{|un|>k}
‖r,q + C
∫
Ω∩{|u0,n|>k}
|Ψ(u0,n)|
(3.9)
To prove (3.9), we multiply the approximate problems (3.7) by hk(un),
where
hk(s) = χ{|s|>k}(s)sign (s)
(
e|γ(s)|−γ(k) − 1
)
,
and integrate over QT . If we define
φk(s) =
s∫
0
hk(σ) dσ ,
A. Dall’Aglio, D. Giachetti and S. Segura de León 529
we obtain, using the assumptions (2.3) and (2.5),
∫
Ω∩{|un(T )|>k}
φk(un(T )) −
∫
Ω∩{|u0,n|>k}
φk(u0,n)+
+ C1
∫
{|un|>k}
(1 + |un|)λ|∇un|2e|γ(un)|−γ(k)
≤ C1
∫
{|un|>k}
(1 + |un|)λ|∇un|2
(
e|γ(un)|−γ(k) − 1
)
+
∫
{|un|>k}
g
(
e|γ(un)|−γ(k) − 1
)
.
Dropping positive terms, this implies
C1
∫
{|un|>k}
(1 + |un|)λ|∇un|2 ≤
∫
{|un|>k}
ge|γ(un)| +
∫
Ω∩{|u0,n|>k}
φk(u0,n)
≤
∥∥g χ{|un|>k}
∥∥
r,q
∥∥∥e|γ(un)|
∥∥∥
r′,q′
+
∫
Ω∩{|u0,n|>k}
φk(u0,n) .
From Lemma 3.1 it follows that
e|γ(s)| ≤ C
(
1 + [Ψ(s)]2
)
,
so that ∥∥∥e|γ(un)|
∥∥∥
r′,q′
≤ C
(
1 + ‖Ψ(un)‖2
2r′,2q′
)
.
It is easy to check that the exponent ρ = 2r′ and σ = 2q′ satisfy
N
σ
+
2
ρ
>
N
2
.
Therefore, applying the usual embeddings between Lebesgue spaces and
Remark 3.2, the last norm is bounded. Moreover 0 ≤ φk(u0,n) ≤ |Ψ(u0,n)|,
therefore (3.9) is completely proved.
4. Proof of main results
Proof of Theorem 2.1. By Proposition 3.1, the sequence {vn} is bounded
in L∞(0, T ;L2(Ω)) and in L2(0, T ;H1
0 (Ω)). Moreover, by the equation,
530 Semilinear Parabolic Equations with...
{(vn)t} is bounded in L2(0, T ;H−1(Ω)) + L1(QT ). Using standard com-
pactness results for evolution spaces (see for instance [13]), we can extract
a subsequence (still denoted by {vn}) which converges to some func-
tion v strongly in L2(QT ) ∩ C0([0, T ];W−1,s(Ω)), for every s < N/(N −
1), and weakly in L2(0, T ;H1
0 (Ω)). It is easy to pass to the limit in
the weak formulation of (3.1), thus showing that v solves (2.1) in the
sense of distributions. Moreover, since there is strong convergence in
C0([0, T ];W−1,s(Ω)), the initial datum has a meaning.
Proof of Theorem 2.2. As in the previous proof, taking Propositions 3.2
and 3.3 into account, we can assume that, passing to a subsequence,
un ⇀ u weakly in L2(0, T ;H1
0 (Ω)),
un → u strongly in L2(QT ) ∩ C0([0, T ];W−1,s(Ω)), for every s <
N
N − 1
,
Ψ(un) ⇀ Ψ(u) weakly in L2(0, T ;H1
0 (Ω))
and ∗-weakly in L∞(0, T ;L2(Ω)).
Moreover it can be proved (exactly as in Propositions 6.2 and 6.3 of
[7], where a technique introduced in [12] is developed) that
∇Tkun → ∇Tku a. e. and in L2(QT ), for every k > 0,
∇un → ∇u a. e. and in Lq(QT ), for every q such that 1 ≤ q < 2,
and, using Proposition 3.3,
TnB(x, t, un,∇un) → B(x, t, u,∇u) in L1(QT ).
The existence result follows easily. We remark that the initial datum has
sense since u ∈ C([0, T ];W−1,s(Ω)).
The authors have recently proved that the solutions obtained in Theo-
rem 2.1 and Theorem 2.2 are bounded if the initial datum is also bounded.
This fact agrees with the result by Aronson-Serrin (see Remark 2.2) and
its proof will be published elsewhere.
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Contact information
A. Dall’Aglio and
D. Giachetti
Dipartimento di Metodi e Modelli Mate-
matici, Università di Roma “La Sapienza”,
Via Antonio Scarpa 16,
I-00161 Roma,
Italy
E-Mail: aglio@dmmm.uniroma1.it,
giachetti@dmmm.uniroma1.it
S. Segura de León Departament d’Anàlisi Matemàtica,
Universitat de València,
Dr. Moliner 50, 46100 Burjassot, València,
Spain
E-Mail: Sergio.Segura@uv.es
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