The Cauchy problem for degenerate parabolic equations with source and damping
We prove optimal estimates for the decay of mass of solutions to the Cauchy problem for a wide class of quasilinear parabolic equations with damping terms. In the degenerate case, we also prove estimates for the finite speed of propagation. When the equation contains also a blow up term, we discuss...
Збережено в:
| Дата: | 2004 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2004
|
| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/124607 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Cauchy problem for degenerate parabolic equations with source and damping / D. Andreucci, A. F. Tedeev, M. Ughi // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 1-20. — Бібліогр.: 30 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-124607 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1246072017-10-01T03:02:34Z The Cauchy problem for degenerate parabolic equations with source and damping Andreucci, D. Tedeev, A.F. Ughi, M. We prove optimal estimates for the decay of mass of solutions to the Cauchy problem for a wide class of quasilinear parabolic equations with damping terms. In the degenerate case, we also prove estimates for the finite speed of propagation. When the equation contains also a blow up term, we discuss existence and nonexistence of global solutions. 2004 Article The Cauchy problem for degenerate parabolic equations with source and damping / D. Andreucci, A. F. Tedeev, M. Ughi // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 1-20. — Бібліогр.: 30 назв. — англ. 1810-3200 2000 MSC. 35B40, 35B33, 35K65, 35K55. http://dspace.nbuv.gov.ua/handle/123456789/124607 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We prove optimal estimates for the decay of mass of solutions to the Cauchy problem for a wide class of quasilinear parabolic equations with damping terms. In the degenerate case, we also prove estimates for the finite speed of propagation. When the equation contains also a blow up term, we discuss existence and nonexistence of global solutions. |
| format |
Article |
| author |
Andreucci, D. Tedeev, A.F. Ughi, M. |
| spellingShingle |
Andreucci, D. Tedeev, A.F. Ughi, M. The Cauchy problem for degenerate parabolic equations with source and damping Український математичний вісник |
| author_facet |
Andreucci, D. Tedeev, A.F. Ughi, M. |
| author_sort |
Andreucci, D. |
| title |
The Cauchy problem for degenerate parabolic equations with source and damping |
| title_short |
The Cauchy problem for degenerate parabolic equations with source and damping |
| title_full |
The Cauchy problem for degenerate parabolic equations with source and damping |
| title_fullStr |
The Cauchy problem for degenerate parabolic equations with source and damping |
| title_full_unstemmed |
The Cauchy problem for degenerate parabolic equations with source and damping |
| title_sort |
cauchy problem for degenerate parabolic equations with source and damping |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/124607 |
| citation_txt |
The Cauchy problem for degenerate parabolic equations with source and damping / D. Andreucci, A. F. Tedeev, M. Ughi // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 1-20. — Бібліогр.: 30 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT andreuccid thecauchyproblemfordegenerateparabolicequationswithsourceanddamping AT tedeevaf thecauchyproblemfordegenerateparabolicequationswithsourceanddamping AT ughim thecauchyproblemfordegenerateparabolicequationswithsourceanddamping AT andreuccid cauchyproblemfordegenerateparabolicequationswithsourceanddamping AT tedeevaf cauchyproblemfordegenerateparabolicequationswithsourceanddamping AT ughim cauchyproblemfordegenerateparabolicequationswithsourceanddamping |
| first_indexed |
2025-07-09T01:42:47Z |
| last_indexed |
2025-07-09T01:42:47Z |
| _version_ |
1837131752933425152 |
| fulltext |
Український математичний вiсник
Том 1 (2004), № 1, 1 – 20
The Cauchy problem for degenerate parabolic
equations with source and damping
Daniele Andreucci, Anatoli F. Tedeev, Maura Ughi
(Presented by I. V. Skrypnik 25.01.2003 )
Abstract. We prove optimal estimates for the decay of mass of solu-
tions to the Cauchy problem for a wide class of quasilinear parabolic
equations with damping terms. In the degenerate case, we also prove es-
timates for the finite speed of propagation. When the equation contains
also a blow up term, we discuss existence and nonexistence of global
solutions.
2000 MSC. 35B40, 35B33, 35K65, 35K55.
Key words and phrases. Cauchy problem, quasilinear parabolic equa-
tions, source and damping terms, decay of a mass, blow-up phenomena.
1. Introduction
We study the following Cauchy problem in QT = R
N × (0, T ), T > 0,
ut − div
(
um−1|Du|λ−1Du
)
= −ǫ|Duν |q + δup , in QT , (1.1)
u(x, 0) = u0(x) ≥ 0 , x ∈ R
N . (1.2)
We assume throughout the paper that ε, δ ≥ 0, m + λ − 2 ≥ 0, λ > 0,
1 < q < λ + 1, νq > m + λ − 1, p > 1, and that u0 ∈ L1(RN ), with
u0 ≥ 0, u0 6≡ 0.
The special case of (1.1)
ut = ∆u − ε|Du|q + δup , (1.3)
has been studied by many authors (see [1, 2, 9–13, 16, 18, 19, 21, 24–29]).
The equation (1.3) was introduced in [16] in order to investigate the effect
of the damping term |Du|q on the existence (or nonexistence) of a global
solution to the Cauchy–Dirichlet problem with blow up sources. Equation
(1.3) has also been proposed as a model in population dynamics, where
the damping term accounts for the presence of predators, attracted by
the flow of preys (see [26]).
Received 15.01.2003
The second author was supported by INTAS:00136 and 01.07./00130 DFFD of Ukraine
ISSN 1810 – 3200. c© Iнститут математики НАН України
2 The Cauchy problem for degenerate parabolic equations
From the mathematical point of view, the interest of (1.1), when
δ = 0, is to understand how the damping gradient term changes the long
time behaviour of solutions. We show that a critical exponent q∗ exists
such that if q ≤ q∗ the influence of the damping term is determinant (for
the semilinear case see [16]). For example the total mass of the solution
decays as t → ∞. When q > q∗ this is not the case (see Theorem 1.4),
and the behaviour of solutions, from our point of view, is the same as in
the homogeneous case when ε, δ = 0.
When δ > 0, blow up of solutions in a finite time may take place. We
identify the critical threshold p∗ such that for p > p∗, q < q∗ solutions
corresponding to small initial data are globally bounded, while for p = p∗,
q < q∗ all non trivial solutions blow up in a finite time. Note that p∗ is
strictly less than the corresponding critical exponent in the case ε = 0,
due to the effect of the damping term (see [2, 9, 11, 16]).
To the best of our knowledge, the problem (1.1)–(1.2) has never been
treated in the range of parameters we consider. Our goal here is to obtain
first optimal bounds for the mass of a solution to (1.1)–(1.2) without blow
up term. This decay estimate leads to the optimal bound of the L∞ norm
of the solution as well. Next, we prove the results about global solvability
and nonexistence of global solutions.
Our approach is based on the natural development of the ideas in [6–
8], and relies on energetical arguments. The methods are flexible enough
to apply even to higher order equations, how we point out briefly in
Section 7. Moreover, as it follows from the proofs, they apply to equations
of more general form, like
ut − diva(x, t, u, Du) = f(x, t, u, Du) ,
under structure assumptions that preserve the key features of (1.1).
Definition 1.1. A nonnegative function u ∈ L∞
loc
(QT ) is a (weak) solu-
tion to (1.1) if
u ∈ C(0, T ; L2
loc(R
N )); |Duσ|λ+1, ε|Duν |q, δup ∈ L1
loc(QT ),
where σ = (m + λ − 1)/λ, and if for any η ∈ C1
0 (QT )
∫∫
QT
{
−uηt + um−1|Du|λ−1Du·Dη + ε|Duν |qη
}
dxdt = δ
∫∫
QT
upη dxdt.
Moreover u is a solution to (1.1)–(1.2) if u(·, t) → u0 as t → 0 in L1(RN ).
Remark 1.1. A solution to (1.1)–(1.2) is locally Hölder continuous in
QT . Although this result is not contained explicitly in [23, 30], the tech-
niques employed therein can be easily adapted to the case with a damping
term. Then, using also the arguments in [4], one can prove existence of
a solution to (1.1)–(1.2), at least if λ = 1. Such a a solution is global in
time if δ = 0. The question of existence and uniqueness of solutions, in
the general case λ 6= 1, however, is not trivial, and will be discussed in a
forthcoming paper.
D. Andreucci, A. F. Tedeev, M. Ughi 3
We use in the following the notation
K = N(m + λ − 2) + λ + 1 , q∗ =
K + N
Nν + 1
,
H = (λ + 1)(νq − 1) − q(m + λ − 2) , A =
q∗ − q
H (Nν + 1) .
Here K is the Barenblatt’s exponent for the homogeneous problem ε,
δ = 0. The constant H is positive under our assumptions. The role of
the parameters introduced above will be commented upon in Remark 1.2.
Moreover we set
‖u‖1,Ω =
∫
Ω
|u| dx, ‖u‖1 = ‖u‖1,RN .
The symbols γ, γ∗, . . . , denote generic positive constants depending on
N , m, λ, q, ν, p.
We begin by stating a result that gives optimal estimates for the finite
speed of propagation in the degenerate case.
Theorem 1.1. Let u be a weak solution of (1.1)–(1.2) with δ = 0, ǫ = 1,
m+λ−2>0, supp u0⊂Bρ0, ρ0 <∞. Then we have for all large enough t
Z(t) := inf{r > 0 | u(x, t) = 0 a.e. |x| > r } ≤ γt
νq−(m+λ−1)
H . (1.4)
Remark 1.2. The exponent q∗ is critical under many respects, as shown
by the results below. We assume here q < q∗.
The growth rate in (1.4) can be heuristically obtained by looking for
self-similar radial solutions in the form t−αf(ρt−β). One finds that β
equals the power in (1.4), that is
νq − m − λ + 1
H ∼ speed of propagation.
Moreover α equals the power given in (1.9), that is
−λ + 1 − q
H ∼ L∞ decay rate.
The constant A will be relevant to us, as (see Theorem 1.2)
−A ∼ L1 decay rate.
Remark 1.3. The estimate
Z(t) ≤ γ
(
‖u0‖
m+λ−2
K
1 t
1
K + ρ0
)
, for all t > 0, (1.5)
is classical under the assumptions of Theorem 1.1 (see e. g., [6]), for
solutions of the homogeneous equation where ε, δ = 0. Obviously it is
valid for u under the assumptions of Theorem 1.1, since u is a subsolution
to the homogeneous equation. For t large enough, when q < q∗, (1.4)
gives better results than (1.5), while the latter is optimal for q > q∗. Also
note that the bound in (1.4) does not depend on the initial data.
4 The Cauchy problem for degenerate parabolic equations
Exploiting the estimate (1.4) we are able to prove our first results on
decay of mass.
Theorem 1.2. Let u be a weak solution of (1.1)–(1.2) with δ = 0, ǫ = 1,
m + λ − 2 > 0, supp u0 ⊂ Bρ0, ρ0 < ∞. Then we have for all large
enough t
if q < q∗ then ‖u(t)‖1 ≤ γt−A , (1.6)
if q = q∗ then ‖u(t)‖1 ≤ γ[ln t]−
1
νq−1 . (1.7)
Remark 1.4. The sup estimate
‖u(t)‖∞,RN ≤ γ sup
t
2
<τ<t
‖u(τ)‖
λ+1
K
1 t−
N
K , t > 0 , (1.8)
is well known for subsolutions of the homogeneous equation ε, δ = 0 (see
e. g., [6]). Thus, it holds for solutions of our equation with δ = 0.
Combining (1.8) with the decay results of, for example, (1.6), we
obtain for large t the sup estimate
‖u(t)‖∞,RN ≤ γt−
λ+1−q
H , (1.9)
valid under the assumptions of Theorem 1.2. Note that (1.9) implies a
faster decay rate, for large t, than (1.8) if and only if q < q∗. For q = q∗,
one can still obtain a decay rate better than t−N/K by substituting (1.7)
in (1.8).
Even dropping the assumption of compactly supported initial data,
and of degeneracy of the equation, we prove the following result of decay
of mass.
Theorem 1.3. Let u be a weak solution of (1.1)–(1.2) with δ = 0, ǫ = 1,
q < q∗. Then for any t large enough we have
‖u(t)‖1 ≤ γ
∫
|x|>R(t)
u0 dx + γt−A , (1.10)
where R(t) = t
νq−(m+λ−1)
H , provided either (a) ν = σ = (m + λ− 1)/λ, or
(b) λ = 1, N ≥ 2.
In case (b) of Theorem 1.3 we need the special structure of the porous
media equation, since we integrate twice by parts in the proof.
The threshold q < q∗ in Theorem 1.3 is optimal, as we show in our
next result.
Theorem 1.4. Let u be a weak solution of (1.1)–(1.2) with q > q∗,
δ = 0, ǫ = 1. Then for all t > 0 we have
‖u(t)‖1 ≥ c > 0 , (1.11)
where c is a positive constant depending on u0.
D. Andreucci, A. F. Tedeev, M. Ughi 5
Remark 1.5. In the case ν, m, λ = 1 the critical exponent q∗ = (N +
2)/(N + 1) is well known, as the critical p∗ appearing in the blow up re-
sults below, which in the mentioned case equals q/(2−q) (see [2, 9, 11, 16]).
However, Theorem 1.3 seems to be new even in that case.
Next we state our results concerning the blow up problem δ = 1. First
we prove, for supercritical p, a priori estimates which are instrumental in
the proof of existence of global solutions.
Theorem 1.5. Let u be a solution to (1.1)–(1.2) which can be a.e. ap-
proximated by globally bounded subsolutions. Let δ = ǫ = 1, ν = σ,
p > m + λ − 1. We also assume that
p > p∗ :=
q(ν(λ + 1) − (m + λ − 1))
λ + 1 − q
, q < q∗ . (1.12)
Moreover assume that a constant M > 0 exists so that for all ρ > 1,
∫
|x|>ρ
u0(x) dx ≤ Mρ
− AH
σq−(m+λ−1) . (1.13)
Assume also that for β > N(p−m−λ+1)/(λ+1) we have ‖u0‖β,RN < σ1,
where σ1(N, λ, m, q, p, M) > 0 is small enough. Then the following a
priori bound holds
‖u(t)‖∞,RN ≤ γt−
λ+1−q
H , t > 1 . (1.14)
Some extra regularity assumption, like u0 ∈ Lβ(RN ), is needed, even
for existence of local in time solutions, when the equation contains non
linear sources (see [3]).
Remark 1.6. If λ = 1, the a priori estimates of Theorem 1.5 are enough
to prove existence of global solutions to (1.1)–(1.2), applying the argu-
ments in [4] (see also Remark 1.1).
Remark 1.7. The p∗ appearing in (1.12) is critical because for p > p∗
∫ ∞
‖u(t)‖p−1
∞,RN dt < ∞ ,
according to (1.14) (we need q < q∗ here, too), a fact which is exploited
in the proof of Theorem 1.5.
Remark 1.8. When q > q∗, estimate (1.14) does not provide any fur-
ther information than estimate (1.8). However, in this case, it follows
immediately from the methods of [3, 6] that global a priori bounds may
be obtained for p > m + λ − 1 + (λ + 1)/N .
Finally, let us state our blow up result, which shows that the restric-
tion on p of Theorem 1.5 is optimal.
Theorem 1.6. Let δ = 1 and 0 < ε < ε̄, where ǭ = ε̄(N, ν, λ, m, q) > 0.
Assume also p = p∗, q < q∗. Then any non trivial solution to (1.1)–(1.2)
blows up in a finite time.
6 The Cauchy problem for degenerate parabolic equations
Remark 1.9. The requirement ε < ε̄ in Theorem 1.6 does not seem
to be only a technical restriction, in view of the existence of a (formal)
stationary solution u(x) = |x|−θ. Here we must assume p = p∗, θ =
(λ+1−q)/(νq−m−λ+1), and ε is a suitable positive constant depending
on the other parameters.
The material is organized as follows: Section 2 is devoted to the
proof of the finite speed of propagation property, Theorem 1.1. Section 3
contains the proofs of the Theorems 1.2 and 1.3 about decay of mass,
while the connected result Theorem 1.4 is proven in Section 4. The proof
of the global existence Theorem 1.5 is given in Section 5, and the proof
of the blow up Theorem 1.6 is given in Section 6. Finally, Section 7 is
devoted to some remarks about higher order equations.
2. Proof of Theorem 1.1
We proceed as in [6–8]. Consider the sequence ri = 2ρ(1−2−i−1), i =
0, 1, . . . , where ρ > 2ρ0. Let r̃i = (ri + ri+1)/2, and let ζi be a cut off
function in C1(RN ) such that ζi ≡ 0 when |x| < ri, ζi ≡ 1 when |x| > r̃i,
and |Dζi| ≤ γ2iρ−1. Then routine calculations give
sup
0<τ<t
∫
Ũi
uθ+1 dx +
∫ t
0
∫
Ũi
um+θ−2|Du|λ+1 dx dτ
+
∫ t
0
∫
Ũi
uθ|Duν |q dx dτ ≤ γ
2i(λ+1)
ρλ+1
∫ t
0
∫
Ui\Ũi
um+λ+θ−1 dx dτ , (2.1)
where Ui = {|x| > ri}, Ũi = {|x| > r̃i}.
Define v = (uζ̃i)
ω, ω = (m + λ + θ − 1)/(λ + 1), β = (θ + 1)/ω > 1
(due to the choice of θ), where ζ̃i is a cut off function analogous to ζi,
but such that ζ̃i ≡ 0 for |x| < r̃i, ζ̃i ≡ 1 for |x| > ri+1.
Next apply Nirenberg–Gagliardo inequality to find
∫
RN
vλ+1 dx ≤ γ
(
∫
RN
|Dv|λ+1 dx
)α(
∫
RN
vβ dx
)
(1−α)(λ+1)
β
, (2.2)
where α = N(m + λ − 2)/Kθ, Kθ = N(m + λ − 2) + (θ + 1)(λ + 1).
Integrating (2.2) in time and using also Hölder’s inequality, we obtain
∫ t
0
∫
RN
vλ+1 dx dτ ≤ γt1−αy
1+(1−α)(λ+1
β
−1)
i , (2.3)
where
yi := sup
0<τ<t
∫
Ui
uθ+1 dx +
∫ t
0
∫
Ui
um+θ−2|Du|λ+1 dx dτ
+
∫ t
0
∫
Ui
uθ|Duν |q dx dτ +
2i(λ+1)
ρλ+1
∫ t
0
∫
Ui\Ui+1
um+λ+θ−1 dx dτ .
D. Andreucci, A. F. Tedeev, M. Ughi 7
Therefore, recalling the definition of v, and (2.1),
yi+1 ≤ γ
2i(λ+1)
ρλ+1
t
(1+θ)(λ+1)
Kθ y
1+
(λ+1)(m+λ−2)
Kθ
i . (2.4)
Next we use the damping term to find an additional recursive inequal-
ity. Let w = u(νq+θ)/q. We already know that the support of u(·, t) is
bounded, i.e., that Z(t) < ∞ (because u is a subsolution to the homoge-
neous equation; see e.g., [6]). Then Poincaré’s inequality gives
∫
Ui
wq dx ≤ γZq
∫
Ui
|Dw|q dx
(we denote Z = Z(t) for ease of notation). Then, by Hölder’s inequality
∫ t
0
∫
Ui\Ui+1
w
q(m+λ+θ−1)
νq+θ dx dτ
≤ γ
(
∫ t
0
∫
Ui
wq dx dτ
)
m+λ+θ−1
νq+θ
(tρN )
νq−(m+λ−1)
νq+θ
≤ γ(tρN )
νq−(m+λ−1)
νq+θ Z
q m+λ+θ−1
νq+θ y
m+λ+θ−1
νq+θ
i . (2.5)
Thus, (2.1) and (2.5) give
yi+1 ≤ γ2i(λ+1)t
νq−(m+λ−1)
νq+θ ρ
N
νq−(m+λ−1)
νq+θ
−λ−1
Z
q m+λ+θ−1
νq+θ y
m+λ+θ−1
νq+θ
i . (2.6)
Let
a =
Kθ
Kθ + (λ + 1)(m + λ − 2)
, A =
[
t
(1+θ)(λ+1)
Kθ
ρ−λ−1]a
,
b =
νq + θ
m + λ + θ − 1
> 1 , B =
[
(tρN )
νq−(m+λ−1)
νq+θ ρ−λ−1Zq m+λ+θ−1
νq+θ
]b
.
Then from (2.4) and (2.6) it follows that
ya
i+1
A
+
yb
i+1
B
≤ γCiyi ,
for a suitable C > 1. On applying Young’s inequality we obtain
yε1
i+1
Aε1B1−ε1
≤ γCiyi , (2.7)
where ε1 = b/(b + 1 − a) < 1. Therefore from the iterative Lemma 5.6,
Chapter II of [22] we conclude that yi → 0 if
(y0B)
1−a
b A ≤ γ0 . (2.8)
8 The Cauchy problem for degenerate parabolic equations
Of course this would imply that u(x, t) = 0 for |x| ≥ 2ρ.
In order to find the sharp bound of y0 we need to proceed as follows.
Let
y(i)(ρ̃) := sup
0<τ<t
∫
|x|>ρ̃i
u(x, τ)1+θ dx +
∫ t
0
∫
|x|>ρ̃i
um+θ−2|Du|λ+1 dx dτ
+
∫ t
0
∫
|x|>ρ̃i
uθ|Duν |q dx dτ +
2i(λ+1)
ρ̃λ+1
∫ t
0
∫
ρ̃i>|x|>ρ̃i+1
um+λ+θ−1 dx dτ ,
where ρ̃i = ρ̃(1+2−i)/2. Proceeding exactly as in the proof of (2.6) we
get
y(i)(ρ̃) ≤ γbi
1t
νq−(m+λ−1)
νq+θ ρ̃
N
νq−(m+λ−1)
νq+θ
−λ−1
Z
q m+λ+θ−1
νq+θ [y(i+1)(ρ̃)]
m+λ+θ−1
νq+θ .
A simple iterative process leads to the bound
y(0)(ρ̃) ≤ γtρ̃
N(νq−(m+λ−1))−(λ+1)(νq+θ)
νq−(m+λ−1) Z
q m+λ+θ−1
νq−(m+λ−1) .
We check by direct inspection that
y0 ≤ γ[y(0)(ρ) + y(0)(4ρ/3) + y(0)(3ρ/2)]
≤ γtρ
N(νq−(m+λ−1))−(λ+1)(νq+θ)
νq−(m+λ−1) Z
q m+λ+θ−1
νq−(m+λ−1) .
Substituting this estimate in (2.8) one checks (after lenghty but trivial
calculations) that u(x, t) ≡ 0 if |x| ≥ 2ρ, with ρ as in
2ρ
(2ρ
Z
)γ1
= γt
νq−(m+λ−1)
H ,
for a constant γ1 > 0 which we do not reproduce here. Note that this
implies 2ρ ≥ Z by definition of Z. Estimate (1.4) follows immediately.
3. Decay of mass
3.1. Proof of Theorem 1.2
1) First we prove (1.6). We multiply both sides of equation (1.1) by
uθ and integrate over Bρ, with
ρ = R(t) := γt
νq−(m+λ−1)
H ≥ Z(t) ,
for large enough t. We find for 0 < τ < t,
1
θ + 1
d
dτ
∫
Bρ
u(x, τ)1+θ dx ≤ −
∫
Bρ
uθ|Duν |q dx ≤ −γ
∫
Bρ
|Du
νq+θ
q |q dx .
(3.1)
D. Andreucci, A. F. Tedeev, M. Ughi 9
Applying Hölder’s and Poincaré’s inequalities, we get from (3.1)
d
dτ
∫
Bρ
uθ+1(x, τ) dx ≤ −γρ−
q(1+θ)+N(νq−1)
1+θ
(
∫
Bρ
uθ+1 dx
)
νq+θ
1+θ
,
and therefore, integrating over (t/2, t),
∫
BR(t)
uθ+1(x, t) dx ≤ γt
− 1+θ
νq−1 R(t)
N(νq−1)+q(1+θ)
νq−1 . (3.2)
Finally we infer (1.6) from an application of Hölder’s inequality, together
with (1.4) and (3.2).
2) Next we prove (1.7). Integrate the equation (1.1) over R
N to find
d
dt
‖u(t)‖1 = −
∫
RN
|Du(x, t)ν |q dx ≤ −γZ(t)−[N(νq−1)+q]‖u(t)‖νq
1 ,
where we have used also Hölder’s and Poincaré’s inequalities.
Recalling the estimate (1.4), and that q = q∗, we find for large
enough t
d
dt
‖u(t)‖1 ≤ −γt−1‖u(t)‖νq
1 ,
whence the desired result follows by integration.
3.2. Proof of Theorem 1.3
Fix t > 0. Let us split the total mass as follows:
‖u(τ)‖1 =
∫
B2ρ
u(x, τ) dx +
∫
|x|>2ρ
u(x, τ) dx =: E1(ρ, τ) + E2(ρ, τ).
(3.3)
We obtain from Hölder’s, Nirenberg-Gagliardo and Young’s inequalities
E1(ρ, τ) ≤ γ
(
∫
RN
uνq(x, τ) dx
)
1
νq
ρ
N(νq−1)
νq
≤ γ
(
∫
RN
|Duν |q dx
)
α
νq ‖u(τ)‖1−α
1 ρ
N(νq−1)
νq
≤ 1
2
‖u(τ)‖1 + γ
(
∫
RN
|Duν |q dx
)
1
νq
ρ
N(νq−1)+q
νq , (3.4)
where
α = N(νq − 1)/[N(νq − 1) + q].
On the other hand, integrating (1.1) over R
N we get
d
dτ
∫
RN
u dx = −
∫
RN
|Duν |q dx . (3.5)
10 The Cauchy problem for degenerate parabolic equations
Thus (3.4), (3.5) yield
‖u(τ)‖1 ≤ γ
(
− d
dτ
‖u(τ)‖1
)
1
νq
ρ
N(νq−1)+q
νq + 2E2(ρ, τ), (3.6)
In order to bound E2 in (3.6) we distinguish between the two cases
listed in the statement of the Theorem.
Case (a): ν = σ. First we note that if ν = σ = m+λ−1
λ then the
equation (1.1) may be rewritten as
ut = σλdiv
(
|Duσ|λ−1Duσ
)
− |Duσ|q . (3.7)
Moreover, the assumption νq > m + λ − 1 leads to λ < q. Let ζ be a
cut off function vanishing inside Bρ and such that ζ(x) ≡ 1 for |x| > 2ρ,
|Dζ| ≤ γ/ρ. Then, on multiplying both sides of (3.7) by ζs, where
s = q/(q − λ) > λ + 1, and integrating by parts over R
N , we get
d
dτ
∫
RN
ζsu dx +
∫
RN
|Duσ|qζs dx ≤ γ
ρ
∫
ρ<|x|<2ρ
|Duσ|λζs−1 dx . (3.8)
The right hand side in (3.8) can be bound above by
1
2
∫
RN
|Duσ|qζs dx + γρ
N− q
q−λ .
Thus (3.8) yields for 0 < τ < t
E2(ρ, τ) =
∫
|x|>2ρ
u(x, τ) dx ≤
∫
|x|>ρ
u0(x) dx + γtρ
N− q
q−λ
=
∫
|x|>ρ
u0(x) dx + γt−A =: Ẽ(t) , (3.9)
where we have set ρ = R(t) = t
σq−(m+λ−1)
H . Then denoting F (τ) =
‖u(τ)‖1 − 2Ẽ(t), we get from (3.6)
F (τ) ≤
(
− dF
dτ
)
1
νq
R(t)
N(νq−1)+q
νq , 0 < τ < t . (3.10)
If F (t) ≤ 0, estimate (1.10) follows immediately. Also keeping in mind
that F ′ ≤ 0, we may therefore assume that F > 0 in (0, t). Integrating
(3.10) we find
F (t) ≤ γt
− 1
σq−1 R(t)
N(σq−1)+q
σq−1 = γt−A ,
whence (1.10).
Case (b): λ = 1, N ≥ 2. Let ζn be a cut off function such that ζn = 0
for |x| < ρn+1, ζn = 1 for |x| > ρn, where ρn = ρ(1 + 2−n). We may
D. Andreucci, A. F. Tedeev, M. Ughi 11
assume that |Dζn| ≤ 2nγ/ρ, |∆ζn| ≤ 22nγ/ρ2. On using ζs
n, where s > 2,
as a testing function in (1.1), we have
∫
RN
ζs
nu(x, τ) dx +
∫∫
Qτ
ζs
n|Duν |q dx dη
=
∫
RN
ζs
nu0 dx +
∫∫
Qτ
um∆ζs
n dx dη
≤
∫
|x|>ρ
u0 dx + γ
22n
ρ2
∫ τ
0
∫
ρn>|x|>ρn+1
um dx dη =: K0 + 22nK1. (3.11)
Using Hölder’s inequality we get
K1 ≤ γt
νq−m
νq ρ
N(νq−m)+mq
νq
−2
(
∫ τ
0
∫
ρn>|x|>ρn+1
uνq
|x|q dx dη
)
m
νq
. (3.12)
Next we apply Hardy’s inequality, obtaining
∫
|x|>ρn+1
uνq
|x|q dx ≤ γ
∫
|x|>ρn+1
|Duν |q dx . (3.13)
Here we use the fact that q < λ + 1 = 2 ≤ N .
Denote for all fixed τ ∈ (0, t)
Hn(τ) =
∫
|x|>ρn
u(x, τ) dx +
∫ τ
0
∫
|x|>ρn
|Duν |q dx , n ≥ 0 .
Thus, it follows from (3.11)–(3.13) that for n ≥ 0
Hn(τ) ≤ K0 + γ22nt
νq−m
νq ρ
N(νq−m)+mq
νq
−2
Hn+1(τ)
m
νq
≤ K0 + ωHn+1(τ) + Cω2
2nνq
νq−m tρ
N+ mq
νq−m
− 2νq
νq−m . (3.14)
A simple iteration procedure, when 0 < ω < 1 is chosen small enough,
yields
∫
|x|>2ρ
u(x, τ) dx ≤ H0(τ) ≤ K0 + γtρ
N+ mq
νq−m
− 2νq
νq−m , 0 < τ < t ,
where we select ρ = R(t) to obtain
∫
|x|>2R(t)
u(x, τ) dx ≤
∫
|x|>R(t)
u0(x) dx + γt−A =: Ẽ(t) . (3.15)
Denoting again F (τ) = ‖u(τ)‖1 − 2Ẽ(t), the proof can be concluded as
in Case (a) above.
12 The Cauchy problem for degenerate parabolic equations
4. No decay of mass: Proof of Theorem 1.4
As a preliminary step to the proof, we recall the well known bound
‖u(·, t)‖L∞(RN ) ≤ γ‖u0‖
λ+1
K
1 t−
N
K , t > 0 , (4.1)
following from (1.8).
Integrating the equation (1.1) over R
N × (t1, t2) for any 0 < t1 < t2,
we have
‖u(t1)‖1 = ‖u(t2)‖1 +
∫ t2
t1
∫
RN
|Duν |q dx dτ . (4.2)
Using Hölder’s inequality, we find for 0 < θ < 1
∫ t2
t1
∫
RN
|Duν |q dx dτ ≤
[
∫ t2
t1
∫
RN
|Du|λ+1uθ+m−2 dx dτ
]
q
λ+1
×
[
∫ t2
t1
∫
RN
u
q[(λ+1)ν−(m+λ−1)−θ]
λ+1−q dx dτ
]
λ+1−q
λ+1
≡ J
q
λ+1
1 J
λ+1−q
λ+1
2 . (4.3)
We estimate J1 by multiplying the equation by uθ and integrating over
(t1, t2) × R
N ; this yields
∫
RN
uθ+1(x, t2) dx −
∫
RN
uθ+1(x, t1) dx
+ θ(θ + 1)
∫ t2
t1
∫
RN
|Du|λ+1uθ+m−2 dx dτ
+ (θ + 1)
∫ t2
t1
∫
RN
uθ|Duν |q dx dτ = 0 . (4.4)
Next we apply the estimate (4.1) to get
J1 ≤ 1
θ(θ + 1)
∫
RN
uθ+1(x, t1) dx ≤ γ‖u0‖
(λ+1)θ
K
1 ‖u(t1)‖1t
−Nθ
K
1 . (4.5)
The power of u in the integral J2 amounts to (H − θq)/(λ + 1 − q) + 1,
and thus it is greater than 1, under our assumptions, at least for small θ.
Therefore we may apply again (4.1) and estimate
J2 ≤ γ‖u0‖
(H−θq)(λ+1)
K(λ+1−q)
1 ‖u(t1)‖1t
1−
N(H−θq)
K(λ+1−q)
1 . (4.6)
Indeed, we invoke here the obvious statement
t 7→ ‖u(t)‖1 is non increasing, (4.7)
and the fact that the exponent of t1 in (4.6) is negative. The latter, in
turn, is a consequence of q > q∗, provided θ > 0 is small enough.
D. Andreucci, A. F. Tedeev, M. Ughi 13
From (4.2)–(4.6) it follows
‖u(t1)‖1 ≤ ‖u(t2)‖1 + γ1‖u0‖
H
K
1 ‖u(t1)‖1 t
AH
K
1 . (4.8)
Since A < 0, owing again to the assumption q > q∗, we finally obtain
‖u(t2)‖1 ≥ 1
2
‖u(t̄)‖1 , for all t2 > t̄ :=
[
(2γ1)
−1‖u0‖
−H
K
1
]
K
AH . (4.9)
This, also taking into account (4.7), implies the statement, provided we
show that ‖u(t̄)‖1 > 0. This is the content of next lemma.
Lemma 4.1. A solution to (1.1) with δ = 0, ε = 1 cannot satisfy
u(x, t0) ≡ 0 over the whole space R
N for any finite t0 > 0.
Note that the special assumption q > q∗ is not needed in this lemma.
Proof. Assume, by contradiction, that a finite time t0 > 0 exists such
that
‖u(t0)‖1 = 0, ‖u(t)‖1 > 0, t < t0.
Define for 0 < θ < 1
Eθ(t) =
∫
RN
u(x, t)1+θ dx.
By means of calculations similar to the ones performed in the proof of
Theorem 1.4, we obtain
d
dt
‖u(t)‖1 = −
∫
RN
|Duν |q dx
≥ −
(
∫
RN
|Du|λ+1uθ+m−2 dx
)
q
λ+1
(
∫
RN
u
q[(λ+1)ν−(m+λ−1)−θ]
λ+1−q dx
)1− q
λ+1
≥ −
(
− d
dt
Eθ(t)
)
q
λ+1
(
C‖u(t)‖1
)1− q
λ+1 ≥ ω
d
dt
Eθ(t) − Cω‖u(t)‖1 .
(4.10)
Here ω ∈ (0, 1) will be chosen later, and the constants C, Cω depend on
a uniform L∞ bound for u(t), t0/2 < t < t0; such a bound can be found
for example invoking (4.1). Integrate (4.10) over (t, t0), for t close to t0,
to obtain
‖u(t)‖1 ≤ ωEθ(t) + Cω
∫ t0
t
‖u(τ)‖1 dτ
≤ ωC‖u(t)‖1 + Cω(t0 − t)‖u(t)‖1 . (4.11)
Here C is as above, and we have taken advantage of (4.7) (this last detail
is in fact inessential). Obviously, dividing (4.11) by ‖u(t)‖1, and then
letting t → t0 we get an inconsistency, provided ω is so small as ωC < 1
in (4.11).
14 The Cauchy problem for degenerate parabolic equations
5. Proof of Theorem 1.5
We remark preliminarly that under our current assumptions,
N
Kβ
(p − 1) < 1 <
λ + 1 − q
H (p − 1) , (5.1)
where Kβ = N(m + λ − 2) + β(λ + 1).
Define T as the supremum of the times t > 1 such that
τ‖u(τ)‖p−1
∞,RN ≤ 1
2
, 0 < τ < t, (5.2)
‖u(τ)‖∞,RN ≤ c0τ
−λ+1−q
H , 1 < τ < t. (5.3)
Here c0 is a constant depending only on N , m, λ, q, p and M , which is
known a priori, but will be specified later. It is known (see [3, 6]) that,
given any t1 > 1, (5.2) is satisfied for all 0 < t < t1, provided ‖u0‖β,RN
is chosen small enough. Next we recall the estimate (see [3, 6])
‖u(τ)‖∞,RN ≤ γ‖u0‖
λ+1
Kβ
β,RN τ
− N
Kβ , 0 < τ < t, (5.4)
which is valid under assumption (5.2). Indeed the quoted results of [3, 6]
were proven for subsolutions to the equation without damping term, and
thus hold true even in the present case. Note that, for any fixed t1 > 1,
we have
γ‖u0‖
λ+1
Kβ
β,RN τ
− N
Kβ ≤ c0τ
−λ+1−q
H , 1 < τ < t1,
provided ‖u0‖β,RN is small enough in dependence of t1. Then, for 1 <
t < t1, (5.3) is a consequence of (5.2).
Let us summarize what we have obtained: the supremum T defined
above satisfies T ≥ t1 > 1, where the finite number t1 can be chosen as
large as we need, provided the bound for ‖u0‖β,RN is redefined accord-
ingly. We always assume t1 > 2. Our goal is to show that T = ∞.
Let now ζ ∈ C1(RN ) be a cut off function such that ζ ≡ 0 inside Bρ,
ρ > 1, and ζ ≡ 1 in R
N \B2ρ, with |Dζ| ≤ 2/ρ. Taking ζs, s > q/(q−λ),
as a cut off function in (1.1) we get for t < T
sup
0<τ<t
∫
RN
ζsu(x, τ) dx +
∫ t
0
∫
RN
ζs|Duσ|q dx dτ ≤
∫
RN
ζsu0(x) dx
+
2s
ρ
∫ t
0
∫
RN
ζs−1|Duσ|λ dx dτ +
∫ t
0
∫
RN
ζsup dx dτ . (5.5)
Next we proceed to bound the three terms on the right hand side of (5.5).
The first term is bound by means of assumption (1.13). An application
of Young’s inequality yields for the second term the estimate
1
2
∫ t
0
∫
RN
ζs|Duσ|q dx dτ + γtρ
N− q
q−λ .
D. Andreucci, A. F. Tedeev, M. Ughi 15
The third term is more critical: first, we bound it by
∫ t
0
‖u(τ)‖p−1
∞,RN dτ sup
0<τ<t
∫
RN
ζsu(x, τ) dx.
Using both (5.4) (for τ < t1), and (5.3) (for τ > t1) we obtain
∫ t
0
‖u(τ)‖p−1
∞,RN dτ
≤ γ
∫ t1
0
‖u0‖
λ+1
Kβ
β,RN τ
− N
Kβ
(p−1)
dτ + c0
∫ t
t1
τ−λ+1−q
H
(p−1) dτ
≤ γ‖u0‖
λ+1
Kβ
β,RN t
1− N
Kβ
(p−1)
1 +
c0H
(λ + 1 − q)(p − 1) −H t
1−λ+1−q
H
(p−1)
1 ≤ 1
2
,
provided we first select t1 = t1(c0) (recall (5.1)), and next a small enough
‖u0‖β,RN . Of course for t < t1 the estimate above is still valid.
As a consequence of the estimates we have found, the right hand side
of (5.5) can be partially absorbed into the left hand side, leading us to
sup
0<τ<t
∫
|x|>2ρ
u(x, τ) dx ≤ 2Mρ
− AH
σq−m−λ+1 + γtρ
N− q
q−λ =: C(ρ, t). (5.6)
It is important to note that the constants in (5.6) do not depend on t1
or c0; this applies to all the constants appearing below and denoted by
γ. Employing this bound, and reasoning as in (3.3)–(3.4), we arrive at
‖u(τ)‖1 ≤ 2C(ρ, t) + γ
(
∫
RN
|Duσ|q dx
)
1
σq
ρ
N(σq−1)+q
σq . (5.7)
Next, it follows from (5.2) and from the definition of T that for all τ < T ,
d
dτ
‖u(τ)‖1 = −
∫
RN
|Duσ|q dx +
∫
RN
up dx
≤ −
∫
RN
|Duσ|q dx +
1
2τ
‖u(τ)‖1. (5.8)
Setting f(τ) = τ− 1
2 ‖u(τ)‖1, this yields (on multiplying (5.8) by τ−1/2)
f ′(τ) ≤ −τ− 1
2
∫
RN
|Duσ|q dx.
Thus, assuming also t/2 < τ < t < T , (5.7) amounts to
f(τ) ≤ 2
√
2t−
1
2 C(ρ, t) + γ
(
− t
1
2 f ′(τ)
)
1
σq ρ
N(σq−1)+q
σq t−
1
2 . (5.9)
16 The Cauchy problem for degenerate parabolic equations
Define also F (τ) = f(τ)−2
√
2t−
1
2 C(ρ, t). Assume provisionally F (t) > 0,
and therefore, due to the monotonic character of F , F (τ) > 0 for τ < t.
Then (5.9) gives
F ′(τ) ≤ −γt
σq−1
2 F (τ)σqρ−N(σq−1)−q. (5.10)
Then, integrating (5.10) over (t/2, t), we get
F (t) ≤ γt
− σq+1
2(σq−1) ρ
N+ q
σq−1 ,
i.e.,
‖u(t)‖1 ≤ 2
√
2C(ρ, t) + γt
1
2
− σq+1
2(σq−1) ρ
N+ q
σq−1 . (5.11)
If F (t) ≤ 0 we still have (5.11), obviously. Finally we select in (5.11)
ρ = ρ(t) = t(σq−m−λ+1)/H, obtaining
‖u(t)‖1 ≤ γt−A, 1 < t < T. (5.12)
Substituting this estimate in the L1–L∞ estimate of Remark 1.8, which
is valid for our u under assumption (5.2) (see [3, 6]) we have
‖u(t1)‖∞,RN ≤ γ sup
t
2
<τ<t
‖u(τ)‖
λ+1
K
1 t−
N
K ≤ γ∗t
−λ+1−q
H , 2 < t < T. (5.13)
The importance of the estimate (5.13) is twofold. First, owing to well
known compactness results (see Remark 1.1), it can be employed to prove
existence of a solution up to time T , by means of standard approximation
techniques with solutions to smoothed problems.
Second, it permits us to prove that T = ∞. Indeed, choose c0 = 2γ∗
in (5.3): here γ∗ is the constant appearing in (5.13). This can be done
safely, because, as we already remarked, γ∗ does not depend either on c0
or on t1. By this choice, we have actually shown that (5.3) holds with c0
formally replaced by c0/2 up to the time T . Therefore, if T < ∞, (5.2)
must fail for some finite time. But, for t1 < τ < T , taking into account
(5.13) again, we have
τ‖u(τ)‖p−1
∞,RN ≤ γ∗τ
1−λ+1−q
H
(p−1) < γ∗t
1−λ+1−q
H
(p−1)
1 ≤ 1
4
,
where the last inequality is guaranteed by a suitable choice of t1 (recall
(5.1)). Again, this can be done without any danger of circular reasoning,
as γ∗ does not depend on t1.
The proof is concluded.
6. Proof of Theorem 1.6
Following [6, 20], we use as a testing function in the equation ζsu−θ,
where 0 < θ < 1, and s > λ + 1 will be chosen later. Here ζ(x) is a
standard cut off function in B2ρ, such that ζ ≡ 1 in Bρ, and |Dζ| ≤ γρ−1.
We have
D. Andreucci, A. F. Tedeev, M. Ughi 17
1
1 − θ
d
dt
∫
RN
ζsu1−θ dx = −
∫
RN
um−1|Du|λ−1Du · D(ζsu−θ) dx
− ε
∫
RN
|Duν |qu−θζs dx +
∫
RN
up−θζs dx =: I1 + I2 + I3. (6.1)
Next we write I1 as
I1 = θ
∫
RN
um−θ−2|Du|λ+1ζs dx
− s
∫
RN
um−1|Du|λ−1Du · Dζζs−1u−θ dx =: θI4 − sI5. (6.2)
Applying Young’s inequality we get for ε1 > 0
I5 ≤ ε1I4 + γ
Cε1
ρλ+1
∫
B2ρ
ζs−(λ+1)um+λ−θ−1 dx, (6.3)
and by the same token, for ε2 > 0,
1
ρλ+1
∫
B2ρ
ζs−(λ+1)um+λ−θ−1 dx ≤ ε2
∫
B2ρ
ζsup−θ dx + Cε2ρ
N−
(λ+1)(p−θ)
p−(m+λ−1) ,
(6.4)
where s = (λ + 1)(p − θ)/(p − m − λ + 1) > λ + 1 if θ is small enough.
Moreover, for ε3 > 0,
− I2 = ε
∫
B2ρ
|Duν |qu−θζs dx ≤ εε3
∫
B2ρ
ζsum−2−θ|Du|λ+1 dx
+ εCε3
∫
B2ρ
ζsup∗−θ dx. (6.5)
Thus, (6.2)–(6.5) show that we may absorb into I4 all negative terms on
the right hand side of (6.1), excepting only the last term in (6.4), provided
ε, ε1, ε2, ε3 are small enough. Of course we are also using p = p∗ here.
We have proven that
d
dt
∫
B2ρ
ζsu1−θ dx ≥ γ̃
∫
B2ρ
ζsup∗−θ dx − γρ
N−
(λ+1)(p∗−θ)
p∗−(m+λ−1) . (6.6)
Then apply Hölder’s inequality to get
I(t) :=
∫
B2ρ
ζs(x)u1−θ(x, t) dx ≤ γ
(
∫
B2ρ
ζsup∗−θ dx
)
1−θ
p∗−θ
ρ
N p∗−1
p∗−θ .
This and (6.6) imply
d
dt
I(t) ≥ γ0I(t)
p∗−θ
1−θ ρ−N p∗−1
1−θ − γ1ρ
N−
(λ+1)(p∗−θ)
p∗−(m+λ−1) . (6.7)
18 The Cauchy problem for degenerate parabolic equations
Fix any t̄ > 0. If for all ρ > 0 the right hand side of (6.7) is small, i.e., if
I(t̄) ≤
(2γ1
γ0
)
1−θ
p∗−θ
ρ
N−
(λ+1)(1−θ)
p∗−(m+λ−1) , (6.8)
then we get u(x, t̄) ≡ 0 on letting ρ → ∞. Indeed, if q < q∗, the power
of ρ in (6.8) is negative, for small enough θ. As we are dealing with non
trivial solutions, we may therefore assume, for some large ρ,
d
dt
I(t) ≥ γ0
2
I(t)
p∗−θ
1−θ ρ−N p∗−1
1−θ . (6.9)
Note that if (6.9) holds at t = t̄, it holds for all t > t̄, because I(t) is an
increasing function, as (6.9) itself implies. Then, I is a supersolution to
a non linear ordinary differential equation whose positive solutions blow
up in a finite time. The proof is concluded.
7. Higher order equations
The approach introduced above can be applied to find optimal decay
estimates for higher order parabolic problems like
(
|v|β−1v
)
t
+ (−1)l
∑
|α|=l
Dα
(
|Dlv|λ−1Dαv
)
= −|Dvµ|qv, in QT , (7.1)
v(x, 0) = v0(x), in R
N , (7.2)
where suppv0 ⊂ Bρ0 , v0 ∈ Lβ+1(RN ), and l > 1. In (7.1) the sum is
extended to all the derivatives of order l and we denote
|Dlv| =
(
∑
|α|=l
|Dαv|λ+1
)
1
λ+1
.
We assume that
1 < 1 + β < 1 + λ < µq + 1, 1 < q < λ + 1,
and
q < q∗ =
N(λ − 1) + βl(λ + 1)
µN + β
,
and consider energy solutions, which can be defined as in [8, 14], with
slight modifications.
Then one can prove the following bounds for large enough t:
Z(t) ≤ γt(µq+1−λ)/Λ, (7.3)
where Λ = l(λ + 1)(µq + 1 − β) + q(β − λ), and Z has been defined in
Theorem 1.1, and
∫
RN
|v(x, t)|β+1 dx ≤ γt−(1+β)(l(λ+1)−q)/Λ. (7.4)
D. Andreucci, A. F. Tedeev, M. Ughi 19
As a consequence of (7.3), (7.4) one obtains the mass decay rate
∫
RN
|v(x, t)|β dx ≤ γt−(q∗−q)(µN+β)/Λ. (7.5)
The proof follows closely the arguments in Sections 2, 3, borrowing some
techniques from [8].
References
[1] L. Alfonsi, F. B. Weissler, Blow-up in R
N for a parabolic equation with a damp-
ing nonlinear gradient term. Nonlinear Diffusion Equation and Their Equilib-
rium States // 3 (1992) N. G. Lloyd, W. M. Ni, L. A. Peletier, J. Serrin (eds),
Birkhāuser, Boston-Basel-Berlin.
[2] L. Amour, M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi
equations // Nonl. Anal. TMA, 31 (1998), 621–628.
[3] D. Andreucci, E. Di Benedetto, On the Cauchy problem and initial traces for a
class of evolution equations with strongly nonlinear sources // Annali Sc. Normale
Sup. Pisa, 18 (1991), 363–441.
[4] D. Andreucci, Degenerate parabolic equations with initial data measures // Trans.
Amer. Math. Soc. 340 (1997), No. 10, 3911–3923.
[5] D. Andreucci, A. F. Tedeev, Optimal bounds and blow up phenomena for parabolic
problems in narrowing domains // Proceeding of the Royal Soc. of Ed., A, 128
(1998), No. 6, 1163–1180
[6] D. Andreucci, A. F. Tedeev, A Fujita type result for degenerate Neumann problem
in domains with non compact boundary // J. Math. Anal. Appl., 231 (1999), 543–
567.
[7] D. Andreucci, A. F. Tedeev, Sharp estimates and finite speed of propagation for
a Neumann problem in domains narrowing at unfinity // Advances Diff. Eqs., 5
(2000), 833–860.
[8] D. Andreucci, A. F. Tedeev, Finite speed of propagation for the Thin Film equa-
tion and other higher order parabolic equations with general nonlinearity // In-
terfaces and Free Boundaries, 3 (2001), No. 3, 233–264.
[9] S. Benachour, Ph. Laurencot, Global solutions to viscous Hamilton-Jacobi equa-
tions with irregular initial data // Comm. Partial. Diff. Eq., 24 (1999), 1999–2021.
[10] B. Ben-Artzi, Global existence ahd decay for a nonlinear parabolic equation //
Nonlin. Anal. TMA, 19 (1992), 763–768.
[11] B. Ben-Artzi, H. Koch, Decay of mass for a semilinear parabolic equation //
Commun Partial Differ. Eq., 24 (1999), 869–881.
[12] M. Ben-Artzi, Ph. Souplet, F. B. Weissler, Sur la non-existence et la non-unicite
des solutions du probleme de Couchy pour une equation parabolique semi- lin-
eaire // C. R. Acad. Sci. Paris., Serie 1, 329 (1999), 371–376.
[13] M. Ben-Artzi, J. Goodman, A. Levy, Remarks on a nonlinear parabolic equa-
tion // Trans. Amer. Math. Soc., 352 (1999), No. 2, 731–751.
[14] F. Bernis, Finite speed of propagation and asymptotic rates of some nonlinear
higher order parabolic equations with absorbtion.
[15] F. Bernis, Existence results for doubly nonlinear higher order parabolic equations
on unbounded domains // Mathematische Annalen, 279 (1988), 373–394.
[16] M. Chipot, F. B. Weissler, Some blow up results for a nonlinear parabolic equation
with a gradient term // SIAM J.Math. Anal., 20 (1989), 886–907.
[17] K. Deng, H. Levine, The role of critical exponents in blow up theorems: The
sequel // J. Math. Anal. Appl. 243 (2000), 85–126 .
20 The Cauchy problem for degenerate parabolic equations
[18] M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient
term // Proc. Amer. Math. Soc., 111 (1991), 795–801.
[19] M. Fila, P. Quittner, Radial positive solutions for a semilinear elliptic equation
with a gradient term // Adv. Math. Sci. Appl., 2 (1993), 39–45.
[20] Z. Junning, On the Couchy problem and initial traces for the evalution p-laplacian
equations with strongly nonlinear sources // J. Differ. Equat., 121 (1995), 329–
383.
[21] B. Kawohl, L. A. Peletier, Observations on blow up and dead cores for nonlinear
parabolic equation // Math. Z., 202 (1989), 207–217.
[22] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-
linear Equations of Parabolic Type, v.23 of Translations of Mathematical Mono-
graphs. American Mathematical Society, Providence, RI, 1968.
[23] M. M. Porzio, V. Vespri, Hőlder estimates for local solutions of some double
nonlinear degenerate parabolic equations // J. Differ. Equat., 103 (1993), 146–
178.
[24] P. Quittner, Blow up for semilinear parabolic equations with a gradient term //
Math. Meth. Appl. Sci., 14 (1991), 413–417.
[25] P. Quittner, On global existence and stationary solutions for two classes of semi-
linear parabolic problems // Comment. Math. Univ. Carolinae, 34 (1993), No. 1,
105–124.
[26] Ph. Souplet, Finite time blow-up for a nonlinear parabolic equation with a gradient
term and applications // Math. Methods Appl. Sci., 19 (1996), 1317–1333.
[27] Ph. Souplet, S. Tayachi, F. B. Weissler, Exact self-similar blow-up of solutions of
a semilinear parabolic equations with a nonlinear gradient term // Indiana Univ.
Math. J.,48 (1996), 655–682.
[28] Ph. Souplet, F. B. Weissler, Self-similar sub-solutions and blow-up for nonlinear
parabolic equations // J. Math. Anal. Appl., 212 (1997), 60–74.
[29] Ph. Souplet, Geometry of unbounded domains, Poincare inequalities and stability
in semilinear parabolic equations // Comm. Partial Differ. Eq., 24 (1999), 951–
973.
[30] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear
parabolic equations // Manuscripta Math., 75 (1992), 65–80.
Contact information
D. Andreucci Dip. Metodi e Modelli,
Università La Sapienza,
via A. Scarpa 16,
00161 Roma, Italy
E-Mail: andreucci@dmmm.uniroma1.it
A. F. Tedeev Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
R. Luxemburg Str. 74,
83114 Donetsk, Ukraine
E-Mail: tedeev@iamm.ac.donetsk.ua
M. Ughi Dip. Scienze Matematiche,
Università di Trieste Piazzale Europa,
34100 Trieste, Italy
E-Mail: ughi@univ.trieste.it
|