Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term
We obtain the removability result for quasilinear equations of the special form and prove a priori estimates of the Keller–Osserman type.
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Shan, M.A. 2020-06-11T20:22:08Z 2020-06-11T20:22:08Z 2018 Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term / M.A. Shan // Український математичний вісник. — 2018. — Т. 15, № 1. — С. 80-93. — Бібліогр.: 19 назв. — англ. 1810-3200 2010 MSC. 35B40, 35B45 https://nasplib.isofts.kiev.ua/handle/123456789/169389 We obtain the removability result for quasilinear equations of the special form and prove a priori estimates of the Keller–Osserman type. This paper is supported by Ministry of Education and Science of Ukraine, grant number is 0118U003138. en Інститут прикладної математики і механіки НАН України Український математичний вісник Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term Article published earlier |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term Shan, M.A. |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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keller-osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term |
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Shan, M.A. |
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Shan, M.A. |
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2018 |
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Інститут прикладної математики і механіки НАН України |
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We obtain the removability result for quasilinear equations of the special form and prove a priori estimates of the Keller–Osserman type.
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1810-3200 |
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https://nasplib.isofts.kiev.ua/handle/123456789/169389 |
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Keller-Osserman a priori estimates and removability result for the anisotropic porous medium equation with absorption term / M.A. Shan // Український математичний вісник. — 2018. — Т. 15, № 1. — С. 80-93. — Бібліогр.: 19 назв. — англ. |
| work_keys_str_mv |
AT shanma kellerossermanaprioriestimatesandremovabilityresultfortheanisotropicporousmediumequationwithabsorptionterm |
| first_indexed |
2025-11-25T14:44:03Z |
| last_indexed |
2025-11-25T14:44:03Z |
| _version_ |
1850515178968317952 |
| fulltext |
Український математичний вiсник
Том 15 (2018), № 1, 80 – 93
Keller–Osserman a priori estimates and
removability result for the anisotropic porous
medium equation with absorption term
Maria A. Shan
(Presented by A. E. Shishkov)
Abstract. In this article we obtained the removability result for quasi-
linear equations model of which is
ut −
n∑
i=1
(
umi−1uxi
)
xi
+ f(u) = 0, u ≥ 0.
and prove a priori estimates of Keller–Osserman type.
2010 MSC. 35B40, 35B45.
Key words and phrases. Anisotropic porous medium equation, Keller–
Osserman a priori estimates, removability of isolated singularity.
1. Introduction and main results
In this paper we study solutions to quasilinear parabolic equation in
the divergent form
ut − divA(x, t, u,∇u) + a0(u) = 0, (x, t) ∈ ΩT = Ω× (0, T ), (1.1)
satisfying a initial condition
u(x, 0) = 0, x ∈ Ω \ {0}, (1.2)
where Ω is a bounded domain in Rn, n ≥ 2, 0 < T <∞.
We suppose that the functions A = (a1, . . . , an) and a0 satisfy the
Caratheodory conditions and the following structure conditions hold
A(x, t, u, ξ)ξ ≥ ν1
n∑
i=1
|u|mi−1|ξi|2,
Received 02.03.2018
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
M. A. Shan 81
|ai(x, t, u, ξ)| ≤ ν2u
(mi−1) 1
2
n∑
j=1
|u|mj−1|ξj |2
1
2
, i = 1, n, (1.3)
a0(u) ≥ ν1f(u),
with positive constants ν1, ν2 and continuous, positive function f(u) and
min
1≤i≤n
mi > 1− 2
n
, max
1≤i≤n
mi ≤ m+
2
n
, (1.4)
where m = 1
n
n∑
i=1
mi. Without loss of generality we will assume later that
m1 ≤ m2 ≤ . . . ≤ mn.
Many authors studied problems of singularities of solutions of second
order quasilinear elliptic and parabolic equations. Review of these results
can be found in the monograph of Veron [19]. Brezis and Veron [2] proved
that for q ≥ n
n−2 the isolated singularities of solutions to the elliptic
equation
−△u+ uq = 0,
are removable. In [3] Brézis and Friedman proved that for q ≥ n+2
n the
isolated singularities of solutions for the following parabolic equation
∂u
∂t
−△u+ |u|q−1u = 0, (x, t) ∈ ΩT \ {(0, 0)}
are removable. The removability of isolated singularity for solutions of
the nonanisotropic porous medium equation (m = m1 = . . . = mn, )
ut −△
(
|u|m−1u
)
+ |u|q−1u = 0,
has been proved under the assumption q ≥ m+ 2
n by Kamin and Peletier
[5].
Development of the qualitative theory of second order quasilinear el-
liptic and parabolic equations with nonstandart growth conditions has
been observed in recent decades. Some results of [4, 6, 7, 9, 10, 12, 15–18]
we mention here. One of the example of such equations is
∂u
∂t
−
n∑
i=1
∂
∂xi
(
|u|(mi−1)(pi−2)
∣∣∣∣ ∂u∂xi
∣∣∣∣pi−2 ∂u
∂xi
)
= 0, pi ≥ 2,mi ≥ 1, i = 1, n.
The removability result of isolated singularity and a priori estimates of
Keller–Osserman type for this equation was obtained in [9, 12].
82 Keller–Osserman a priori estimates...
We now define a weak solution of the problem (1.1), (1.2) with sin-
gularity at the point (0, 0). We will write V2,m(ΩT ) for the class of func-
tions φ ∈ Cloc(0, T, L
1+m−
loc (Ω)) with
n∑
i=1
∫∫
ΩT
|φ|mi+m
−−2 |φxi |
2 dxdt < ∞,
where m− = min(mn, 1). By a weak solution of the problem (1.1),
(1.2) we mean a function u(x, t) ≥ 0 satisfying the inclusion uψ ∈
V2,m(ΩT ) ∩ L2
loc(0, T ;W
1,2
loc (Ω)) and the integral identity∫
Ω
u(x, τ)ψφdx
+
τ∫
0
∫
Ω
{−u(ψφ)t +A(x, t, u,∇u)∇(ψφ) + a0(u)ψφ} dx dt = 0 (1.5)
holds for any testing function φ∈W 1,2
loc (0, T ;L
2
loc(Ω))∩L2
loc(0, T ;
o
W
1,2
loc(Ω)),
any ψ ∈ C1(ΩT ) vanishing in the neighborhood of {(0, 0)} and for all
τ ∈ (0, T ) .
The result of this paper is the removability of isolated singularities
for solutions of the anisotropic porous medium equation with absorption
term. The proof of this result is based on a priori estimates of Keller–
Osserman type of the solution to the equation (1.1). The main difficulty
lies in the fact that part of mi < 1 (singular case), and another part of
mi > 1 (degenerate case).
Theorem 1.1. Let the conditions (1.3), (1.4) be fulfilled and u be a
nonnegative weak solution to the problem (1.1), (1.2). Assume also that
f(u) = uq and
q ≥ m+
2
n
, (1.6)
then the singularity at the point {(0, 0)} is removable.
Let (x(0), t(0)) ∈ ΩT , for any τ, θ1, θ2, . . . , θn > 0, θ = (θ1, . . . , θn) we
define Qθ,τ (x(0), t(0)) := {(x, t) : |t − t(0)| < τ, |xi − x
(0)
i | < θi, i = 1, n}
and set M(θ, τ) := sup
Qθ,τ (x(0),t(0))
u, F (θ, τ) := sup
Qθ,τ (x(0),t(0))
F (u), F (u)=
u∫
0
sm
−−1
f(s)ds, m+= max(mn, 1).
Theorem 1.2. Let the conditions (1.3), (1.4) be fulfilled and u be a non-
negative weak solution to equation (1.1), assume also that f ∈ C1(R1
+)
and f
′
(u) ≥ 0. Let (x(0), t(0)) ∈ ΩT , fix σ ∈ (0, 1), let Q8θ,8τ (x
(0), t(0))⊂
M. A. Shan 83
ΩT . Set ρ =
{
θn, if mn > 1,
τ
1
2 , if mn < 1,
, then there exist positive number c1, c2
depending only on n, ν1, ν2,m1, . . . ,mn such that either
u(x(0), t(0)) ≤
(
θ2n
τ
) 1
mn−1
+
n−1∑
i=1
(
ρ
θi
) 2
m+−mi
, (1.7)
or
(M(σθ, στ))1−m
−+
n(m−m−)
2 F (M(σθ, στ))
≤ c1(1− σ)−γρ−2(M(θ, τ))m
++1+
n(m−m−)
2 (1.8)
holds true.
We also have, in particular, if
F (εu) ≤ εm
++m−+βF (u), β > 0, (1.9)
then
F (M(θ, τ)) ≤ c2(1− σ)−γMm++m−
(θ, τ)ρ−2, (1.10)
An example of the function f , which satisfies the conditions (1.9) is
f(u) = uq, q ≥ m+ 2
n . Assuming for simplicity that dist(x(0), ∂Ω) = |x(0)|,
and choosing τ, θi, from the conditions
• mn > 1:
(
θ2n
τ
) 1
mn−1
= θ
− 2
q−mn
n , i.e. τ= θ
2(q−1)
q−mn
n ,(
θn
θi
) 2
mn−mi = θ
− 2
q−mn
n , i.e. θi = θ
q−mi
q−mn
n ,
• mn < 1:
(
θ2n
τ
) 1
mn−1
= τ
− 1
q−mn , i.e. τ= θ
2(q−1)
q−mn
n ,(
τ
1
2
θi
) 2
1−mi
= τ
− 1
q−1 , i.e. θi = τ
q−mi
2(q−1) ,
from (1.7), (1.10) we obtain an estimate
u(x(0), t(0)) ≤ c
(
n∑
i=1
|x(0)i |
2
q−mi + (t(0))
1
q−1
)−1
. (1.11)
2. Keller–Osserman a priori estimates
2.1. Auxiliary propositions
LetE(2ρ) = {(x, t) ∈ ΩT : u(x, t) > M(2ρ)}, u(ρ)(x, t) = min
(
M
(ρ
2
)
−
M(2ρ), u(x, t)−M(2ρ)) .
84 Keller–Osserman a priori estimates...
Lemma 2.1. [11] Under the assumptions of Theorem 1.1 following in-
equality holds ∫∫
E(2ρ)
u(ρ)uqψlrdxdt ≤ γ
(
M
(ρ
2
)
−M(2ρ)
)
×
{
F3(r, λ) + (F1(r, λ) + F2(r, λ))
1
2F
1
2
4 (r, λ)
}
, (2.1)
where
F1(r, λ) =
Rλ(r), λ > 0,
ln
q−2
q−1
1
r
, λ = 0, q > 2,
ln ln
1
r
, λ = 0, q = 2,
ln
− 2−q
q−1
1
r
, λ = 0, q < 2
F2(r, λ) =
Rλ(r), λ > 0,
ln
q−2m1
q−m1
1
r
, λ = 0, q > 2m1,
ln ln
1
r
, λ = 0, q = 2m1,
ln
− 2m1−q
1−m1
1
r
, λ = 0, q < 2m1.
F3(r, λ) =
Rλ(r), λ > 0,
ln
− 1
q−1
1
r
, λ = 0,
F4(r, λ) =
Rλ(r), λ > 0,
ln−1 1
R(r)
, λ = 0,
where λ = n− 2
q−m , 0 < r < R0.
Lemma 2.2. [1] Let Ω ⊂ Rn, n ≥ 2 be a bounded domain, v ∈
o
W 1,1(Ω)
and
n∑
i=1
∫
Ω
|v|αi |vxi |pidx <∞, αi ≥ 0, pi > 1. (2.2)
If 1 < p < n, then v ∈ Lq(Ω), q = np
n−p
(
1 + 1
n
n∑
i=1
αi
pi
)
, 1
p = 1
n
n∑
i=1
1
pi
and
the following inequality holds
∥v∥Lq(Ω) ≤ γ
n∏
i=1
∫
Ω
|v|αi |vxi |pidx
1
npi
(
1+ 1
n
n∑
i=1
αi
pi
)
, (2.3)
where the positive constant γ depends only on n, pi, αi, i = 1, n.
M. A. Shan 85
Lemma 2.3. [8, chap. 2] Let {yj}j∈N be a sequence of nonnegative num-
bers such that for any j = 0, 1, 2, . . . the inequality
yj+1 ≤ Cbjy1+εj
holds with positive ε, C > 0, b > 1. Then the following estimate is true
yj ≤ C
(1+ε)j−1
ε b
(1+ε)j−1
ε2
− j
ε y
(1+ε)j
0 .
Particulary, if y0 ≤ C− 1
ε b−
1
ε2 , then lim
j→∞
yj = 0.
2.2. Integral estimates of solutions
Consider a cylinder Qθ,τ (x(0), t(0)) and let (x̄, t̄) be an arbitrary point
in Qσθ,στ (x
(0), t(0)). If u(x(0), t(0)) ≥
(
θ2n
τ
) 1
mn−1
+
n−1∑
i=1
(
ρ
θi
) 2
m+−mi then
M(θ, τ)=max(M(θ,τ),δ(θ,τ))≥(τ−1θn)
1
mn−1 +
n∑
i=1
(θ−1
i ρ)
2
m+−mi , and hence
Qη,s(x̄, t̄)⊂Qθ,τ (x(0), t(0)), where s = (1 − σ)M1−m+
(θ, τ)ρ2, ηi = (1 −
σ)M
mi−m+
2 (θ, τ)ρ, i = 1, n. For fixed k > 0 and l, j = 0, 1, 2 . . . set αl =
1
4(1+ 2−1+ · · ·+2−l), set kj = k(1− 2−j), ηi,j,l = (αl+
1
42
−j−l−1)ηi, i =
1, n, ηj,l = (η1,j,l, . . . , ηn,j,l), sj,l = (αl +
1
42
−j−l−1)s, Qj,l = Qηj,l,sj,l(x̄, t̄),
Akj ,j,l = {x ∈ Qj,l(x̄, t̄) : F (u) > kj}. Let ξj ∈ C∞
0 (Qj,l(x̄, t̄)), 0 ≤ ξj ≤
1, ξj = 1 in Qj+1,l(x̄, t̄),
∣∣∣∂ξj∂t ∣∣∣ ≤ γ2j+ls−1,
∣∣∣ ∂ξj∂xi
∣∣∣ ≤ γ2j+lη−1
i , i = 1, n.
In what follows γ stands for a constant depending only n, ν1, ν2,
m1, . . . , mn which may vary from line to line.
Lemma 2.4. Let u be a nonnegative weak solution to equation (1.1)
and let conditions (1.3), (1.4) hold. Then for any j ≥ 0 the following
inequality holds true
l1−m
−
j
∫
Akj,j,l
(t)
(F (u)− kj)
2
+ξ
2
j dx+
n∑
i=1
lmi−m−
j
∫∫
Akj,j,l
|∇((F (u)− kj)+)|2 ξ2j dxdt
+
∫∫
Akj,j,l
(F (u)− kj)+f
2(u)ξ2j dxdt≤γMm+−m−
(θ, τ)ρ−2
∫∫
Akj,j,l
(F (u)− kj)
2
+dxdt
(2.4)
where lj = F−1(kj), j = 0, 1, 2, . . .
86 Keller–Osserman a priori estimates...
Proof. Testing identity (1.5) by φ = (F (u)−kj)+f(u)ξ2j , using conditions
(1.3) we obtain ∫∫
Akj,j,l
utf(u)(F (u)− kj)+ξ
2
j dxdt
+
n∑
i=1
∫∫
Akj,j,l
umi+m
−−2|uxi |2f2(u)ξ2j dxdt+
∫∫
Akj,j,l
(F (u)− kj)+f
2(u)ξ2j dxdt
≤ γ
n∑
i=1
∫∫
Akj,j,l
u
mi−1
2
(
n∑
l=1
uml−1|uxl |
2
) 1
2
(F (u)− kj)+f(u)ξj
∣∣∣∣∂ξj∂xi
∣∣∣∣ dxdt.
From this, using the Young inequality and the evident inequality
lj < u(x, t) < M(θ, τ) on Akj ,j,l we arrive at the required (2.4).
2.3. Proof of Theorem 1.2
By Lemma 2.2 and the Hölder inequality we obtain
Yj+1,l =
∫∫
Akj+1,j+1,l
(F (u)− kj+1)
2
+dxdt
≤
∣∣Akj+1,j+1,l
∣∣ 2
n+2
∫∫
Akj+1,j+1,l
((F (u)− kj+1)+ξj)
2(n+2)
n dxdt
n
n+2
≤
∣∣Akj+1,j+1,l
∣∣ 2
n+2 ess sup
0<t<T
∫
Akj+1,j+1,l(t)
(F (u)− kj+1)
2
+ξ
2
j dx
2
n+2
×
T∫
0
n∏
i=1
∫
Akj+1,j+1,l(t)
|((F (u)− kj+1)+ξj)xi |
2 dx
1
n
dt
n
n+2
≤
∣∣Akj+1,j+1,l
∣∣ 2
n+2 ess sup
0<t<T
∫
Akj+1,j+1,l(t)
(F (u)− kj+1)
2
+ξ
2
j dx
2
n+2
M. A. Shan 87
×
T∫
0
n∏
i=1
∫
Akj+1,j+1,l(t)
|((F (u)− kj+1)+)xi |
2 ξ2j dx
+
∫
Akj+1,j+1,l(t)
(F (u)− kj+1)
2
+
∣∣∣∣∂ξj∂xi
∣∣∣∣2 dx
1
n
dt
n
n+2
.
Denote Ql = Qαlη,αls,Ml = sup
Ql
u, using (2.4), it follows from Lem-
ma 2.3 that yj,l → 0 as j → ∞, provided k is chosen to satisfy
k2 = γ2lγ l
m−−1+
n(m−−m)
2
j M
(n+2)(m+−m−)
2
l+1 (θ, τ)ρ−n−2
∫∫
Ql+1
F 2(u)dxdt.
From this we obtain
M
1−m−+
n(m−m−)
2
l F 2(Ml)
≤ γ(1− σ)−γ2lγM
(n+2)(m+−m−)
2
l+1 ρ−n−2
∫∫
Ql+1
F 2(u)dxdt.
Denoting M
1−m−
2
+
n(m−m−)
4
l F (Ml) =M
a
2
l F (Ml) = Ψl, we have
Ψ2
l ≤ γ(1− σ)−γ2lγΨl+1M
(n+2)(m+−m−)
2
−a
2
l+1 ρ−n−2
∫∫
Ql+1
F (u)dxdt
≤ εΨ2
l+1 +
1
ε
(1− σ)−γγ2lγ(M(θ, τ))(n+2)(m+−m−)−a
×ρ−2(n+2)
∫∫
Ql+1
F (u)dxdt
2
.
From this by iteration
Ψ2(u(x̄, t̄)) ≤ Ψ2
0 ≤ εlΨ2
l +
1
ε
γ(1− σ)−γ
l−1∑
i=0
(ε2γ)i
×(M(θ, τ))(n+2)(m+−m−)−aρ−2(n+2)
∫∫
Ql+1
F (u)dxdt
2
.
88 Keller–Osserman a priori estimates...
We choose ε = 2−γ−1 and passing to the limit as l → ∞, we obtain
(u(x̄, t̄))1−m
−+
n(m−m−)
2 F (u(x̄, t̄)))
≤ γ(1− σ)−γρ−n−2(M(θ, τ))
(n+2)(m+−m−)
2
∫∫
Q η
2 , s2
(x̄,t̄)
f(u)um
−
dxdt. (2.5)
To estimate the integral on the right-hand side of (2.5) we test integral
identity by φ = um
−
ζ2, using conditions (1.4) and the Hölder inequality,
we obtain∫∫
Q η
2 , s2
(x̄,t̄)
f(u)um
−
ζ2dxdt≤γ
∫∫
Q η
2 , s2
(x̄,t̄)
um
−+1|ζt|ζdxdt+ γ
n∑
i=1
∫∫
Q η
2 , s2
(x̄,t̄)
umi+m
−|ζxi |2dxdt
≤ γρ−2Mm++m−
(θ, τ)|Q η
2
, s
2
(x̄, t̄)| ≤ γρnMm−+1+m−m+
2
n(θ, τ). (2.6)
Since (x̄, t̄) is an arbitrary point in Qσθ,στ (x(0), t(0)) from (2.5), (2.6)
we arrive at
(M(σθ, στ))1−m
−+
n(m−m−)
2 F (M(σθ, στ))
≤ γ(1− σ)−γρ−2(M(θ, τ))m
++1+
n(m−m−)
2 . (2.7)
For j = 0, 1, 2 . . . define the sequences {σj}, {θj}, {τj}, {Mj} by
σj := 1−2−j−1
1−2−j−2 , θj := (θ1j , θ2j , . . . , θnj), θij = θi
(
1 + 1
2 + · · ·+ 1
2j
)
,
i = 1, n, τj = τ
(
1 + 1
2 + · · ·+ 1
2j
)
, Mj := sup
Qθj ,τj
(x(o))
,
Γ(Mj) =
[
F (Mj)
Mm++m−
j
] 1
m++1+
n(m−m−)
2 .
We write (2.7) for the pair of boxes Qθj ,τj (x
(0), t(0)) and
Qθj+1,τj+1
(x(0), t(0)). This gives
MjΓ(Mj) ≤ γ(1− σ)−γ2jγρ
−2
m++1+
n(m−m−)
2 Mj+1.
Using the following inequality which is an immediate consequence of
our choice of Γ
Γ(u)v ≤ ε−1Γ(u)u+ Γ(εv)v, ε, u, v > 0, (2.8)
indeed if v ≤ ε−1u, then Γ(u)v ≤ ε−1Γ(u)u, and if v ≥ ε−1u, then
Γ(u)v ≤ Γ(εv)v, and in both cases (2.8) holds.
M. A. Shan 89
If ε ∈ (0, 1), µ = β
m++1+
n(m−m−)
2
then
Γ(Ml) ≤ Γ(εMl+1) +
1
ε
Γ(Ml)Ml
Ml+1
≤ Γ(εMl+1) + ε−1γ(1− σ)−γ2lγρ
−2
m++1+
n(m−m−)
2
≤ εµΓ(Ml+1) + ε−1γ(1− σ)−γ2lγρ
−2
m++1+
n(m−m−)
2 .
From this by iteration
Γ(M0) ≤ εlµΓ(Mi+1) + ε−1γ(1− σ)−γ
l∑
i=0
(εiµ2iγ)ρ
−2
m++1+
n(m−m−)
2 .
We choose εµ = 2−γ−1 and passing to the limit as l → ∞, we obtain
Γ(u(x(0), t(0))) ≤ γ(1− σ)−γρ
−2
m++1+
n(m−m−)
2 .
Return to the previous notation
F (u(x(0), t(0))) ≤ γ(1− σ)−γ(M(θ, τ))m
++m−
ρ−2. (2.9)
Thus Theorem 1.2 is proved.
3. Proof of Theorem 1.1
3.1. Pointwise estimates of solutions
Let
Qr =
(x, t) ∈ ΩT :
(
t
κ(λ)
κ1(λ) +
n∑
i=1
|xi|
κi(λ)
κ1(λ)
)κ1(λ)
< r,
,
where κ(λ) = 1
2+(n−λ)(m−1) , κi(λ) = 2
2+(n−λ)(m−mi)
, i = 1, n, λ = n −
2
q−m . For 0 < r < ρ < R0
2 (R0 : QR0 ⊂ ΩT ) we set M(r) = sup
QR0
\Qr
u(x, t)
and u2ρ = u(x, t) − M(2ρ) ≤ M
(ρ
2
)
− M(2ρ) for (x, t) ∈ QR0 \ Q ρ
2
.
For fixed k > 0 and j = 0, 1, . . . set ρj = ρ
4
(
1 + 1
2j
)
, kj = k(1 − 2−j),
Akj ,j = {(x, t) ∈ Qρj : u2ρ > kj}. Let ζj ∈ C∞
(
Q ρj+1+ρj
2
)
, 0 ≤ ζj ≤ 1,
ζj = 1 outside Qρj , ζj = 0 in Qρj+1 , and
∣∣∣∂ζj∂t ∣∣∣ ≤ γ2jγρ
− 1
κ(λ) ,
∣∣∣ ∂ζj∂xi
∣∣∣ ≤
90 Keller–Osserman a priori estimates...
γ2jγρ
− 2
κi(λ) , i = 1, n. Let i0 be the number such that mi ≤ 1, i = 1, . . . i0
and mi > 1, i = i0+1, . . . n, m′
= 1
n
i0∑
i=1
mi, m
′′
= 1
n
n∑
i=i0+1
mi. Note that
i0 = 0 if mi > 1, i = 1, n, and i0 = n if mi ≤ 1, i = 1, n.
Testing identity (1.4) by φ = (u2ρ−kj)+ζ2j , using conditions (1.4) we
obtain
ess sup
∫
Akj,j
(t)
(u2ρ − kj)
2
+ζjdx+
i0∑
i=1
Mmi−1
(ρ
2
)∫∫
Akj,j
|uxi |2ζ2j dxdt
+
n∑
i=i0+1
kmi−1
j
∫∫
Akj,j
|uxi |2ζ2j dxdt+
∫∫
Akj,j
(u2ρ − kj)+u
qζ2j dxdt
≤ γ
(
M2
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi+1
(ρ
2
)
ρ
− 2
κi(λ)
)
|Akj ,j |. (3.1)
By Lemma 2.2, the Hölder inequality and estimate (3.1), we obtain
Yj+1 =
∫∫
Akj+1,j+1
(u2ρ − kj+1)
2
+dxdt
≤
∣∣Akj+1,j+1
∣∣ 2
n+2
∫∫
Akj+1,j+1
((u2ρ − kj+1)+ζj)
2+ 4
ndxdt
n
n+2
≤
∣∣Akj+1,j+1
∣∣ 2
n+2 ess sup
0<t<T
∫
Akj+1,j+1(t)
(u2ρ − kj+1)
2
+ζ
2
j dx
2
n+2
×
t∫
0
n∏
i=1
∫
Akj+1,j+1(t)
|((u2ρ − kj+1)+ζj)xi |
2 dx
1
n
dτ
n
n+2
≤ γM
(1−m
′
)i0
n+2
(ρ
2
)
k
(1−m
′′
)(n−i0)
n+2
j+1
×
(
M2
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi+1
(ρ
2
)
ρ
− 2
κi(λ)
)
|Akj+1,j+1|1+
2
n+2 .
From this by the evident inequality (u2ρ− kj)+ ≥ k
2j+1 on Akj+1,j , we
obtain estimate
M. A. Shan 91
Yj+1 ≤ γ2jγM
(1−m
′
)i0
n+2
(ρ
2
)
k
(1−m
′′
)(n−i0)
n+2
j+1
(
M2
(ρ
2
)
ρ
− 1
κ(λ)
+
n∑
i=1
Mmi+1
(ρ
2
)
ρ
− 2
κi(λ)
)
Y
1+ 2
n+2
j . (3.2)
It follows from Lemma 2.3 that
(M(ρ)−M(2ρ))
(m
′′
−1)(n−i0)
2
+n+4 ≤ γ2jγM
(1−m
′
)i0
n+2
(ρ
2
)
×
(
M2
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi+1
(ρ
2
)
ρ
− 2
κi(λ)
)∫∫
Q ρ
2
u22ρdxdt. (3.3)
By the Hölder inequality and Lemma 2.1 we get
(M(ρ)−M(2ρ))
(m
′′
−1)(n−i0)
2
+n+4 ≤ γ2jγM
(1−m
′
)i0
n+2
(ρ
2
)
×
(
M2
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi+1
(ρ
2
)
ρ
− 2
κi(λ)
)
×
{
F3(r, λ) + (F1(r, λ) + F2(r, λ))
1
2F
1
2
4 (r, λ)
}
|Q ρ
2
|
q−1
q+1 . (3.4)
Similarly to [11], we obtain the following estimate
M(ρ)−M(2ρ) ≤ 0,
iterating last inequality we get for any ρ ≤ R0
2
M(ρ) ≤M(R0),
this proves the boundedness of solutions.
3.2. End of the proof of Theorem 1.1
Let K be a compact subset in Ω, and ξ = 0 in ∂Ω× (0, T ), such that
ξ = 1 for (x, t) ∈ K × (0, T ). Testing (1.5) by φ = um
−
ξ2ψr, ψ = ψr,
using conditions (1.3), the Young inequality, the boundedness of u and
passing to the limit r → 0 we get
sup
0<t<T
∫
K
um
−+1dx+
n∑
i=1
T∫
0
∫
K
umi+m
−−2|uxi |2dxdt+
T∫
0
∫
K
uq+m
−
dxdt ≤ γ.
(3.5)
92 Keller–Osserman a priori estimates...
Testing (1.5) by φψr, using (1.3), the boundedness of solution, and
passing to the limit r → 0, we obtain the integral identity (1.5) with an
arbitrary φ ∈ W 1,2
loc (0, T ;L
2
loc(Ω)) ∩ L2
loc(0, T ;
o
W
1,2
loc(Ω)) and ψ ≡ 1. Thus
Theorem 1.1 is proved.
Acknowledgements
This paper is supported by Ministry of Education and Science of
Ukraine, grant number is 0118U003138.
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Contact information
Maria Alekseevna
Shan
Vasyl’ Stus Donetsk National University,
Vinnytsia, Ukraine
E-Mail: shan_maria@ukr.net
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