Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term
The removability of an isolated singularity for solutions to the quasilinear equation is proved.
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term / M.A. Shan // Український математичний вісник. — 2016. — Т. 13, № 3. — С. 350-360. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1450772019-01-15T01:23:28Z Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term Shan, M.A. The removability of an isolated singularity for solutions to the quasilinear equation is proved. 2016 Article Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term / M.A. Shan // Український математичний вісник. — 2016. — Т. 13, № 3. — С. 350-360. — Бібліогр.: 14 назв. — англ. 1810-3200 2010 MSC. 35B40 http://dspace.nbuv.gov.ua/handle/123456789/145077 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The removability of an isolated singularity for solutions to the quasilinear equation is proved. |
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Article |
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Shan, M.A. |
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Shan, M.A. Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term Український математичний вісник |
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Shan, M.A. |
| author_sort |
Shan, M.A. |
| title |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
| title_short |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
| title_full |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
| title_fullStr |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
| title_full_unstemmed |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
| title_sort |
removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/145077 |
| citation_txt |
Removability of isolated singularity for solutions of anisotropic porous medium equation with absorption term / M.A. Shan // Український математичний вісник. — 2016. — Т. 13, № 3. — С. 350-360. — Бібліогр.: 14 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT shanma removabilityofisolatedsingularityforsolutionsofanisotropicporousmediumequationwithabsorptionterm |
| first_indexed |
2025-07-10T20:47:49Z |
| last_indexed |
2025-07-10T20:47:49Z |
| _version_ |
1837294399402278912 |
| fulltext |
Український математичний вiсник
Том 13 (2016), № 3, 350 – 360
Removability of isolated singularity
for solutions of anisotropic porous
medium equation with absorption term
Maria A. Shan
(Presented by I. I. Skripnik)
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.
2010 MSC. 35B40.
Key words and phrases. Quasilinear parabolic equations, removable
isolated singularity.
1. Introduction and main result
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 , (1.1)
satisfying a initial condition
u(x, 0) = 0, x ∈ Ω \ {(0, 0)} (1.2)
in ΩT = Ω × (0, T ), 0 < T < ∞, where Ω is a bounded domain in
Rn, n > 2.
The qualitative behaviour of solution to elliptic equations was in-
vestigated by many authors starting from the seminal papers of Serrin
(see [4–8] ). In [1] Brezis and Veron proved that for q ≥ n
n−2 the isolated
singularities of solutions to the elliptic equation
Received 05.09.2016
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
M. A. Shan 351
−△u+ uq = 0,
are removable. The result on the removability of an isolated singularity
for the following parabolic equation
∂u
∂t
−△u+ |u|q−1u = 0, (x, t) ∈ ΩT \ {(0, 0)}
was obtained by Brézis and Friedman [2] in the case q ≥ n+2
n . The
anisotropic elliptic equation with absorption
−
n∑
i=1
(
|uxi |pi−2uxi
)
xi
+ |u|q−1u = 0
was studied in [12]. It was proved that the isolated singularity for solution
of the this equation is removable if
q ≥ n(p− 1)
n− p
, 1 ≤ p1 ≤ . . . ≤ pn ≤ n− 1
n− p
p.
For quasilinear elliptic and parabolic equations of special form with
absorption similar questions were treated by many authors. A survey
of their results and references can be found in Veron’s monograph [14].
The removability of isolated singularities for more general elliptic and
parabolic equations with absorption were established in [10] and [11].
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,
|ai(x, t, u, ξ)| ≤ ν2u
mi−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, max
1≤i≤n
mi ≤ 1 +
κ
n
, p < n, (1.4)
where κ = n(m − 1) + 2, d = 1
n
n∑
i=1
mi
2 , and assume without loss, that
mn = max
1≤i≤n
mi.
352 Removability of isolated singularity for solutions...
We will write V2,m(ΩT ) for the class of functions φ ∈ C(0, T, L2(Ω))
with
n∑
i=1
∫∫
ΩT
|φ|mi−1 |φxi |
2 dxdt <∞.
We say that u is a weak solution to the problem (1.1), (1.2) if for
an arbitrary ψ ∈ C1(ΩT ), vanishing in a neighborhood of {(0, 0)}, we
have an inclusion uψ ∈ V2,m(ΩT ) and for any interval (t1, t2) ⊂ [0, T ) the
integral identity
∫
Ω
uφdx
∣∣∣∣∣∣
t2
t1
+
t2∫
t1
∫
Ω
{−uφt +A(x, t, u,∇u)∇φ+ a0(u)φ} dx dt = 0 (1.5)
holds for φ = ζψ with an arbitrary ζ ∈
o
V 2,m(ΩT ).
We say that solution u to the problem (1.1), (1.2) has a removable
singularity at {(0, 0)} if u can be extended to {(0, 0)} so that the extension
ũ of u satisfies (1.5) with ψ ≡ 1 and ũ ∈ V2,m(ΩT ).
Remark 1.1. Condition (1.4) implies the local boundedness of weak
solutions to the equation (1.1) ([3]).
The main result of this paper is the following theorem.
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.
The rest of the paper contains the proof of Theorem 1.1.
2. Integral estimates of solutions
For 0 ≤ λ < n we define the following numbers
κ(λ) =
1
2 + (n− λ)(m− 1)
, κi(λ) =
2
2 + (n− λ)(m−mi)
, i = 1, n.
Let
ρλ(x, t) =
(
t
κ(λ)
κ1(λ) +
n∑
i=1
|xi|
κi(λ)
κ1(λ)
)κ1(λ)
,
assume that Dλ(r) = {(x, t) : ρλ(x, t) < r}, Dλ(R0) ⊂ ΩT and for
0 < r < R0 we set M(r, λ) = sup
Dλ(R0)\Dλ(r)
u(x, t), E(r, λ) = {(x, t) ∈
M. A. Shan 353
ΩT : u(x, t) > M(r, λ)}, ur(r, t, λ) = (u(x, t) −M(r, λ))+ and consider
the function ψr(x, t) = ηr(ρλ(x, t)), where ηr : R1 → R1 is a function
taking the following values: ηr(z) = 0 if z ≤ r, ηr(z) = 1 if z ≥ R(r),
ηr(z) =
[
(1 − ε) ln ln 1
r
]−1 (
ln ln 1
r− ln ln 1
z
)
, if r ≤ z ≤ R(r), here ε is a
number from the interval (0, 1) specified in what follows and R(r) defined
by the equality
ln
1
R(r)
= lnε
1
r
. (2.1)
Note that by the evident equalities 1
q−1 = (n− λ)κ(λ), 2
q−mi = (n−
λ)κi(λ), i = 1, n, with λ ≥ 0 defined by
λ = n− 2
q −m
, (2.2)
the Keller–Osserman estimate yields
M(r, λ) ≤ γrλ−n, r > 0. (2.3)
This estimate is received from Theorems 4.1, 4.2 (Appendix) in the case
p1 = p2 = ... = pn = 2.
Consider the functions F1(r, λ), F2(r, λ) defined by the following eq-
ualities
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 , 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 , λ = 0, q < 2m1.
To simplify the following calculations we will write M(r), E(r),
ur(x, t) instead of M(r, λ), E(r, λ), ur(x, t, λ).
Lemma 2.1. Let the assumptions of Theorem 1.1 be fulfilled, then for
every l ≥ 2q
q−mn and for every 2r < ρ ≤ R0
2 the following estimate holds
354 Removability of isolated singularity for solutions...
sup
0<t<T
∫
E( ρ2 )×{t}
u∫
M( ρ2 )
ln+
s
M
(ρ
2
)dsψlr dx+
n∑
i=1
∫∫
E( ρ2 )
umi−2|uxi |2ψlrdxdt
+
∫∫
E( ρ2 )
uq ln
u
M
(ρ
2
)ψlrdxdt ≤ γ (F1(r, λ) + F2(r, λ)) . (2.4)
Proof. Testing (1.5) by φ = ln+
u
M( ρ2 )
ψlr, using (1.3) and the Young
inequality we get
sup
0<t<T
∫
E( ρ2 )×{t}
u∫
M( ρ2 )
ln+
s
M
(ρ
2
)dsψlr dx+
n∑
i=1
∫∫
E( ρ2 )
umi−2|uxi |2ψlrdxdt
+
∫∫
E( ρ2 )
uq ln
u
M
(ρ
2
)ψlrdxdt ≤ γ
∫∫
E( ρ2 )
u ln
u
M
(ρ
2
) ∣∣∣∣∂ψr∂t
∣∣∣∣ψl−1
r dxdt
+ γ
n∑
i=1
∫∫
E( ρ2 )
umi ln2 u
M
(ρ
2
) ∣∣∣∣∂ψr∂xi
∣∣∣∣2 ψl−2
r dxdt.
From this, by the Young inequality we obtain
sup
0<t<T
∫
E( ρ2 )×{t}
u∫
M( ρ2 )
ln+
s
M
(ρ
2
)dsψlr dx+
n∑
i=1
∫∫
E( ρ2 )
umi−2|uxi |2ψlrdxdt
+
∫∫
E( ρ2 )
uq ln
u
M
(ρ
2
)ψlrdxdt ≤ γ
∫∫
E( ρ2 )
ln
u
M
(ρ
2
) ∣∣∣∣∂ψr∂t
∣∣∣∣ q
q−1
dxdt
+γ
∫∫
E( ρ2 )
ln
2q−mi
q−mi
u
M
(ρ
2
) ∣∣∣∣∂ψr∂xi
∣∣∣∣ 2q
q−mi
dxdt = γ (J1 + J2) . (2.5)
By (2.3) we have
J1 + J2 ≤ γ
∫∫
Dλ(R(r))\Dλ(r)
ln
− 1
q−1
1
ρλ
ρ
− 1
κ(λ)
q
q−1
λ dxdt
+ γ
n∑
i=1
∫∫
Dλ(R(r))\Dλ(r)
ln
− mi
q−mi
1
ρλ
ρ
− 2q
κi(λ)(q−mi)
λ dxdt
M. A. Shan 355
≤γ
R(r)∫
r
ln
− 1
q−1
1
z
zλ−1dz + γ
R(r)∫
r
ln
− m1
q−m1
1
z
zλ−1dz ≤γ (F1(r, λ) + F2(r, λ)) .
(2.6)
Combining (2.5), (2.6) we obtain (2.4), which completes the proof of the
lemma.
Define a function u(ρ)(x, t) and a set E
(ρ
2 , 2ρ
)
as follows
u(ρ)(x, t) = min
(
M
(ρ
2
)
−M(2ρ), u2ρ(x, t)
)
,
E
(ρ
2
, 2ρ
)
= {x ∈ E(2ρ) : u < M
(ρ
2
)
}.
Lemma 2.2. Under the assumptions of Lemma 2.1 next inequality 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)
}
, (2.7)
where
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
, λ = 0.
Proof. Testing (1.5) by φ = u(ρ)ψlr, using (1.3) and the Young inequality
we get∫∫
E(2ρ)
u(ρ)uqψlrdxdt ≤ γ
∫∫
E(2ρ)
u(ρ)
∣∣∣∣∂ψr∂t
∣∣∣∣ q
q−1
dxdt
+γ
n∑
i=1
∫∫
E(2ρ)
n∑
j=1
umj−1|uxj |2
1
2
u
mi−1
2 u(ρ)
∣∣∣∣∂ψr∂xi
∣∣∣∣ψl−1
r dxdt
= γ (J3 + J4) . (2.8)
By the Hölder inequality, (2.3) and Lemma 2.1 the integrals in the
right-hand side of (2.8) are estimated as follows
J3 ≤ γ
(
M
(ρ
2
)
−M(2ρ)
)∫∫
E(2ρ)
∣∣∣∣∂ψr∂t
∣∣∣∣ q
q−1
dxdt
356 Removability of isolated singularity for solutions...
≤ γ
(
M
(ρ
2
)
−M(2ρ)
) ∫
Dλ(R(λ))\Dλ(r)
ln
− q
q−1
1
ρλ
ρ
− q
(q−1)κ(λ)
λ dxdt
≤γ
(
M
(ρ
2
)
−M(2ρ)
)R(λ)∫
r
ln
− q
q−1
1
z
zλ−1dz≤γ
(
M
(ρ
2
)
−M(2ρ)
)
F3(r, λ). (2.9)
Similarly
J4 ≤ γ
(
M
(ρ
2
)
−M(2ρ)
) n∑
i=1
n∑
j=1
∫∫
E(2ρ)
umj−2|uxj |2ψlrdxdt
1
2
×
∫∫
E(2ρ)
umi
∣∣∣∣∂ψr∂xi
∣∣∣∣2 ψlrdxdt
1
2
≤ γ
(
M
(ρ
2
)
−M(2ρ)
)
×
× (F1(r, λ) + F2(r, λ))
1
2
n∑
i=1
∫∫
Dλ(R(λ))\Dλ(r)
ln−2 1
ρλ
ρ
−mi(n−λ)− 2
κi(λ)
λ dxdt
1
2
≤ γ
(
M
(ρ
2
)
−M(2ρ)
)
(F1(r, λ) + F2(r, λ))
1
2
R(r)∫
r
ln−2 1
z
zλ−1dz
1
2
≤ γ
(
M
(ρ
2
)
−M(2ρ)
)
(F1(r, λ) + F2(r, λ))
1
2 F
1
2
4 (r, λ). (2.10)
Combining (2.8)–(2.10) we arrive at the required (2.7), this proves
the lemma.
2.1. Pointwise estimates of solutions
Similarly to [13], using the De Giorgi type iteration, we prove the
following estimate
(M(ρ) −M(2ρ)1+m+mn+2
2
≤ γ
(
M
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi
(ρ
2
)
ρ
− 2
κi(λ)
)n+2
2 ∫∫
Dλ(R0)\Dλ( ρ2 )
u1+m2ρ dxdt.
M. A. Shan 357
Since u2ρ ≤ M
(ρ
2
)
−M(2ρ) for (x, t) ∈ Dλ(R0) \Dλ
(ρ
2
)
by the Hölder
inequality and Lemma 2.2 we get
(M(ρ) −M(2ρ)1+m+mn+2
2
≤ γM
m+1
q+1
(ρ
2
)(
M
(ρ
2
)
ρ
− 1
κ(λ) +
n∑
i=1
Mmi
(ρ
2
)
ρ
− 2
κi(λ)
)n+2
2
×
{
F3(r, λ) + (F1(r, λ) + F2(r, λ))
1
2F
1
2
4 (r, λ)
}
|Dλ(R0)|
q−m
q+1 . (2.11)
In the inequality (2.11) we will pass to the limit as r → 0. By (2.1)
the following relations are valid for λ = 0
F1(r, 0)F4(r, 0) = ln
q−2
q−1
1
r
ln−1 1
R(r)
= ln
q−2
q−1
−ε 1
r
, if q > 2,
F2(r, 0)F4(r, 0) = ln
q−2m1
q−m1
1
r
ln−1 1
R(r)
= ln
q−2m1
q−m1
−ε 1
r
, if q > 2m1,
choose ε from the condition max
(
1
2 ,
q−2
q−1 ,
q−2m1
q−m1
)
< ε < 1, now passing
to the limit as r → 0 in (2.11) we obtain for any ρ ≤ R0
2
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. 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 = uξ2ψr, 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
u2dx+
n∑
i=1
T∫
0
∫
K
umi−1|uxi |2dxdt+
T∫
0
∫
K
uq+1dxdt ≤ γ. (3.1)
Testing (1.5) by φψr, where φ is an arbitrary function which belongs
to
o
V 2,m(ΩT ), using (3.1), the boundedness of solution, and passing to
the limit r → 0, we obtain the integral identity (1.5) with an arbitrary
φ ∈
o
V 2,m(Ωt) and ψ ≡ 1. Thus Theorem 1.1 is proved.
358 Removability of isolated singularity for solutions...
4. Appendix
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, δ(θ, τ) := sup
Qθ,τ (x(0),t(0))
δ(u),
Φ(θ, τ) := sup
Qθ,τ (x(0),t(0))
Φ(u),Φ(u) =
u∫
0
φ(s)ds, φ(s) = smn−1f(s).
We say that nondecreasing continuous function ψ satisfies the condi-
tion (A) if for any ε ∈ (0, 1) there exists u0(ε) ≥ 1 such that
ψ(εu) ≤ εµψ(u), (A)
with some µ > 0 and for all u ≥ u0(ε).
Theorem 4.1 ([9]). Let the conditions (1.3), (1.4) be fulfilled and u be
a nonnegative 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), τ ∈
(0,min(θpnn , t(0), T−t(0))), θi ∈ (0, θn) for i ∈ I
′
= {i = 1, n : mi(pi−1) <
mn(pn − 1)} and θi = θn for i ∈ I
′′
= {i = 1, n : mi(pi − 1) =
mn(pn − 1)}, then there exist positive numbers c8, c9 depending only on
n, ν1, ν2,m1, . . . ,mn, p1, . . . , pn such that either
u(x(0), t(0)) ≤ (τ−1ρpn)
1
mn(pn−1)−1 +
∑
i∈I′
(θ−1
i θ
pn
pi
n )
pi
mn(pn−1)−mi(pi−1) , (4.1)
or
Φ(σθ, στ) ≤ c8(1 − σ)−c9θ−pnn δ(θ, τ)Mmnpn−1(θ, τ). (4.2)
On the other hand, if I
′
is empty, i.e. m1(p1 − 1) = m2(p2 − 1) = · · · =
mn(pn − 1), then either
u(x(0), t(0)) ≤ (τ−1θpnn )
1
mn(pn−1)−1 , (4.3)
or (4.2) holds true.
Theorem 4.2 ([9]). Let the conditions (1.3), (1.4) be fulfilled, u be a
nonnegative weak solution to (1.1), f ∈ C1(R1
+) and f
′
(u) ≥ 0. Let ∂ΩT
be the parabolic boundary of ΩT , assume also that lim
(x,t)→∂ΩT
u(x, t) = +∞
and with some 0 ≤ a ≤ 1 and c > 0 there holds
δ(u) ≤ cua.
M. A. Shan 359
Let ψ(u) = u−1Φ
1
mnpn+a−1 (u) satisfies condition (A). Let (x(0), t(0)) ∈ ΩT
and 8ρ = dist(x(0), ∂Ω). Fix τ ∈ (0,min(ρpn , t(0), T−t(0))) and θi ∈ (0, ρ)
for i ∈ I
′
, then there exists a positive number c10 depending only on
n, ν1, ν2,m1, ...,mn, p1, ..., pn and c, such that either (4.1) holds, or
Φ(u(x(0), t(0))) ≤ c10θ
−pn
n umnpn+a−1(x(0), t(0)). (4.4)
On the other hand if I
′
is empty, i.e. m1(p1 − 1) = m2(p2 − 1) = ... =
mn(pn − 1) and ψ(u) = u−1Φ
1
mnpn+a−1 (u) satisfies condition (A), then
either (4.3) holds, or (4.4) holds true.
Acknowledgements
This work is supported by grant of Ministry of Education and Science
of Ukraine (project number is 0115 U 000 136) and it is based on the re-
search provided by the grant support of the State Fund For Fundamental
Research (project number is 0116U007160).
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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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