Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials
For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.
Gespeichert in:
| Datum: | 2019 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2019
|
| Schriftenreihe: | Український математичний вісник |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/169430 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials / K.O. Buryachenko // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 28-45. — Бібліогр.: 23 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169430 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1694302020-06-14T01:26:21Z Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials Buryachenko, K.O. For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation. 2019 Article Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials / K.O. Buryachenko // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 28-45. — Бібліогр.: 23 назв. — англ. 1810-3200 2010 MSC. 35B40, 35B45, 35J62, 35K59 http://dspace.nbuv.gov.ua/handle/123456789/169430 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 |
For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation. |
| format |
Article |
| author |
Buryachenko, K.O. |
| spellingShingle |
Buryachenko, K.O. Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials Український математичний вісник |
| author_facet |
Buryachenko, K.O. |
| author_sort |
Buryachenko, K.O. |
| title |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| title_short |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| title_full |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| title_fullStr |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| title_full_unstemmed |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| title_sort |
local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169430 |
| citation_txt |
Local sub-estimates of solutions to double phase parabolic equations via nonlinear parabolic potentials / K.O. Buryachenko // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 28-45. — Бібліогр.: 23 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT buryachenkoko localsubestimatesofsolutionstodoublephaseparabolicequationsvianonlinearparabolicpotentials |
| first_indexed |
2025-07-15T04:15:01Z |
| last_indexed |
2025-07-15T04:15:01Z |
| _version_ |
1837684913967464448 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 1, 28 – 45
Local sub-estimates of solutions to double
phase parabolic equations via nonlinear
parabolic potentials
Kateryna O. Buryachenko
(Presented by I. I. Skrypnik)
Dedicated to the memory of Professor Bogdan Bojarski
Abstract. For parabolic equations with nonstandard growth condi-
tions we prove local boundedness of weak solutions in terms of nonlinear
parabolic potentials of right-hand side of the equation.
2010 MSC. 35B40, 35B45, 35J62, 35K59.
Key words and phrases. Double phase parabolic equations, weak
solutions, parabolic potentials, local boundedness, local sub-estimetes.
1. Introduction
In this paper we consider a class of parabolic equations with nonstan-
dard growth condition and singular lower order term. Let Ω be a domain
in Rn, T > 0, set ΩT = Ω× (0, T ). We study solution to the equation
ut − divA(x, t, u,∇u) = f(x, t), (x, t) ∈ ΩT . (1.1)
Throughout the paper we suppose that the functions A(·, ·, u, ξ) are Le-
besgue measurable for all u ∈ R1, ξ ∈ Rn, A(x, t, ·, ·) are continuous
for almost all (x, t) ∈ ΩT . We also assume that the following structure
conditions are satisfied
A(x, t, u, ξ)ξ ≥ c1(|ξ|p + a(x, t)|ξ|q),
|A(x, t, u, ξ)| ≤ c2(|ξ|p−1 + a(x, t)|ξ|q−1), (1.2)
Received 28.03.2019
This work is supported by grants of Ministry of Education and Science of Ukraine,
project numbers are 0118U003138, 0119U100421.
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
K. O. Buryachenko 29
where c1, c2 are positive constants, a(x, t) ≥ 0, a(x, t) ∈ Cα,
α
2 (ΩT ) with
some positive α ∈ (0, 1], f ∈ L1(ΩT ), and
2n
n+ 1
< p ≤ q < p+ α. (1.3)
The main goal of this paper is to establish local boundedness of solu-
tions to equation (1.1) in terms of parabolic potential of the right-hand
side. This fact is basically characterized by the different types of degener-
ate behavior according to the size of a coefficient a(x, t) that determines
the “phase”. Indeed, on the set a(x, t) = 0 equation (1.1) has growth of or-
der p with respect to the gradient (this is the “p-phase”), and at the same
time this growth is of order q when a(x, t) > 0 (this is the “(p, q)-phase”).
Before formulating the main results, let us say a few words concerning
the history of the problem. In the standard case p = q, the class of equa-
tions (1.1) has numerous application for several decades (see e.g. [5–7]
and references therein). Starting from the seminal papers by P. Mar-
cellini [18, 19], V. V. Zhikov [23] and G. Lieberman [14] during the last
decade there has been growing interest and substantial development in
the quasilinear elliptic and parabolic equations. The interest grows not
only from the calculus of variations but also from a number of recent ap-
plications in modeling electrorheological fluids, image processing, theory
of elasticity (see e.g. [20]). The basic prototypes of elliptic equations with
nonstandard growth conditions are
−div
(
g(|∇u|) ∇u
|∇u|
)
= f,
(
t
τ
)p−1
≤ g(t)
g(τ)
≤
(
t
τ
)q−1
, t ≥ τ ≥ 0,
(1.4)
−div(|∇u|p−2∇u+ a(x)|∇u|q−2∇u) = f, a(x) ≥ 0. (1.5)
The qualitative theory of parabolic equations with nonstandard growth
conditions has not been developed yet to the same extend. Local bound-
edness of the gradient of solutions to quasilinear parabolic equations of
the type
ut − div
(
g(| ∇u |) ∇u
| ∇u |
)
= f,
(s
τ
)p−1
≤ g(s)
g(τ)
≤
( s
τ
)q−1
, s ≥ τ > 0,
(1.6)
ut − div(|∇u|p−2∇u+ a(x, t)|∇u|q−2∇u) = f, a(x, t) ≥ 0 (1.7)
30 Local sub-estimates of solutions to double phase...
were obtained in [1, 22], Hölder continuity of solutions to equation (1.6)
was proved in [8–10].
To describe our results let us remind the reader the definition of a weak
solution to equation (1.1). For ξ ∈ Rn set ga(|ξ|) := |ξ|p−1 + a(x, t)|ξ|q−1
and Ga(|ξ|) = |ξ|ga(|ξ|). We will write W 1,Ga(ΩT ) for a class of functions
which are weakly differentiable with
∫∫
ΩT
Ga(|∇u|)dxdt <∞. We say that
u is a weak solution to (1.1) if u ∈ V (ΩT ) := C(0, T ;L2(Ω)) ∩W 1,Ga(ΩT )
and for any interval (t1, t2) ⊂ (0, T ) the integral identity
∫
Ω
uϕdx |t2t1 +
t2∫
t1
∫
Ω
(−uϕt+A(x, t, u,∇u)∇ϕ)dxdt =
t2∫
t1
∫
Ω
ϕfdxdt (1.8)
holds true for any testing function ϕ ∈
0
W
1,Ga
(ΩT ) with ϕ,ϕt ∈ L∞(ΩT ).
Note that the assumptions that the testing function ϕ and its deriva-
tive ϕt must be bounded guarantee the time derivative and the right-hand
side of (1.8) are well defined. To formulate our first main result, we define
the local parabolic potential.
Let (x0, t0) ∈ ΩT for ρ, θ > 0 and let Qρ,θ(x0, t0) := Q−
ρ,θ(x0, t0) ∪
Q+
ρ,θ(x0, t0), Q
−
ρ,θ(x0, t0) := Bρ(x0)× (t0 − θ, t0), Q
+
ρ,θ(x0, t0) := Bρ(x0)×
(t0 + θ, t0). For m > 2n
n−1 , ρ > 0 define
Dm(ρ;x0, t0) := inf
τ>0
1
τm−2
+ ρ−n
∫∫
Q
ρ,ρmτm−2 (x0,t0)
|f |dxdt
. (1.9)
Note that the above infimum is attained at some τ ∈ (0,+∞] since the
function under the infimum is continuous for τ . Moreover D2(ρ;x0, t0)
=
∫∫
Q
ρ,ρ2(x0,t0)
|f |dxdt.
Now for j = 0, 1, 2, ... set ρj := 2−jρ. Following [16] we define the
parabolic potential
P fm(ρ;x0, t0) :=
∞∑
j=0
Dm(ρj;x0, t0). (1.10)
Particularly, there exists γ > 1 such that
1
γ
P f2 (ρ;x0, t0) ≤
ρ∫
0
r−n
∫∫
Qρ,ρ2(x0,t0)
|f |dxdtdr
r
≤ γP f2 (ρ;x0, t0).
K. O. Buryachenko 31
So that for m = 2 the introduced potential is equivalent to the truncated
Riesz potential used in [2, 4, 12]. Note also that for m > 2 and for a
time-independent f the minimum in the the definition of Dm(ρ;x0, t0) is
attained at
τ = (m− 2)−
1
m−1
ρm−n
∫
Bρ(x0)
|f |dx
1
m−1
,
so
Dm(ρ;x0, t0) = (m− 1)(m− 2)
1
m−1
ρm−n
∫
Bρ(x0)
|f |dx
1
m−1
and P fm(ρ;x0, t0) = W f
1,m(ρ;x0), where W f
1,m(ρ;x0) is Wolff potential
defined by the formula
W f
1,m(ρ;x0) =
∞∑
j=0
ρm−n
j
∫
Bρj
(x0)
f dx
1
m−1
, ρj =
ρ
2j
, j = 0, 1, ..
Remark 1.1. We can estimate P fm by the Lebesgue norm as follows.
Let f ∈ Lr(0, T ;Ls(Ω)) for 1
r +
n
ms < 1. Then
ρ−n
∫
Qρ,ρmτm−2 (x0,t0)
|f |dx ≤ γτ (m−2)(1− 1
r
)ρm(1− 1
r
− n
ms
)||f ||s,r
and
Dm(ρ;x0, t0) ≤ γ(ρm(1− 1
r
− n
ms
)||f ||s,r)
1
1+(m−2)(1− 1
r ) .
Hence if 1
r +
n
ms < 1, then
P fm(ρ;x0, t0) ≤ γ(ρm(1− 1
r
− n
ms
)||f ||s,r)
1
1+(m−2)(1− 1
r )
and limρ→0 sup
(x0,t0)∈ΩT
P fm(ρ;x0, t0) = 0.
The main result of the paper is the local boundedness of the solutions.
As it has already mentioned before the behavior of the solution in a neigh-
borhood of a point (x0, t0) depends on the value of the function a(x0, t0).
In what follows we will distinguish two cases: sup
Qρ,ρ2(x0,t0)
a(x, t) ≥ 2[a]αρ
α
32 Local sub-estimates of solutions to double phase...
(so called (p, q)-phase) and sup
Qρ,ρ2(x0,t0)
a(x, t) ≤ 2[a]αρ
α(so called p-phase),
here [a]α := sup
(x,t),(y,τ)∈ΩT
(x,t) 6=(y,τ)
|a(x,t)−a(y,τ)|
(|x−y|+|t−τ |)α .
Theorem 1.1. (Local boundedness of solution in the (p, q)-phase). Let
u be a solution of equation (1.1) and assumptions (1.2), (1.3) be fulfilled,
q 6= 2. Fix a point (x0, t0) ∈ ΩT such that a0 := a(x0, t0) > 0. Let
R := ( a0
2[a]α
)
1
α and Qρ,θ(x0, t0) ⊂ QR,R2(x0, t0) ⊂ Q8R,(8R)2(x0, t0) ⊂ ΩT .
Then for any 0 < λ < p
nq the following estimate
|u(x0, t0)| ≤ γ
(
ρq
a0θ
) 1
q−2
+γ
(
a0
ρn+q
∫∫
Qρ,θ(x0,t0)
|u|q−1+λ(q−1)dxdt
) 1
1+λ(q−1)
+γ
(
1
ρn+p
∫∫
Qρ,θ(x0,t0)
|u|p−1+λ(q−1)dxdt
) 1
1+λ(q−1)
+γ(1 + a
− 1
q−2
0 )P fq (2ρ;x0; t0) (1.11)
holds true with a constant γ > 0 depending only on n, p, q, c1, c2, [a]α
and λ.
Theorem 1.2. (Local boundedness of solution in the p-phase). Let u
be a solution of equation (1.1) and assumptions (1.2), (1.3) be fulfilled,
and assume also that q < pn+1
n . Fix a point (x0, t0) ∈ ΩT such that
a0 = a(x0, t0) = 0. Then for any 0 < λ < p−n(q−p)
nq the following estimate
|u(x0, t0)| ≤ γ
(
ρp
θ
) 1
p−2
+ γ
1
ρn+p
∫∫
Qρ,θ(x0,t0)
|u|p−1+λ(q−1)dxdt
1
1+λ(q−1)
+γ
1
ρn+p
∫∫
Qρ,θ(x0,t0)
|u|(q−1)(1+λ)dxdt
p
p−n(q−p)+λp(q−1)
+ γP fp (2ρ;x0, t0)
(1.12)
hold true with a constant γ depending only on n, p, q, c1, c2, [a]α and λ.
The proof of Theorems 1.1, 1.2 is based on the adaption of the
Kilpeläinen–Malý technique [11] to the parabolic equations using ideas
from [16].
K. O. Buryachenko 33
2. Local boundedness of solutions. Proof of
Theorems 1.1, 1.2
2.1. Integral estimates of the solutions
For 0 < λ < min(1,m − 1),m > 1, set Wm(s) :=
∫ s
0 (1 + z)−
1+λ
m dz =
m
m−1−λ((1 + s)
m−1−λ
m
−1) for any ε ∈ (0, 1) evidently we have
Wm(s) ≤
m
m− 1− λ
s
m−1−λ
m , s ≤ ε+ γ(ε)W
m
m−1−λ (s) (2.1)
with a constant γ(ε) depending only on ε,m, λ. In what follows we shall
also need the following simple inequality.
s ≤ ε+ γ(ε)
∫ s
0
(1− (1 + z)−λ)dz, ε, λ ∈ (0, 1) (2.2)
with a constant γ(ε) depending only on ε, λ.
The next two lemmas are Cacciopolli type estimates adapted to the
Kilpeläinen–Maly technique.
Lemma 2.1. (p, q-phase). Let the conditions of Theorem 1.1 be ful-
filled. Then there exists γ > 0 depending only on the data such that
for any λ ∈ (0, 1), k > q, l, δ > 0, any cylinder Q
(δ)
r := Qr, rq
a0
δ2−q ⊂
Qρ,θ(x0, t0) ⊂ QR,R2(x0, t0) and any ζ ∈ C∞
0 (Q
(δ)
r ), such that 0 ≤ ζ ≤
1, |∇ζ| ≤ γr−1, |ζt| ≤ γa0r
−qδq−2 one has
sup
0<t<T
δ−1
∫
L(t)
∫ u
l
(
1−
(
1 +
z − l
δ
)−λ)
dzζkdx
+ δp−2
∫∫
L
∣∣∣∣∇Wp
(
u− l
δ
)∣∣∣∣
p
ζkdxdt
+ δq−2a0
∫∫
L
∣∣∣∣∇Wq
(
u− l
δ
)∣∣∣∣
q
ζkdxdt
≤ γa0
δq−2
rq
∫∫
L
(
1 +
u− l
δ
)q−1+λ(q−1)
ζk−qdxdt
+ γ
δp−2
rp
∫∫
L
(
1 +
u− l
δ
)p−1+λ(q−1)
ζk−qdxdt
+ γδ−1
∫∫
Q
(δ)
r
|f |dxdt, (2.3)
where L := Q
(δ)
r ∩ {u > l}, L(t) := L ∩ {τ = t}.
34 Local sub-estimates of solutions to double phase...
Proof. First note that by our choice of R we have a0
2 = a0 − [a]αR
α ≤
a(x, t) ≤ a0+[a]αR
α = 3
2a0 for any (x, t) ∈ Q
(δ)
r ⊂ QR,R2(x0, t0). Testing
identify (1.8) by ϕ = (1 − (1 + (u−lδ )+)
−λ)ζk, using conditions (1.2) we
obtain
sup
0<t<T
∫
L(t)
u∫
l
(
1−
(
1 +
z − l
δ
)−λ
)
dzζkdx
+δ−1
∫∫
L
(
1 +
u− l
δ
)−1−λ
|∇u|pζkdxdt
δ−1a0
∫∫
L
(
1 +
u− l
δ
)−1−λ
|∇u|qζkdxdt ≤ γa0
δq−1
rq
∫∫
L
u− l
δ
ζk−1dxdt+ γr−1
∫∫
L
|∇u|p−1ζk−1dxdt
+γa0r
−1
∫∫
L
|∇u|q−1ζk−1dxdt+ γ
∫∫
Q
(δ)
r
|f |dxdt.
From this using the Young inequality and by our choice of Wp(
u−l
δ ),
Wq(
u−l
δ ) we arrive at the required (2.3).
Lemma 2.2. (p-phase). Let the conditions of Theorem 1.2 be fulfilled.
Then there exists γ > 0 depending only on the data such that for any λ ∈
(0, 1), k ≥ q, l > 0, δ ≥ rσ1 , any cylinder Q
(δ)
r := Q
rδ
p−2
p ,rpδ2−p
(x0, t0) ⊂
Qρ,θ(x0, t0) and any ζ ∈ C∞
0 (Q
(δ)
r ), such that 0 ≤ ζ ≤ 1, |∇ζ| ≤
γr−1, |ζt| ≤ γr−pδp−2 one has
sup
0<t<T
∫
L(t)
∫ u
l
(
1−
(
1 +
z − l
δ
)−λ
)
dzζkdx
+δp−2
∫∫
L
∣∣∣∣∇Wp
(
u− l
δ
)∣∣∣∣
p
ζkdxdt
≤ γδp−2r−p
∫∫
L
(
1 +
u− l
δ
)p−1+λ(q−1)
ζk−qdxdt
+γδq−2r−p
∫∫
L
(
1 +
u− l
δ
)q−1+λ(q−1)
ζk−qdxdt+ γδ−1
∫∫
Q
(δ)
r
|f |dxdt.
(2.4)
K. O. Buryachenko 35
Proof. Note that by our choice of δ we have an inclusionQ
(δ)
r ⊂ Qr,r2(x0, t0).
Therefore for any (x, t) ∈ Q
(δ)
r we have a(x, t) ≤ [a]αr
α ≤ [a]αr
q−p (we
have p, q > 2).
Testing (1.8) by ϕ = (1− (1+(u−lδ )+)
−λ)ζk, using condition (1.2) we
obtain
sup
0<t<T
∫
L(t)
∫ u
l
(
1−
(
1 +
z − l
δ
)−λ
)
dzζkdx
+δ−1
∫∫
L
a(x, t)
(
u− l
δ
)−1−λ
|∇u|qζkdxdt ≤ γ
δp−1
rp
∫∫
L
u− l
δ
ζk−1dxdt
+γr−1
∫∫
L
|∇u|p−1ζk−1dxdt+ γr−1
∫∫
L
a(x, t)|∇u|q−1ζk−1dxdt
+γ
∫∫
Q
(δ)
r
|f |dxdt.
Using the Young inequality we arrive at the required (2.4).
2.2. Proof of Theorem 1.1
Fix a number æ ∈ (0, 1) depending only on the data and λ, which
will be specified later. For j = 0, 1, 2, ... positive numbers lj and δj are
defined inductively as follows.
δ−1 :=
(
ρq
a0θ
) 1
q−2
+
a0
æρn+q
∫∫
Qρ,θ(x0,t0)
uq−1+λ(q−1)dxdt
1
1+λ(q−1)
+
1
æρn+p
∫∫
Qρ,θ(x0,t0)
up−1+λ(q−1)dxdt
1
1+λ(q−1)
(2.5)
and l0 = 0. For j = 0, 1, 2, ..., given δj−1 and lj we define δj and lj+1 as
follows. We denote rj := ρ2−j and τj := sup{τ : 1
τ+r
−n
j
∫∫
Q
rj,r
q
j
τq−2 (x0,t0)
|f |dxdt =
Dq(rj ;x0, t0)}, where Dq(rj;x0, t0) is as in (1.9). For δ ≥ 1
2δj−1 we define
Bj := Brj(x0), Q
(δ)
j := Q
rj ,
r
p
j
a0
δ2−q
(x0, t0). Let ζj ∈ C∞
0 (Q
(δ)
j ) be such that
36 Local sub-estimates of solutions to double phase...
0 ≤ ζj ≤ 1, ζj = 1 in 1
4Q
(δ)
j and |∇ζj| ≤ γr−1
j , |∂ζj∂t | ≤ γa0r
−q
j δq−2. Set
Aj(δ) := a0
δq−2
rn+qj
∫∫
L
(δ)
j
(
u− lj
δ
)q−1+λ(q−1)
ζqj dxdt
+
δp−2
rn+pj
∫∫
L
(δ)
j
(
u− lj
δ
)p−1+λ(q−1)
ζqj dxdt, (2.6)
here L
(δ)
j := Q
(δ)
j ∩ {u > lj}.
If Aj(
1
2δj−1) ≤ æ, we set δj =
1
2δj−1 and δj = lj+1− lj. Since Aj(δ) is
continuous and decreasing as a function of δ, then if Aj(
1
2δj−1) > æ there
exists δ̂ > 1
2δj−1 such that Aj(δ̂) = æ. In this case we set δj = δ̂ and
lj+1 = lj+δj. Further we set Qj = Q
(δj)
j , Lj = L
(δj)
j . By our choice of δ−1
and δj , j = 0, 1, 2, ... we have an inclusion Qj ⊂ Qj−1 ⊂ Q0 ⊂ Qρ,θ(x0, t0)
for j = 1, 2, ... and in particular ζj−1 ≡ 1 on Qj , j = 1, 2, ..., and moreover
Aj(δj) ≤ æ, j = 1, 2, ... (2.7)
Claim. Set B = 2n+q, then for any j = 0, 1, 2, ...
δj ≤ Bδj−1. (2.8)
We establish the claim by induction. By our choice of δ−1 we have for
j = 0
A0(Bδ−1) =
a0δ
−1−λ(q−1)
−1
ρn+qB1+λ(q−1)
∫∫
Q0
uq−1+λ(q−1)ζq0dxdt
+
δ
−1−λ(q−1)
−1
ρn+pB1+λ(q−1)
∫∫
Q0
up−1+λ(q−1)ζq0dxdt
≤ B−1−λ(q−1)
a0δ
−1−λ(q−1)
−1
ρn+q
∫∫
Qρ,θ(x0,t0)
uq−1+λ(q−1)dxdt
+
δ
−p−1−λ(q−1)
−1
ρn+p
∫∫
Qρ,θ(x0,t0)
up−1+λ(q−1)dxdt
≤ B−1æ < æ.
K. O. Buryachenko 37
If δ0 = 1
2δ−1 ≤ Bδ−1, and if A0(δ0) = æ > A0(Bδ−1), and since A0δ
is decreasing, then δ0 ≤ Bδ−1, and in both cases we obtain δ0 ≤ Bδ−1.
Assume that (2.8) holds for i = 1, 2, ..., j − 1, then
Aj(Bδj−1) = a0
(
2
rj−1
)n+q δq−2
j−1
B1+λ(q−1)
∫∫
Lj
(
u− lj
δj−1
)q−1+λ(q−1)
ζqj dxdt
+
(
2
rj−1
)n+p δp−2
j−1
B1+λ(q−1)
∫∫
Lj
(
u− lj
δj−1
)p−1+λ(q−1)
ζqj dxdt
≤ 2n+qB−1
a0
δq−2
j−1
rn+qj−1
∫∫
Lj
(
u− lj−1
δj−1
)q−1+λ(q−1)
ζqj−1dxdt
+
δp−2
j−1
rn+pj−1
∫∫
Lj
(
u− lj−1
δj−1
)p−1+λ(q−1)
ζqj dxdt
≤ 2n+qB−1Aj−1(δj−1) ≤ æ2n+qB−1 ≤ æ.
If δj = 1
2δj−1 ≤ Bδj−1, Aj(δj) = æ ≥ Aj−1(Bδj−1), and since Aj(δ) is
decreasing, then δj ≤ Bδj−1, and in both cases we obtain δj ≤ Bδj−1,
which proves the claim.
The following lemma is a key in the Kilpeläinen–Malý technique.
Lemma 2.3. Let the conditions of Theorem 1.1 be fulfilled. Then for
any j ≥ 1 there exists γ > 0 depending only on the data and λ, such that
δj ≤
1
2
δj−1 + γ(1 + a
− 1
q−2
0 )Dq(rj ;x0, t0). (2.9)
Proof. We shall assume later that
δj >
1
2
δj−1, δj > a
− 1
q−2
0
1
τj
, (2.10)
since otherwise (2.9) is evident. The first inequality in (2.10) guarantees
that Aj(δj) = æ. First note the inequality
δq−2
j
rn+qj
|Lj |+
δp−2
j−1
rn+pj−1
|Lj | ≤ γæ, j = 1, 2, ... (2.11)
Indeed, by (2.7) and (2.8) we have
a0
δq−2
j
rn+qj
|Lj |+
δp−2
j−1
rn+pj−1
|Lj |
38 Local sub-estimates of solutions to double phase...
= a0
δq−2
j
rn+qj
∫∫
Lj
(
lj − lj−1
δj−1
)q−1+λ(q−1)
ζqj−1dxdt
+
δp−2
j−1
rn+pj−1
∫∫
Lj
(
lj − lj−1
δj−1
)p−1+λ(q−1)
ζqj−1dxdt
≤ γ(B)
a0
δq−2
j
rn+qj
∫∫
Lj−1
(
u− lj−1
δj−1
)q−1+λ(q−1)
ζqj−1dxdt
+
δp−2
j−1
rn+pj−1
∫∫
Lj−1
(
u− lj−1
δj−1
)p−1+λ(q−1)
ζqj−1
dxdt
≤ γ(B)Aj−1(δj−1) ≤ γ(B)æ.
By (2.1) and (2.11) we have for any ε ∈ (0, 1)
æ = a0
δq−2
j
rn+qj
∫∫
Lj
(
u− lj
δj
)q−1+λ(q−1)
ζqj dxdt+
δp−2
j
rn+pj−1
∫∫
Lj
(
u− lj
δj
)p−1+λ(q−1)
ζqj dxdt ≤ a0γε
q−1+λ(q−1)δq−2
j r−n−qj |Lj |
+γεp−1+λ(q−1)δ
p−2
r
−n−q
j
j |Lj |+ γ(ε)J1 ≤ εγæ + γ(ε)J1, (2.12)
where
J1 = a0
δq−2
j
rn+qj
∫∫
Lj
W q
q
(
u− lj
δj
)(
u− lj
δj
)λq
ζqj dxdt
+
δp−2
j
rn+qj
∫∫
Lj
W p
p
(
u− lj
δj
)(
u− lj
δj
)λq
ζqj dxdt.
Further we shall assume that λ satisfies the condition 0 < λ < p
nq . By
the Sobolev embedding theorem and our choice of λ we obtain
J1 ≤ a0γ
δq−2
j
rn+qj
sup
0<t<T
∫
Lj(t)
u− lj
δj
ζqj dx
q
n ∫∫
Lj
∣∣∣∣∇
(
Wq
(
u− lj
δj
)
ζj
)∣∣∣∣
q
dxdt
+γ
δp−2
j
rn+qj
sup
0<t<T
∫
Lj(t)
u− lj
δj
ζqj dx
p
n ∫∫
Lj
∣∣∣∣∇
(
Wp
(
u− lj
δj
)
ζj
)∣∣∣∣
p
dxdt. (2.13)
K. O. Buryachenko 39
By (2.2) and Lemma 2.1 we obtain for every ε1 ∈ (0, 1)
sup
0<t<T
∫
Lj(t)
u− lj
δj
ζqj dx ≤ ε1|Bj |
+ γ(ε1)δ
−1
j sup
0<t<T
∫
Lj(t)
∫ u
lj
(
1−
(
1 +
z − lj
δj
)−λ
)
dzζqj dx
≤ |Bj |
ε1 + γ(ε1)a0
δq−2
j
rn+qj
∫∫
Lj
(
1 +
u− lj
δj
)q−1+λ(q−1)
dxdt
+ γ(ε1)
δp−2
j
rn+qj
∫∫
Lj
(
1 +
u− lj
δj
)p−1+λ(q−1)
dxdt
+ γ(ε1)δ
−1
j r−nj
∫∫
Qj
|f |dxdt. (2.14)
Further by (2.7), (2.8), (2.10), (2.11) and our choice of ζj we obtain
a0
δq−2
j
rn+qj
∫∫
Lj
(
1 +
u− lj
δj
)q−1+λ(q−1)
dxdt
+
δp−2
j
rn+qj
∫∫
Lj
(
1 +
u− lj
δj
)p−1+λ(q−1)
dxdt ≤ γAj−1(δj−1) ≤ γæ. (2.15)
Therefore, inequalities (2.13)–(2.15) and Lemma 2.1 imply
æ ≤ εγæ+ γ(ε)
æ+ δ−1
j r−nj
∫∫
Qj
|f |dxdt
×
ε1 + γ(ε1)æ + δ−1
j r−nj
∫∫
Qj
|f |dxdt
q
n
+
ε1 + γ(ε1)æ + δ−1
j r−nj
∫∫
Qj
|f |dxdt
p
n
. (2.16)
Now choose ε = 1
16γ , ε1 =
1
16γ(ε) and æ such that γ(ε, ε1)æ
p
n+γ(ε, ε1)æ
q
n =
1
16 . From (2.16) it follows that there exists γ > 0 such that δ−1
j r−nj
∫∫
Qj
|f |dxdt ≥
40 Local sub-estimates of solutions to double phase...
γæ, hence δj ≤ γr−nj
∫∫
Qj
|f |dxdt. By the second inequality in (2.10) we
have an inclusion Qj ⊂ Qrj ,rqj τ
q−2
j
(x0, t0), so
δj ≤ γr−nj
∫∫
Q
rj,r
q
j
τ
q−2
j
(x0,t0)
|f |dxdt ≤ γDq(rj ;x0, t0).
Such a way inequality (2.9) is proved, which completes the proof of
Lemma 2.3.
Summing up inequality (2.9) for j = 1, 2, ..., J − 1 by (2.8) we obtain
lJ ≤ γδ0 + γ(1 + a
− 1
q−2
0 )
∞∑
j=1
Dq(rj ;x0, t0)
≤ γδ−1 + γ(1 + a
− 1
q−2
0 )P fq (2ρ;x0, t0). (2.17)
Hence we can pass to the limit J → ∞ in (2.17). Let l̄ = limj→∞ lj, from
(2.6), (2.7) we conclude that r−n−qj
∫∫
Qj
(u−l̄)q−1+λ(q−1)dxdt ≤ γδ
1+λ(q−1)
j →
0, j → ∞. Choosing (x0, t0) as a Lebesgue point of the function (u −
l̄)q−1+λ(q−1) we conclude that u(x0, t0) ≤ l̄ and hence u(x0, t0) is esti-
mated from above by the righthand side of (2.17). This completes the
proof of Theorem 1.1.
2.3. Proof of Theorem 1.2
The proof of Theorem 1.2 is similar to that of Theorem 1.1. We note
only the differences arising here.
Fix a number æ ∈ (0, 1) depending only on the data and λ, which
will be specified later. For j = 0, 1, 2, ... positive numbers lj and δj are
defined inductively as follows.
δ−1 :=
(
ρp
θ
) 1
p−2
+
1
æρn+p
∫∫
Qρ,θ(x0,t0)
up−1+λ(q−1)dxdt
1
1+λ(q−1)
+
(
1
æρn+p
∫∫
Qρ,θ(x0,t0)
uq−1+λ(q−1)dxdt
) p
p−n(q−p)+λp(q−1)
, (2.18)
K. O. Buryachenko 41
and l0 = 0. We denote rj := ρ2−j and
τj := sup
τ :
1
τ
+ r−nj
∫∫
Q
rj,r
p
j
τp−2(x0,t0)
|f |dxdt
= Dp(rj ;x0, t0), (2.19)
where Dp(rj;x0, t0) is defined by (1.9). For δ ≥ 1
2δj−1 we define Bj :=
Brj(x0), Q
(δ)
j := Qrj ,rpj δ2−p(x0, t0) and let ζj ∈ C∞
0 (Q
(δ)
j ) be such that
0 ≤ ζj ≤ 1, ζj = 1 in 1
4Q
(δ)
j and |∇ζj| ≤ γr−1
j , |∂ζj∂t | ≤ γr−pj δp−2. Set
Aj(δ) :=
δp−2
rn+pj
∫∫
L
(δ)
j
(
u− lj
δ
)p−1+λ(q−1)
ζqj dxdt
+
δq−2
rn+pj
∫∫
L
(δ)
j
(
u− lj
δ
)q−1+λ(q−1)
ζqj dxdt, (2.20)
where L
(δ)
j := Q
(δ)
j ∩ {u > lj}.
If Aj(
1
2δj−1) ≤ æ, we set δj =
1
2δj−1 and δj = lj+1 − lj. Since Aj(δ)
is continuous and decreasing as a function of δ, then Aj(
1
2δj−1) > æ
and there exists δ̂ > 1
2δj−1 such that Aj(δ̂) = æ. In this case we set
δj = δ̂. Further we set Qj = Q
(δj)
j and Lj = L
(δj )
j . By our choice of
δj , j = 0, 1, 2, ... we have an inclusion Qj ⊂ Qj−1 ⊂ Q0 ⊂ Qρ,θ(x0, t0) for
j = 1, 2, ..., in particular, ζj−1 ≡ 0 on Qj, j = 1, 2, ... and
Aj(δj) ≤ æ, j = 1, 2, ... (2.21)
Similarly to (2.8) we prove
δj ≤ Bδj−1, j = 0, 1, 2, ... (2.22)
where B = 2σ3 , σ3 =
(n+p)p
p−n(q−p) .
The next Lemma is a key in the Kilpeläinen–Malý technique in the
p-phase.
Lemma 2.4. Let the conditions of Theorem 1.2 be fulfilled. Then for
any j ≥ 1 there exists γ > 0 depending only on the data and λ such that
δj ≤
1
2
δj−1 + γDp(rj ;x0, t0). (2.23)
42 Local sub-estimates of solutions to double phase...
Proof. We will assume that
δj >
1
2
δj−1, δj >
1
τj
,
since otherwise inequality (2.23) is evident. First, similarly to (2.11) we
obtain
(δp−2
j + δq−2
j )r−n−pj |Lj | ≤ γæ, j = 1, 2, ... (2.24)
By (2.1) and (2.24) we have for any ε ∈ (0, 1)
æ =
δp−2
j
rn+pj
∫∫
Lj
(
u− lj
δj
)p−1+λ(q−1)
ζqj dxdt
+
δq−2
j
rn+pj
∫∫
Lj
(
u− lj
δj
)q−1+λ(q−1)
ζqj dxdt ≤ εæ+ γ(ε)J2, (2.25)
where
J2 =
δp−2
j
rn+pj
∫∫
Lj
W p
p (
u− lj
δj
)(
u− lj
δj
)λqζqj dxdt
+
δq−2
j
rn+pj
∫∫
Lj
W p
p (
u− lj
δj
)(
u− lj
δj
)q−p+λqζqj dxdt.
Assuming that λ satisfies the condition 0 < λ < p−n(q−p)
nq and using
the Sobolev embedding theorem we obtain
J2 ≤ γ
(
δp−2
j + δ
q−2+n
p
(q−p)
j
)
r−n−pj
×
sup
0<t<T
∫
Lj(t)
u− lj
δj
ζqj dx
p
n ∫∫
Lj
∣∣∣∣∇
(
Wp
(
u− lj
δj
)
ζj
)∣∣∣∣
p
dxdt
= γ
(
δp−2
j + δ
q−2+n
p
(q−p)
j
)
r−n−pj J3. (2.26)
K. O. Buryachenko 43
By (2.2) and Lemma 2.2 we obtain for every ε, ε1 ∈ (0, 1)
γ(ε)δ
q−2+n
p
(q−p)
j r−n−pj J3
≤ γ(ε)
ε1 + γ(ε1)æ + δ
− p−n(q−p)
p
j r−nj
∫∫
Qj
|f |dxdt
p
n
×
æ+ δ
− p−n(q−p)
p
j r−nj
∫∫
Qj
|f |dxdt
. (2.27)
Similarly, by (2.2) and Lemma 2.2 we have for any ε, ε1 ∈ (0, 1)
γ(ε)δ
− p−n(q−p)
p
j r−n−pj J3
≤ γ(ε)
ε1 + γ(ε1)æ + δ
− p−n(q−p)
p
j r−nj
∫∫
Qj
|f |dxdt
p
n
×
æ+ δ
− p−n(q−p)
p
j r−nj
∫∫
Qj
|f |dxdt
. (2.28)
Choose ε = 1
16γ , ε1 = 1
16γ(ε) and æ such that γ(ε, ε1)æ
p
n = 1
16 . From
(2.25)–(2.28) it follows
δj ≤ γ
r−nj
∫∫
Qj
|f |dxdt
+ γ
r−nj
∫∫
Qj
|f |dxdt
p
p−n(q−p)
.
Since δj >
1
τj
we have an inclusion Qj ⊂ Qrj ,rpj τ
p−2
j
(x0, t0). From the
previous we obtain
δj ≤ γr−nj
∫∫
Q
rj,r
p
j
τ
p−2
j
(x0,t0)
|f |dxdt ≤ γDj(rj ;x0, t0),
which proves the lemma.
Summing inequalities (2.23) for j = 1, 2, ..., J − 1, using (2.22) and
passing to the limit J → ∞, we arrive at (1.12). Here (x0, t0) is a
Lebesgue point of the function (u− l̄)p−1+λ(q−1), where l̄ = lim
j→∞
lj. This
completes the proof of Theorem 1.2.
44 Local sub-estimates of solutions to double phase...
References
[1] V. Bögelein, F. Duzaar, P. Marcellini, Parabolic equations with p, q− growth //
J. Math.Pures Appl., 100 (2013), 535–563.
[2] V. Bögelein, F. Duzaar, U. Gianazza, Porous medium type equations with measure
data and potential estimates // SIAM J. Math.Anal. (6), 45 (2013), 3283–3330.
[3] K. O. Buryachenko, I. I. Skrypnik, Pointwise estimates of solutions to the double-
phase elliptic equations // Journal of Mathematical Sciences, 222 (2017), 772–
786.
[4] K. O. Buryachenko, I. I. Skrypnik, Riesz potentials and pointwise estimates of so-
lutions to anisotropic porous medium equation // Nonlinear Analysis, 178 (2019),
56–85.
[5] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York,
1993.
[6] E. DiBenedetto, J. M. Urbano, V. Vespri, Current issues on singular and de-
generate evolutoin equations, in: C. Dafermos, E. FEireisi (Eds.), Evolutionary
Equations. Handl. Differ. Equat., Vol. 1, Elsevier, 2004, 169–286.
[7] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s inequality for degenerate and
sinqular parabolic equations, Mongraphs in Mathematics, Springer-Verlag, New
York, 2012.
[8] S. Hwang, Hölder regularity of solutions of generalized p-Laplacian type parabolic
equations, PhD thesis, Iowa State Univ., 2012.
[9] S. Hwang, G. M. Lieberman, Hölder continuity of a bounded weak solution of
generalized parabolic p-Laplacian equations II: singular case // Elect. J. Diff.
Eq., 2015 (2015), No. 288, 1–24.
[10] S. Hwang, G. M. Lieberman, Hölder continuity of a bounded weak solution of
generalized parabolic p-Laplacian equations I: degenerate case // Elect. J. Diff.
Eq., 2015 (2015), No. 287, 1–32.
[11] T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear
elliptic equations // Acta Math., 172 (1994), No. 1, 137–161.
[12] T. Kuusi, G. Mingione, Riesz potentials and non-linear parabolic equations //
ARMA. (2), 212 (2014), 727–780.
[13] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and Quasilinear
Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, AMS, Providence,
1967.
[14] G. Lieberman, The natural generalizations of the natural conditions of Ladyzhen-
skaya and Ural’tseva for elliptic equations // Comm. in PDE, 16 (1991), No. 2–3,
311–361.
K. O. Buryachenko 45
[15] V. Liskevich, I. I. Skrypnik, Harnack inequality and continuity of solutions
to quasilinear degenerate parabolic equations with coefficients from Kato-type
classes // Journal of Differential Equations, 247 (2009), 2740–2777.
[16] V. Liskevich, I. I. Skrypnik, Z. Sobol, Estimates of solutions for the parabolic p-
Laplacian equation with measure via parabolic nonlinear potentials, Comm. Pure
Appl. Anal., 12 (2013), No. 4, 1731–1744.
[17] V. Liskevich, I. I. Skrypnik, Poitwise estimates for solutions to the porous medium
equation with measure as a forcing term // Israel J. Math., 194 (2013), 259–275.
[18] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations
with non standard growth conditions, Arch. Rat. Mech. Analys., 105 (1989), No. 3,
267–284.
[19] P. Marcellini, Regularity and existence of solutions of elliptic equations with (p; q)-
growth conditions // J. Diff. Equa., 90 (1991), No. 1, 1–30.
[20] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory,
Springer, Berlin, 2000.
[21] T. Singer, Parabolic equationswith p, q− growth: the subquadratic case // The
Quarterly J. of Math., 66 (2015), No. 2, 707–742.
[22] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity
theory // Izv. Akad. Nauk SSSR, Ser. Mat., 50 (1986), 675–710.
[23] V. V. Zhikov, On Lavreatiev’s phenomenon // Russ. J. of Math. Physics, 3 (1995),
264–269.
Contact information
Kateryna
Olexandrivna
Buryachenko
Vasyl’ Stus Donetsk National University,
Vinnytsia, Ukraine
E-Mail: katarzyna_@ukr.net
|