HNUE JOURNAL OF SCIENCE
DOI: 10.18173/23541059.20190025
Natural Science, 2019, Volume 64, Issue 6, pp. 311
This paper is available online at http://stdb.hnue.edu.vn
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS
OF QUASILINEAR DEGENERATE PARABOLIC EQUATIONS
Tran Thi Quynh Chi and Le Thi Thuy
Faculty of Mathematics, Electric Power University
Abstract. In this paper we prove the existence and uniqueness of weak solutions
to a class of quasilinear degenerate parabolic equations involving weighted
pLaplacian operators by combining compactness and monotonicity methods.
Keywords: Quasilinear degenerate parabolic equation, weighted pLaplacian
operator, weak solution, compactness method, monotonicity method.
1.
Introduction
In this paper we consider the following parabolic problem:
p−2
ut − div(a(x)∇u ∇u) + f (u) = g(x), x ∈ Ω, t > 0,
u(x, t) = 0,
x ∈ ∂Ω, t > 0,
u(x, 0) = u0 (x),
x ∈ Ω,
(1.1)
where Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂Ω, 2 ≤ p ≤ N,
u0 ∈ L2 (Ω) given, the coefficient a(·), the nonlinearity f and the external force g satisfy
the following conditions:
(H1) The function a : Ω → R satisfies the following assumptions: a ∈ L1loc (Ω) and
a(x) = 0 for x ∈ Σ, and a(x) > 0 for x ∈ Ω \ Σ, where Σ is a closed subset of Ω
with meas(Σ) = 0. Furthermore, we assume that
1
N
Ω
[a(x)] α
dx < ∞ for some α ∈ (0, p);
(1.2)
(H2) f : R → R is a C 1 function satisfying
C1 uq − C0 ≤ f (u)u ≤ C2 uq + C0 ,
f ′ (u) ≥ −ℓ,
for some q ≥ 2,
(1.3)
(1.4)
where C0 , C1, C2 , ℓ are positive constants;
Received March 11, 2019. Revised June 5, 2019. Accepted June 12, 2019.
Contact Tran Thi Quynh Chi, email address: chittq@epu.edu.vn
3
Tran Thi Quynh Chi and Le Thi Thuy
q
pN
.
,
q − 1 (N + 1)p − N + α
(H3) g ∈ Ls (Ω), where s ≥ min
The degeneracy of problem (1.1) is considered in the sense that the measurable,
nonnegative diffusion coefficient a(x) is allowed to vanish somewhere. The physical
motivation of the assumption (H1) is related to the modeling of reaction diffusion
processes in composite materials, occupying a bounded domain Ω, in which at some
points they behave as perfect insulator. Following [1, p. 79], when at some points the
medium is perfectly insulating, it is natural to assume that a(x) vanishes at these points.
As mentioned in [2], the assumption (H1) implies that the degenerate set may consist of
an infinite many number of points, which is different from the weight of CaldiroliMusina
type in [3, 4] that is only allowed to have at most a finite number of zeroes. A typical
example of the weight a is dist(x, ∂Ω).
Problem (1.1) contains some important classes of parabolic equations, such as the
semilinear heat equation (when a = 1, p = 2), semilinear degenerate parabolic equations
(when p = 2), the pLaplacian equations (when a = 1, p = 2), etc. It is noticed that the
existence and longtime behavior of solutions to (1.1) when p = 2, the semilinear case,
have been studied recently by Li et al. in [2]. We also refer the interested reader to [411]
for related results on degenerate parabolic equations.
2.
Preliminary results
To study problem (1.1), we introduce the weighted Sobolev space W01,p (Ω, a),
defined as the closure of C0∞ (Ω) in the norm
u
W01,p (Ω,a)
p
:=
1
p
a(x)∇u dx
,
Ω
′
and denote by W −1,p (Ω, a) its dual space.
We now prove some embedding results, which are generalizations of the
corresponding results in the case p = 2 of Li et al. [2].
Proposition 2.1. Assume that Ω is a bounded domain in RN , N ≥ 2, and a(·) satisfies
(H1). Then the following embeddings hold:
(i) W01,p (Ω, a) ֒→ W01,β (Ω) continuously if 1 ≤ β ≤ NpN
;
+α
(ii) W01,p (Ω, a) ֒→ Lr (Ω) continuously if 1 ≤ r ≤ p∗α , where p∗α =
pN
.
N −p+α
(iii) W01,p (Ω, a) ֒→ Lr (Ω) compactly if 1 ≤ r < p∗α .
Proof. Applying the H¨older inequality, we have
pN
pN
N
1
N+α ∇u N+α dx
∇u N+α dx =
N [a(x)]
Ω
Ω [a(x)] N+α
1
≤
N
Ω
4
α
N+α
[a(x)] α
p
dx
a(x)∇u dx
Ω
N
α
.
Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations
Using the assumption (H1), we complete the proof of (i).
The conclusions (ii) and (iii) follow from (i) and the wellknown embedding results
for the classical Sobolev spaces.
Putting
Lp,a u = −div(a(x)∇up−2∇u), u ∈ W01,p (Ω, a).
The following proposition, its proof is straightforward, gives some important properties
of the operator Lp,a .
Proposition 2.2. The operator Lp,a maps W01,p (Ω, a) into its dual W −1,p (Ω, a).
Moreover,
(i) Lp,a is hemicontinuous, i.e., for all u, v, w ∈ W01,p (Ω, a), the map λ → Lp,a (u+
λv), w is continuous from R to R;
(ii) Lp,a is strongly monotone when p ≥ 2, i.e.,
′
Lp,a u − Lp,a v, u − v ≥ δ u − v
3.
p
W01,p (Ω,a)
for all u, v ∈ W01,p (Ω, a).
Existence and uniqueness of global weak solutions
Denote
ΩT = Ω × (0, T ),
V = Lp (0, T ; W01,p(Ω, a)) ∩ Lq (0, T ; Lq (Ω)),
′
′
′
′
V ∗ = Lp (0, T ; W −1,p (Ω, a)) + Lq (0, T ; Lq (Ω)).
Definition 3.1. A function u is called a weak solution of problem (1.1) on the interval
(0, T ) if
du
∈ V ∗,
dt
= u0 a.e. in Ω,
u ∈ V,
ut=0
and
ΩT
∂u
η + a(x)∇up−2∇u∇η + f (u)η − gη dxdt = 0,
∂t
(3.1)
for all test functions η ∈ V .
du
It is known (see e.g. [4]) that if u ∈ V and
∈ V ∗ , then u ∈ C([0, T ]; L2 (Ω)).
dt
This makes the initial condition in problem (1.1) meaningful.
Lemma 3.1. Let {un } be a bounded sequence in Lp (0, T ; W01,p(Ω, a)) such that {u′n } is
bounded in V ∗ . If (H1) and (H3) hold, then {un } converges almost everywhere in ΩT up
to a subsequence.
5
Tran Thi Quynh Chi and Le Thi Thuy
Proof. By Proposition 2.1, one can take a number r ∈ [2, p∗α ) such that
W01,p (Ω, a) ֒→֒→ Lr (Ω).
(3.2)
Since r ′ ≤ 2, we have
′
Lp (Ω) ∩ Lq (Ω) ֒→ Lr (Ω),
and therefore,
′
′
Lr (Ω) ֒→ Lp (Ω) + Lq (Ω).
(3.3)
Using Proposition 2.1 once again and noticing that p ≤ p∗α since α ∈ (0, p), we see that
W01,p (Ω, a) ֒→ Lp (Ω).
This and (3.3) follow that
′
′
Lr (Ω) ֒→ W −1,p (Ω, a) + Lq (Ω).
Now with (3.2), we have an evolution triple
W01,p (Ω, a) ֒→֒→ Lr (Ω) ֒→ W −1,p (Ω, a) + Lq (Ω).
′
′
The assumption of {u′n } in V ∗ implies that
′
′
{u′n } is also bounded in Ls (0, T ; W −1,p (Ω, a) + Lq (Ω)), where s = min{p′ , q ′}.
Thanks to the wellknown AubinLions compactness lemma (see [12, p. 58]), {un } is
precompact in Lp (0, T ; Lr (Ω)) and therefore in Lt (0, T ; Lt (Ω)), t = min(p, r), so it has
an a.e. convergent subsequence.
The following lemma is a direct consequence of Young’s inequality and the
∗
pN
embedding W01,p (Ω, a) ֒→ Lpα (Ω), where p∗α = N −p+α
, which is frequently used later.
Lemma 3.2. Let condition (H3) hold and u ∈ W01,p (Ω, a) ∩ Lq (Ω). Then for any ε > 0,
we have
gudx ≤
Ω
ε u
ε u
p
+ C(ε) g sLs(Ω)
W01,p (Ω,a)
q
s
Lq (Ω) + C(ε) g Ls (Ω)
if s ≥
if s ≥
pN
,
(N +1)p−N +α
q
.
q−1
The following theorem is the main result of the paper.
Theorem 3.1. Under assumptions (H1) − (H3), for each u0 ∈ L2 (Ω) and T > 0 given,
problem (1.1) has a unique weak solution on (0, T ). Moreover, the mapping u0 → u(t) is
continuous on L2 (Ω).
6
Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations
Proof. (i) Existence. Consider the approximating solution un (t) in the form
n
unk (t)ek ,
un (t) =
k=1
1,p
q
2
where {ej }∞
j=1 is a basis of W0 (Ω, a) ∩ L (Ω), which is orthogonal in L (Ω). We get un
from solving the problem
dun
, ek + Lp,a un , ek + f (un), ek = g, ek ,
dt
(u (0), e ) = (u , e ), k = 1, . . . , n.
n
0
k
k
By the Peano theorem, we obtain the local existence of un .
We now establish some a priori estimates for un . Since
1d
un (t)
2 dt
2
L2 (Ω)
a(x)∇un p dx +
+
f (un )un dx =
Ω
Ω
gun dx.
Ω
Using (1.3) and Lemma 3.2, we have
d
un
dt
2
L2 (Ω)
a(x)∇un p dx +
+C
Ω
un q dx
≤ C( g
Integrating from 0 to t, 0 ≤ t ≤ T and using the fact that un (0)
obtain
t
un (t)
2
L2 (Ω)
L2 (Ω)
≤ u0
L2 (Ω) ,
we
t
a(x)∇un p dxdt + C
+C
0
≤
Ls (Ω) , Ω).
Ω
Ω
2
u0 L2 (Ω)
un q dxdt
0
+ T C( g
Ω
Ls (Ω) , Ω).
It follows that
• {un } is bounded in L∞ (0, T ; L2 (Ω));
• {un } is bounded in Lp (0, T ; W01,p(Ω, a));
• {un } is bounded in Lq (0, T ; Lq (Ω)).
The H¨older inequality yields
T

T
a(x)∇un p−2 ∇un ∇vdxdt
Lp,a un , v dt = 
0
0
T
Ω
≤
a(x)
0
≤ un
p−1
p
1
∇un p−1)(a(x) p ∇v dxdt
Ω
p
p′
Lp (0,T ;W01,p (Ω,a))
v
Lp (0,T ;W01,p (Ω,a)) ,
7
Tran Thi Quynh Chi and Le Thi Thuy
for any v ∈ Lp (0, T ; W01,p(Ω, a)).
Using the boundedness of {un } in
′
′
1,p
p
L (0, T ; W0 (Ω, a)), we infer that {Lp,a un } is bounded in Lp (0, T ; W −1,p (Ω, a)). From
(1.3), we have
f (u) ≤ C(up−1 + 1).
Hence, since {un } is bounded in Lq (0, T ; Lq (Ω)), one can check that {f (un )} is bounded
′
′
in Lq (0, T ; Lq (Ω)). Rewriting (1.1) in V ∗ as
u′n = g − Lp,a un − f (un )
(3.4)
and using the above estimates, we deduce that {u′n } is bounded in V ∗ .
From the above estimates, we can assume that
• u′n ⇀ u′ in V ∗ ;
′
′
• Lp,a un ⇀ ψ in Lp (0, T ; W −1,p (Ω, a));
′
• f (un ) ⇀ χ in Lq (ΩT ).
By Lemma 3.1, un → u a.e. in ΩT , so f (un ) → f (u) a.e. in ΩT since f (·) is continuous.
Thus, χ = f (u) thanks to Lemma 1.3 in [12]. Now taking (3.4) into account, we obtain
the following equation in V ∗ ,
u′ = g − ψ − f (u).
(3.5)
We now show that ψ = Lp,a u. We have for every v ∈ Lp (0, T ; W01,p (Ω, a)),
T
Xn :=
Lp,a un − Lp,a v, un − v ≥ 0.
0
Noticing that
T
T
a(x)∇un p dxdt
Lp,a un , un dt =
0
0
Ω
T
(gun − f (un )un − u′n un )dxdt
=
0
Ω
T
=
(gun − f (un )un )dxdt +
0
Ω
1
un (0)
2
2
L2 (Ω)
−
1
un (T )
2
(3.6)
Therefore,
T
Xn =
(gun − f (un )un )dxdt +
0
Ω
T
−
8
1
un (0)
2
2
L2 (Ω)
−
1
un (T )
2
T
Lp,aun , v dt −
0
2
L2 (Ω) .
Lp,a v, un − v dt.
0
2
L2 (Ω)
Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations
It follows from the formulation of un (0) that un (0) → u0 in L2 (Ω). Moreover, by the
lower semicontinuity of . L2 (Ω) we obtain
u(T )
L2 (Ω)
≤ lim inf un (T )
n→∞
L2 (Ω) .
(3.7)
Meanwhile, by the Lebesgue dominated theorem, one can check that
T
T
(gu − f (u)u)dxdt = lim
0
(gun − f (un )un )dxdt.
n→∞
Ω
0
Ω
This fact and (3.6), (3.7) imply that
T
lim sup Xn ≤
n→∞
(gu − f (u)u)dxdt +
0
Ω
T
−
1
u(0)
2
2
L2 (Ω)
−
1
u(T )
2
2
L2 (Ω)
(3.8)
T
ψ, v dt −
0
Lp,av, u − v dt.
0
In view of (3.5), we have
T
(gu − f (u)u)dxdt +
0
Ω
1
u(0)
2
2
L2 (Ω)
−
1
u(T )
2
T
2
L2 (Ω)
=
ψ, u dt.
0
This and (3.8) deduce that
T
ψ − Lp,a v, u − v dt ≥ 0.
(3.9)
0
Putting v = u − λw, w ∈ Lp (0, T ; W01,p(Ω, a)), λ > 0. Since (3.9) we have
T
ψ − Lp,a (u − λw), w dt ≥ 0.
λ
0
Then
T
ψ − Lp,a (u − λw), w dt ≥ 0.
0
Taking the limit λ → 0 and noticing that Lp,a is hemicontinuous, we obtain
T
ψ − Lp,a u, w dt ≥ 0,
0
for all w ∈ Lp (0, T ; W01,p(Ω, a)). Thus, ψ = Lp,a u.
We now prove u(0) = u0 . Choosing some test function ϕ ∈ C 1 ([0, T ]; W01,p (Ω, a)∩
q
L (Ω)) with ϕ(T ) = 0 and integrating by parts in t in the approximate equations, we have
T
T
′
− un , ϕ dt +
0
Lp,a un , ϕ dt +
0
(f (un )ϕ − gϕ)dxdt = (un (0), ϕ(0)).
ΩT
9
Tran Thi Quynh Chi and Le Thi Thuy
Taking limits as n → ∞, we obtain
T
T
′
− u, ϕ dt +
Lp,a u, ϕ dt +
0
(f (u)ϕ − gϕ)dxdt = (u0 , ϕ(0)),
0
(3.10)
ΩT
since un (0) → u0 . On the other hand, for the ”limiting equation”, we have
T
T
− u, ϕ′ dt +
0
Lp,au, ϕ dt +
0
(f (u)ϕ − gϕ)dxdt = (u(0), ϕ(0)).
(3.11)
ΩT
Comparing (3.10) and (3.11), we get u(0) = u0 .
(ii) Uniqueness and continuous dependence. Let u, v be two weak solutions of
problem (1.1) with initial data u0 , v0 in L2 (Ω). Then w := u − v satisfies
dw
+ (Lp,a u − Lp,a v) + (f (u) − f (v)) = 0,
dt
w(0) = u − v .
0
0
Hence
1d
w
2 dt
2
L2 (Ω)
+ Lp,a u − Lp,a v, u − v +
(f (u) − f (v))(u − v)dx = 0.
Ω
Using (1.4) and the monotonicity of the operator Lp,a , we have
d
w
dt
2
L2 (Ω)
≤ 2ℓ w
2
L2 (Ω) .
Applying the Gronwall inequality, we obtain
w(t)
L2 (Ω)
≤ w(0)
2ℓt
L2 (Ω) e
for all t ∈ [0, T ].
This completes the proof.
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