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Well-posedness problem of an anisotropic parabolic equation
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Well-posedness problem of an anisotropic parabolic equation Huashui Zhan a , Zhaosheng Feng b,∗ a School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China b Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Received 16 October 2018; revised 24 April 2019; accepted 15 August 2019
Abstract In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂ and the others are positive on , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain , is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions. © 2019 Elsevier Inc. All rights reserved. MSC: 35B35; 35G31; 35K55 Keywords: Anisotropic parabolic equation; Characteristic function; Partial boundary value condition; Stability
1. Introduction As we know, the anisotropic parabolic equation ut =
N N ∂ ∂bi (u, x, t) , (x, t) ∈ QT , ai (x)|uxi |pi −2 uxi + ∂xi ∂xi i=1
* Corresponding author.
E-mail address: [email protected] (Z. Feng). https://doi.org/10.1016/j.jde.2019.08.014 0022-0396/© 2019 Elsevier Inc. All rights reserved.
i=1
(1.1)
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plays an important role in mathematical modelling of various physical processes, such as flows of incompressible turbulent fluids, gases in pipes and filtration in glaciology [1–3]. Here, pi > 1, ai (x) ∈ C(), i ∈ {1, 2, · · · , N }, QT = × (0, T ), and is a bounded domain with a smooth boundary ∂. We consider equation (1.1) under the initial value condition: u(x, 0) = u0 (x), x ∈ ,
(1.2)
and the partial boundary value condition: u(x, t) = 0, (x, t) ∈ 1 × (0, T ),
(1.3)
where 1 is a relatively open subset of ∂ to be determined. Set 2 = ∂ \ 1 , and let p0 = min{p1 , p2 , · · · , pN−1 , pN } > 1, p 0 = max{p1 , p2 , · · · , pN−1 , pN }. Assume that ai (x) > 0, i ∈ {1, 2, · · · , N } when x ∈ , and ai1 (x) ≥ c1 > 0, ai2 (x) ≥ c2 > 0, · · · , aik (x) ≥ ck > 0, x ∈ , aj1 (x) = 0, aj2 (x) = 0, · · · , ajl (x) = 0, x ∈ ∂,
(1.4)
where {i1 , i2 , · · · , ik , j1 , j2 , · · · , jl } is a permutation of {1, 2, · · · , N } and ci s are positive constants. For a smooth bounded domain ⊂ RN , the function φ(x) is said to be a general characteristic function of if it is a nonnegative continuous function and satisfies φ(x) > 0, x ∈ and φ(x) = 0, x ∈ ∂, and |φxir (x)|pir
[φ(x)]pir −1 is bounded when r = 1, 2, · · · , k. The main feature which distinguishes this paper from other related works lies in the fact that we propose an explicit formula of 1 in terms of a general characteristic function of : |φxir (x)|pir 1 = x ∈ ∂ : = 0, r = 1, 2, · · · , k . (1.5) [φ(x)]pir −1 For example, if a1 (x)|x∈∂ = 0, ai (x) > 0, x ∈ , i ≥ 2, we can choose φ(x) = a1 (x), and then (1.5) becomes
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a1x pi 1 = x ∈ ∂ : a1 i = 0, i ≥ 2 . a1
3
(1.6)
Before summarizing our main results, let us briefly introduce the definitions of weak solutions. Definition 1. A function u(x, t) is said to be a weak solution of equation (1.1) with the initial value (1.2), provided that p 0 0 u ∈ L∞ (QT ), ai (x) uxi i ∈ L1 (QT ), ut ∈ Lp 0, T ; W −1,p () ,
where p 0 =
p0 , p 0 −1
and for any function ϕ ∈ C01 (QT ) there holds
N
p −2 < ut , ϕ > + ai (x) uxi i uxi ϕxi + bi (u, x, t) · ϕxi dxdt = 0. i=1 Q
T
Here, (1.2) is true in the sense of |u(x, t) − u0 (x)|dx = 0.
lim
t→0
(1.7)
Definition 2. The function u(x, t) is said to be a weak solution of equation (1.1) with the initialboundary conditions (1.2) and (1.3), if u(x, t) satisfies Definition 1 and the partial boundary value condition (1.3) is satisfied in the sense of trace. Theorem 1.1. Suppose that bi (u, x, t) is a C 1 function on R × × [0, T ] and u0 ∈ L∞ (), |∇u0 |p ∈ L1 (). 0
Then equation (1.1) with the initial value (1.2) has a weak solution. Moreover, when 1 ≤ s ≤ l − p 1−1 and ajs js (x)dx < ∞, then the initial-boundary value problem (1.1)-(1.2)-(1.3) has a weak solution. Theorem 1.2. Let u(x, t) and v(x, t) be two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively, and with the same partial boundary value condition (1.3). Suppose that φ(x) is a general characteristic function of and 1 is given by (1.5). If ⎛ l 1⎜ ⎝ η s=1
⎞ ⎟ ajs (x)|φxjs (x)|pjs dx ⎠
1 pjs
≤ c,
(1.8)
\η
where η = {x ∈ : φ(x) > η}, and 1 p
|bi (u, x, t) − bi (v, x, t)| ≤ cai i |u − v|, i = 1, 2, · · · , N,
(1.9)
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then we have
|u(x, t) − v(x, t)|dx ≤ c
|u0 (x) − v0 (x)|dx.
(1.10)
Here and in the sequel, c may have different values in different formulae. When b = {bi (u, x, t)} is independent of the diffusion coefficients {ai (x)}, i.e. without condition (1.9), we have the following result regarding the stability of weak solutions of equation (1.1). Theorem 1.3. Suppose that is a normal smooth boundary, and u(x, t) and v(x, t) are two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively, and with the same partial boundary value condition (1.3), where 1 is given by (1.5). Then the stability (1.10) of weak solutions of equation (1.1) is true, provided that condition (1.8) holds and ∂bi (s, x, t) ∂bi (s, x, t) ≥ 0 or ≤ 0 for i ≥ 1. ∂s ∂s If the condition ∂bi (s,x,t) ≥ 0 or ∂bi (s,x,t) ≤ 0 in Theorem 1.3 becomes invalid, the stability ∂s ∂s of weak solutions of equation (1.1) under the partial boundary value condition (1.5) can still be achieved. For example and for simplicity, here we only consider the case of l = 1: a1 (x) = 0, x ∈ ∂,
(1.11)
while ai (x) ≥ ci > 0 for i > 1. The discussions for other cases can be processed in an analogous manner. Theorem 1.4. Let be a normal smooth boundary, and u(x, t) and v(x, t) be two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively. The partial boundary value condition (1.3) holds while 1 is given by (1.6). Suppose that ai (x) ≥ ci > 0 for i > 1, and a1 (x) satisfies (1.11) and a1xi pi ≤ c, a1 (x) a (x) 1
⎛ 1⎜ ⎝ η
⎞ ⎟ a1 (x)|a1x1 (x)|p1 dx ⎠
1 p1
≤ c,
(1.12)
η
where c is a constant. If there exists a function g1 (x) such that |b1 (·, x, t)| ≤ cg1 (x),
(1.13)
|bi (u, x, t) − bi (v, x, t))| ≤ c|u − v|, i ≥ 2,
(1.14)
q1 1 (a1 (x))− p1 g1 (x) dx < ∞,
(1.15)
and
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then the stability (1.10) of weak solutions is true, where a1x1 (x) = 1 . η = {x ∈ : a1 (x) > η}, and q1 = p1p−1
5
∂a1 (x) ∂x1 ,
a1xi (x) =
∂a1 (x) ∂xi ,
It is notable that the expression of 1 = φ , as given by (1.5), depends on the choice of the general characteristic function φ(x). For a smooth domain ⊂ RN , since the general characteristic function φ(x) is not unique, if one chooses a different general characteristic function φ1 (x), the corresponding partial boundary φ1 might be different from φ . Thus the optimal partial boundary 0 in (1.3), should satisfy 0 ⊆ {φ }. We wish to make a conjecture that 0 =
φ .
φ
Whether this conjecture is true or false, the proof is a challenging problem worthy of being investigated in the future. The idea and the technique described in this study are conveniently generalized to the other parabolic equations (e.g. [4–18]), even to those with variable exponents and nonlinearity (e.g. [19–23]). The remainder of this paper is structured as follows. In Section 2, we consider the existence of weak solutions of equation (1.1). Sections 3 and 4 are devoted to the stability of weak solutions of equation (1.1) under the proposed partial boundary condition. In Section 5, we briefly give a comment on the construction of the explicit formula of 1 . 2. Existence of weak solutions Let us recall the approximate problem N p −2 0 uεt − εdiv |∇uε |p −2 ∇uε + ai (x) uεxi i uεxi i=1
−
N ∂bi (uε , x, t) i=1
∂xi
xi
(2.1)
= 0, (x, t) ∈ QT ,
uε (x, t) = 0, (x, t) ∈ ∂ × (0, T ),
(2.2)
uε (x, 0) = uε,0 (x), x ∈ ,
(2.3)
p 0 where uε,0 ∈ C0∞ (), |uε,0 |L∞ () ≤ |u0 |L∞ () , and ∇uε,0 is uniformly convergent to 0 |∇u0 (x)|p in L1 (). It is well-known that the above problem has a unique solution uε ∈ 0
1,p 0
Lp (0, T ; W0
()) [19].
− p 1−1 Lemma 2.1. [5] If ai i (x)dx < ∞, and u is a weak solution of equation (1.1) with the initial value (1.2). Then for any given t ∈ [0, T ) there holds
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|uxi |dx ≤ c, i = 1, 2, · · · , N,
and the trace of u on the boundary ∂ can be defined in the traditional way. − p 1−1 In view of condition (1.4), the inequality ai i (x)dx < ∞ given in Lemma 2.1 can be − p 1−1 replaced by ajs js (x)dx < ∞ for 1 ≤ s ≤ l. Proof of Theorem 1.1. By the maximum principle [19] we know that uε L∞ (QT ) c.
(2.4)
We multiply both sides of equation (2.1) by uε and integrate it over Qt = × [0, t] (t < T ). Since t
1 uε uεt dxdt = 2
0
t
∂u2ε dxdt ∂t
0
1 = 2
u2ε (x, t)dx
1 − 2
u2ε (x, 0)dx
and N ∂bi (uε , x, t) − uε dx ∂x i i=1 N ∂uε bi (uε , x, t) dx = ∂xi i=1 uε N uε N ∂ ∂ bi (s, x, t)dsdx − bi (s, x, t)dsdx = ∂xi ∂xi i=1
0
uε N ∂ bi (s, x, t)dsdx = ∂xi i=1
i=1 0
0
≤c, we have 1 2
u2ε (x, t)dx + ε
0
|∇uε |p dxdt + Qt
N i=1 Q t
ai (x)|uεxi |pi dxdt c, ∀t < T .
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This implies that ai (x)|uεxi |pi dxdt ≤ c
(2.5)
QT
and
0
|∇uε |p ≤ c.
ε
(2.6)
QT 0
1,p 0
For any v ∈ Lp (0, T ; W0
()) and v
1,p 0
0
Lp (0,T ;W0
have
())
= 1, by Young’s inequality, we
pi −2 ai (x)|uεxi | uεxi vxi dxdt QT ai (x) |uεxi |pi + |vxi |pi dxdt ≤c QT
≤c
0 ai (x) |uεxi |pi + |vxi |p + 1 dxdt
QT
ai (x)|uεxi |pi dxdt + c
≤c QT
≤c, and ⎡ ⎢ | < uεt , v > | ≤ c ⎣ε
0
|∇uε |p dxdt +
N i=1 Q
QT
⎤ ⎥ ai (x)|uεxi |pi dxdt + 1⎦
T
≤ c. Then, we obtain uεt
0
0
Lp (0,T ;W −1,p ()
≤ c.
(2.7)
For any ϕ ∈ C01 () and 0 ≤ ϕ ≤ 1, it is easy to see that (ϕuε )t
0
0
Lp (0,T ;W −1,p ())
≤ c.
(2.8)
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For a fixed s such that s > 0
N 2
0
+ 1, it has H0s () → W 1,p (), which implies that
W −1,p () → H −s (). As a result, we have (ϕuε )t
0
≤c
(2.9)
≤ c.
(2.10)
Lp (0,T ;H −s ())
and ϕuε 1,p0
Note that W0
0
1,p0
Lp (0,T ;W0
())
() → Lp0 () → H −s (), one can employ Aubin’s compactness theorem 0
[7,8] to derive that ϕuε → ϕu strongly in Lp (0, T ; Lp0 ()). Thus, ϕuε → ϕu a.e. in QT . In particular, due to the arbitrariness of ϕ, we have uε → u a.e. in QT . − → In view of (2.4)-(2.7), there exists a function u and an n−dimensional vector function ζ = (ζ1 , · · · , ζn ) such that p 0 −2
ε|∇uε |
∇uε 0 in L
p0 p 0 −1
(QT ),
pi
u ∈ L∞ (QT ), |ζi | ∈ L pi −1 (QT ), uε u, weakly-star in L∞ (QT ), bi (uε , x, t) → bi (u, x, t) a.e. in QT , p
i uεxi uxi in Lloc (QT ),
and pi p −2 ai (x) uεxi i uεxi ζi in L pi −1 (QT ).
By a similar argument to the usual evolutionary p−Laplacian [19], we can obtain (1.7) and −−−−−−−−→ − → p−2 ζ · ∇ϕdxdt a |∇u| ∇u · ∇ϕdxdt = QT
QT
for any given function ϕ ∈ C01 (QT ), where −−−−−−−−→ a |∇u|p−2 ∇u = ai (x)|uxi |pi −2 uxi . The desired result follows Lemma 2.1 immediately.
2
(2.11)
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3. Stability under partial boundary condition For small η > 0, let s Sη (s) =
hη (τ )dτ, hη (s) = 0
2 |s| . 1− η η +
(3.1)
Clearly, hη (s) ∈ C(R), and hη (s) ≥ 0, | shη (s) |≤ 1, | Sη (s) |≤ 1, lim Sη (s) = sgns. η→0
If we denote Hη (s) =
s 0
(3.2)
Sη (s)ds, then
lim Sη (s) = sgn(s), lim Hη (s) = |s|, s ∈ (−∞, +∞),
(3.3)
lim hη (s)s = 0.
(3.4)
η→0
η→0
and η→0
Proof of Theorem 1.2. Let u(x, t) and v(x, t) be two weak solutions of equation (1.1) with the initial values u(x, 0) and v(x, 0) respectively, and with the partial homogeneous boundary value condition: u(x, t) = v(x, t) = 0, (x, t) ∈ 1 × [0, T ).
(3.5)
Let η = {x ∈ : φ(x) > η} and φη (x) =
1, φ(x) η ,
if x ∈ η , if x ∈ \ η .
φx (x)
Obviously, φηxi = iη when x ∈ \ η . Otherwise, it is identically zero. Choosing χ[τ,s] φη Sη (u − v) as the test function, we deduce that s φη Sη (u − v)
∂(u − v) dxdt ∂t
τ
+
N s i=1 τ
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi (uxi − vxi )hη (u − v)φη (x)dxdt
(3.6)
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10
+
k s
air (x) |uxir |pir −2 uxir − |vxir |pir −2 vxir Sη (u − v)φηxir dxdt
r=1 τ
+
l s
ajr (x) |uxjr |pjr −2 uxjr − |vxjr |pjr −2 vxjr Sη (u − v)φηxjr dxdt
r=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt
i=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
i=1 τ
= 0.
(3.7)
For the first term on the left hand side of (3.7), following [4, Lemma 3.1] we find s φη (x)Sη (u − v)
lim
η→0
τ
(3.8)
|u − v|(x, s)dx −
=
∂(u − v) dxdt ∂t
|u − v|(x, τ )dx.
For the second term on the left hand side of (3.7), it is easy to see that
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi (uxi − vxi )hη (u − v)φη (x)dx ≥ 0.
(3.9)
To evaluate the third term on the left hand side of (3.7), in consideration of (1.3) and (1.6), by a straightforward calculation we obtain ai (x) |ux |pir −2 ux − |vx |pir −2 vx Sη (u − v)φηx dx r ir ir ir ir ir air (x) |uxir |pir −2 uxir − |vxir |pir −2 vxir Sη (u − v)φηxir dx = \η air (x) |uxir |pir −1 + |vxir |pir −1 |Sη (u − v)φηxir |dx ≤ \η
≤ \η
! pir −1 pir 1 air (x)φ(x) |uxir |pir −1 + |vxir |pir −1 η
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·(air (x)) pir ⎛
dx ⎞
⎜ ≤⎝
1 1 pir [φ(x)]xir Sη (u − v) η pir −1 [φ(x)] pir
11
⎟ 1 φ(x)air (x) |uxir |pir + |vxir |pir dx ⎠ η
1 qir
\η
⎛ ⎜ ·⎝
⎞
|pir
\η
⎛ ⎜ ≤ c⎝
|[φ(x)]xir 1 ⎟ dx ⎠ air (x)|Sη (u − v)|pir η [φ(x)]pir −1 ⎞
⎟ air (x) |uxir |pir + |vxir |pir dx ⎠
1 pir
1 qir
\η
⎛ ⎜1 ·⎝ η
⎞
|Sη (u − v)|pir \η
where qir = Denote
|[φ(x)]xir
|pir
[φ(x)]pir −1
⎟ dx ⎠
1 pir
(3.10)
,
pir pir −1 .
η1 = {x ∈ \ η : dist(x, 2 ) > dist(x, 1 )}, η2 = {x ∈ \ η : dist(x, 2 ) ≤ dist(x, 1 )}. Then we have 1 η
|Sη (u − v)|pir | \η
1 ≤ η
|Sη (u − v)|pir
η1
1 + η
|Sη (u − v)|pir η2
|[φ(x)]xir |pir
dx
|[φ(x)]xir |pir
dx
[φ(x)]pir −1
[φ(x)]pir −1
|[φ(x)]xir |pir [φ(x)]pir −1
dx.
By making use of u(x, t) = v(x, t) = 0, (x, t) ∈ 1 × (0, T ), we have
(3.11)
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1 η→0 η
|Sη (u − v)|pir
lim
η1
=
sign(u − v)
|[φ(x)]xir |pir [φ(x)]pir −1
|[φ(x)]xir |pir
1
[φ(x)]pir −1
dx (3.12)
d
=0. Moreover, by using the identity |[φ(x)]xir |pir
[φ(x)]pir (x) −1
= 0, x ∈ 2 ,
we derive that 1 η→∞ η
|Sη (u − v)|pir
lim
η2
1 ≤ lim η→∞ η = 2
η2
|[φ(x)]xir |pir [φ(x)]pir −1
|(φ(x))xir |pir [φ(x)]pir −1
|[φ(x)]xir |pir [φ(x)]pir −1
dx
dx (3.13)
d
=0. From (3.10)-(3.13), we obtain pir −2 pir −2 lim air (x) |uxir | uxir − |vxir | vxir Sη (u − v)φηxir dx = 0. η→0
(3.14)
To evaluate the fourth term on the left hand side of (3.7), by a direct calculation we deduce that aj (x) |ux |pjr −2 ux − |vx |pjr −2 vx Sη (u − v)φηx dx r jr jr jr jr jr pjr −2 pjr −2 ajr (x) |uxjr | uxjr − |vxjr | vxjr Sη (u − v)φηxjr dx = \η ajr (x) |uxjr |pjr −1 + |vxjr |pjr −1 [φ(x)]xjr Sη (u − v)|dx ≤ \η
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⎛ ⎜ ≤ c⎝
⎞
⎟ ajr (x) |uxjr |pjr + |vxjr |pjr dx ⎠
13
1 qjr
\η
⎛ 1⎜ · ⎝ η
⎞ ⎟ ajr (x)|[φ(x)]xi |pjr dx ⎠
1 pjr
.
\η
Using condition (1.8), we have
pjr −2 pjr −2 lim ajr (x) |uxjr | uxjr − |vxjr | vxjr Sη (u − v)φηxjr dx = 0. η→0
(3.15)
To evaluate the fifth term in (3.7), in view of condition (1.9), it follows Hölder’s inequality that
≤c
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx 1 pi a (u − v)x φη (u − v)hη (u − v) dx i i
⎛ ⎞1 ⎛ ⎞1 pi qi p p q i i i ≤c ⎝ ai (x)(|uxi | + |vxi | )dx ⎠ ⎝ |(u − v)hη (u − v)| dx ⎠ ,
where qi =
pi pi −1 .
By (3.2), we have
s [bi (u, x, t) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt = 0.
lim
η→0 τ
(3.16)
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14
For the sixth term in (3.7), using condition (1.9) again leads to N lim [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx η→0 i=1 N [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx = lim η→0 i=1 \ η ≤ lim
η→0
≤ lim
η→0
N 1
1 p
ai i |φxi (x)||Sη (u − v)(u − v)|dx
η
i=1
k 1
η→0
1 p
air ir |φxir (x)||Sη (u − v)(u − v)|dx
η
r=1
+ lim
(3.17)
\η
\η
l 1 r=1
1 p
ajrjr |φxjr (x)||Sη (u − v)(u − v)|dx.
η \η
By the partial homogeneous boundary condition (1.3), we have
lim
k 1
η→0
r=1
1 p
air ir |φxir (x)||Sη (u − v)(u − v)|dx
η \η
⎞
⎛ k ⎜1 ≤ lim ⎝ η→0 η r=1
⎡
⎢1 ·⎣ η
|φxir
\η
(x)|pir
|φ(x)|pir −1
air |φ(x)|
pir −1 pir
⎟ |u − v|pir dx ⎠
r ppi−1 ir
1 pir
⎤ pir −1 |Sη (u − v)|
pir −1 pir
⎥ dx ⎦
\η
⎛ ≤ c lim
η→0
k r=1
⎜1 ⎝ η
\η
⎛
⎜ = c⎝
1
⎞ |φxir
(x)|pir
|(φ(x))|pir −1
⎟ |u − v|pir dx ⎠
⎞ |φxir
(x)|pir
|φ(x)|pir −1
⎟ |u − v|pir d⎠
1 pir
1 pir
pir
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⎛
⎞
|φxir
⎜ +c ⎝
2
(x)|pir
|φ(x)|pir −1
⎟ |u − v|pir d⎠
15
1 pir
= 0. In view of condition (1.8), we get
lim
l 1
η→0
s=1
1 p
ajsjs |φxjs (x)||Sη (u − v)(u − v)|dx
η \η
⎛ 1⎜ ≤ lim ⎝ η→0 η
⎞
pjs
ajs (x)|φxjs (x)|
⎟ dx ⎠
1 pjs
⎝
\η
dx ⎠
pjs
(3.18)
⎞ pjs −1
η→0
|Sη (u − v)(u − v)|
pjs pjs −1
⎛ ≤c lim ⎝
⎞ pjs −1
⎛
|Sη (u − v)(u − v)|
pjs pjs −1
dx ⎠
pjs
⎛
≤c ⎝
⎞ pjs −1 pjs
|u − v|dx ⎠
.
In accordance with (3.17)-(3.18), we derive that s [bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
lim
η→0 τ
(3.19)
⎞ pi −1 ⎛ s pi ⎠ ⎝ |u − v|dxdt . ≤c τ
Combining (3.8)-(3.9), (3.14)-(3.16) with (3.19), and letting η → 0 in (3.7) leads to
|u(x, s) − v(x, s)|dx
⎞ pi −1 ⎛ s pi N ⎠ ⎝ |u(x, τ ) − v(x, τ )|dx + c |u − v|dxdt . (3.20)
i=1
τ
Let κ(s) = |u(x, s) − v(x, s)|dx. Without s loss of generality, we assume that there exists τ ∈ [0, T ) and κ(τ ) > 0. For any s > τ and τ k(t)dt > 0, we denote that τ0 τ0 = max{t ∈ [τ, s], κ(t) > 0} and
k(t)dt = c1 . τ
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Then, τ < τ0 ≤ s, and s
τ0 k(t)dt ≥
τ
k(t)dt = c1 . τ
Since u, v ∈ L∞ (QT ), there exist constants Ci > 0 such that c
s
k(t)dt
s
τ
τ
pip−1 i
k(t)dt
≤
c
s τ
k(t)dt c1
pip−1 i
≤ Ci = Ci (c, c1 , T , q), i = 1, 2, · · · , N.
(3.21)
By (3.20) and (3.21), we have " κ(s) − κ(τ ) ≤
N
# s Ci + c
i=1
k(t)dt. τ
Using Gronwall’s inequality, we get
|u(x, s) − v(x, s)| dx c
|u(x, τ ) − v(x, τ )| dx.
Due to the arbitrariness of τ , we obtain
|u(x, s) − v(x, s)| dx c
|u0 (x) − v0 (x)| dx.
2
Proof of Theorem 1.3. From the proof of Theorem 1.2, we see that (3.8)-(3.9), and (3.14) hold too. Let us evaluate the last two formulas involving the convection terms on the left hand side of (3.7). We start with ≤c
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx 1 −1 pi a (u − v)x a pi φη (u − v)hη (u − v) dx i i i
⎞1 ⎛ ⎞1 ⎛ pi qi qi −1 pi pi pi ⎠ ⎠ ⎝ ⎝ ai (x) |uxi | + |vxi | dx , ≤c ai (u − v)hη (u − v) dx
where
(3.22)
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17
⎞1 qi qi −1 pi ⎝ a (u − v)hη (u − v) dx ⎠ i ⎛
⎛
=⎝
−1 pi −1
ai
⎞1
(3.23)
qi
|(u − v)hη (u − v)|qi dx ⎠ .
If {x ∈ : |u − v| = 0} is a measure zero set, by |(u − v)hη (u − v)| ≤ c and
−1 pi −1
ai
(x)dx ≤ c,
we get qi −1 pi a (u − v)hη (u − v) dx i η→0 lim
−1 pi −1
≤c
ai
(x)dx
(3.24)
{:|u−v|=0}
=0. If {x ∈ : |u − v| = 0} has a positive measure, by |(u − v)hη (u − v)| ≤ c and
−1 pi −1
ai
(x)dx ≤ c,
it follows Lebesgue’s dominated convergence theorem that qi −1 pi a (u − v)hη (u − v) dx = 0. i η→0 lim
(3.25)
By (3.22)-(3.25), we derive that lim [bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx = 0. η→0
Meanwhile, if
∂bi (s,x,t) ∂s
≥ 0 for i ≥ 1, we denote D1η = {x ∈ \ η : φxi (x) < 0}.
To evaluate the last term on the left hand side of (3.7), we re-write it as
(3.26)
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18
−
[bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx
[bi (u, x, t) − bi (v, x, t)]φηxi sign(u − v)dx
=−
(3.27)
[bi (u, x, t) − bi (v, x, t)]φηxi [Sη (u − v) − sign(u − v)]dx.
−
Note that lim D1η ⊆ 1 .
η→0
We then have − lim
[bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx
η→0
1 η→0 η
= − lim ≤ lim
η→0
≤c
c η
bi (ζ, x, t)(u − v)(φ(x))xi sign(u − v)dx
|u − v|[−(φ(x))xi ]dx
(3.28)
D1η
|u − v|d
1
=0, where bi (ζ, x, t) = ∂bi (u,x,t) |u=ζ . ∂u If ∂bi (s,x,t) ≤ 0 for i ≥ 1, we denote ∂s D1η = {x ∈ \ η : φxi (x) > 0}. Proceeding in a similar way, we can achieve (3.28) too. Let ρ(x) = dist(x, ∂). Since is a smooth domain, there exists a finite open cover of ∂: ∂ ⊂
$
Uα .
α
Denote αη = {x ∈ \ η : x ∈ Uα } and
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α = αη
19
∂.
Using the flattening technique and the L’Hospital rule leads to lim [bi (u, x, t) − bi (v, x, t)]φηxi [Sη (u − v) − sign(u − v)]dx η→0 [bi (u, x, t) − bi (v, x, t)]φηx [Sη (u − v) − sign(u − v)] dx lim ≤c i α
≤c
α
=c
α
1 + η
η→0 αη
lim
η→0 α
η |u − v||Sη (u − v) − sign(u − v)|dρ 0
%
lim
η→0 α
η
1 d η
|Sη (u − v) − sign(u − v)||u − v| (3.29)
⎤
|u − v||hη (u − v)|dρ ⎦ d
0
=c
α
=c
α
η
1 lim η→0 η lim
|u − v||hη (u − v)|dρd α 0
η→0 α
|u − v||hη1 (u − v)|d
=0, where η1 ∈ [0, η]. By virtue of (3.27)-(3.29), we obtain lim − [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx = 0. η→0
The rest is similar to the one as we show in the proof of Theorem 1.2, and thus we arrive at (1.10). 2 4. Proof of Theorem 1.4 Let η = {x ∈ : a1 (x) > η}, and φη (x) =
1, 1 η a1 (x),
if x ∈ η , if x ∈ \ η .
(4.1)
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20
Choosing χ[τ,s] φη Sη (u − v) as the test function, we have s φη Sη (u − v)
∂(u − v) dxdt ∂t
τ
+
N s
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi )(uxi − vxi hη (u − v)φη (x)dxdt
i=1 τ
s
a1 (x) |ux1 |p1 −2 ux1 − |vx1 |p1 −2 vx1 Sη (u − v)φηx1 dxdt
+ τ
+
N s
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi Sη (u − v)φηxi dxdt
(4.2)
i=2 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt
i=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
i=1 τ
=0. As we show in the preceding section, for the first two terms of the left hand side of (4.2), formulas (3.8) and (3.9) still hold. To evaluate the third term of the left hand side of (4.2), by (1.12) we have p1 −2 p1 −2 lim a1 (x)(|ux1 | ux1 − |vx1 | vx1 )Sη (u − v)φηx1 dx η→0 1 a1 (x) |ux1 |p1 −1 + |vx1 |p1 −1 |(a1 (x))x1 Sη (u − v)|dx ≤ lim η→0 η \η
⎛ ⎜ ≤ lim ⎝
η→0
⎞1
a1 (x) |ux1 |
p1
\η
= 0. Meanwhile, we can deduce that
+ |vx1 |
p1
⎟ dx ⎠
q1
⎛ 1⎜ ⎝ η
\η
⎞ ⎟ a1 (x)|(a1 (x))x1 |p1 dx ⎠
(4.3) 1 p1
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21
N pi −2 pi −2 ai (x) |uxi | uxi − |vxi | vxi Sη (u − v)φηxi dx i=2 \ η ≤
N c i=2
≤c
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
η \η
N c i=2
η
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
\η
→0, as η → 0. This is due to the partial boundary value condition (1.3) with the formula (1.6):
c η
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
\η
1 ≤ η
pi −1 pi
a1
pi −1
|uxi |
pi −1
+ |vxi |
(a1 (x))x Sη (u − v) i pi −1 pi
dx
a1
⎛
⎞ pi −1
⎜1 ≤c ⎝ η
a1 (x) |uxi | + |vxi | pi
pi
⎟ dx ⎠
pi
\η
⎛ ⎜1 ·⎝ η
⎜1 ≤c ⎝ η
|a1xi |pi p −1
\η
⎛
⎞1
a1 i
|a1xi |pi a1 i
⎛
⎜ →c ⎝
1
⎟ |Sη (u − v)| dx ⎠
|pi
p −1 a1 i
pi
pi
⎞1 |a1xi
pi
⎞1 p −1
\η
⎟ |Sη (u − v)| dx ⎠ pi
⎟ |sign(u − v)|pi d⎠
pi
⎛ ⎞1 pi p i ⎜ |a1xi | ⎟ pi +c⎝ |sign(u − v)| d⎠ p −1 a1 i 2
=0. For the fifth term of the left hand side of (4.2), it follows Hölder’s inequality that N [bi (u, x, t)) − bi (v, x, t)](u − v)x φη hη (u − v)dx i i=1
(4.4)
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22
≤ [b1 (u, x, t)) − b1 (v, x, t)](u − v)x1 φη hη (u − v)dx N [bi (u, x, t)) − bi (v, x, t)](u − v)x φη hη (u − v)dx + i i=2 ≤ c |g1 (x)(u − v)x1 φη hη (u − v)|dx
+
N
|(u − v)xi φη (u − v)hη (u − v)|dx.
(4.5)
i=2
Using (1.15), we have |g1 (x)(u − v)x1 φη hη (u − v)|dx
⎛ ⎞1 ⎛ ⎞1 p1 q1 1 − p1 p1 q1 p ⎝ ⎠ ⎠ ⎝ 1 ≤c a1 (x) |ux1 | + |vx1 | dx |(a1 (x)) g1 (x)(u − v)hη (u − v)| dx
→0, as η → 0. (4.6) Let us recall (1.14): |bi (u, x, t) − bi (v, x, t)| ≤ c|u − v|, i ≥ 2, and thus get N
|(u − v)xi φη (u − v)hη (u − v)|dx
i=2
⎞1 ⎛ ⎞1 ⎛ pi qi N ⎝ ≤c |uxi |pi + |vxi |pi dx ⎠ ⎝ |(u − v)hη (u − v)|qi dx ⎠ i=2
→0, as η → 0. For the last term of the left hand side of (4.2), we have N [bi (u, x, t) − bi (v, x, t)]φηx Sη (u − v)dx lim i η→0 i=1
(4.7)
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23
= lim [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx η→0 i=1 \ η N
≤ lim
η→0
N c i=1
\η
⎛ ≤ lim
η→0
N i=1
|(a1 (x))xi ||Sη (u − v)(u − v)|dx
η
1⎜ ⎝ η
⎞1 ⎛ ⎟ |(a1 (x))xi |pi dx ⎠
pi
⎜ ⎝
\η
⎞ p1 −1
|Sη (u − v)(u − v)|
p1 p1 −1
⎟ dx ⎠
p1
\η
→ 0, as η → 0.
(4.8)
Now, we let η → 0 in (4.2). By (4.3)-(4.8), we have
|u(x, s) − v(x, s)|dx
|u(x, τ ) − v(x, τ )|dx.
Due to the arbitrariness of τ , we obtain |u(x, s) − v(x, s)|dx |u0 (x) − v0 (x)|dx.
5. Explicit formula of 1 Let us recall the explicit expression of 1 as constructed in (1.5), i.e. 1 = x ∈ ∂ :
|φxir (x)|pir
[φ(x)]pir −1
= 0, r = 1, 2, · · · , k ,
where φ(x) ∈ C 1 () is a nonnegative function satisfying φ(x) > 0, x ∈ and φ(x) = 0, x ∈ ∂,
(5.1)
such that |φxir (x)|pir
[φ(x)]pir −1 is bounded for every r = 1, 2, · · · , k. Since the boundary ∂ is a smooth (N − 1)-dimensional manifold, the distance ρ(x) = dist(x, ∂) is smooth when x is near the boundary ∂. If we choose φ = ρ α (x), only when α > p 0 , then |φxir (x)|pir
[φ(x)]pir −1
= 0, r = 1, 2, · · · , k.
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24
If pir = p for all 1 ≤ r ≤ k, we choose φ = ρ α (x) with α = p, then |φxir (x)|pir
[φ(x)]pir −1
=
|αρ α−1 ρxir |p = |αρxir |p ρ α(p−1)
is bounded. In fact, there always exists a nonnegative φ(x) ∈ C 1 () satisfying (5.1) such that is bounded. To see this, without loss of generality, we can assume that i1 ≤ i2 ≤ nonnegative φ(x) ∈ C 1 () satisfies (5.1) but φ
pik
|φxi (x)|pir r
[φ(x)]pir −1
|φxi (x)|pir r
[φ(x)]pir −1 · · · ≤ ik . If the
is unbounded, then the function φ1 =
satisfies (5.1) and pi
|φxirk (x)|pir
[φ(x)]pik (pir −1)
= 0, 1 ≤ r < k,
and pi
|φxikk (x)|pir [φ(x)]pik (pik − 1) is bounded. References [1] S.V. Antontsev, J.I. Diaz, S. Shmarev, Energy Methods for Free Boundary Problems, Applications to Nonlinear PDES and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 48, Birkhäuser, Boston, MA, 2002. [2] J.I. Diaz, J. Padial, Uniqueness and existence of a solution in BVt (q) space to a doubly nonlinear parabolic problem, Publ. Mat. 40 (1996) 527–560. [3] J.I. Diaz, F. Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (1994) 1085–1111. [4] H. Zhan, The uniqueness of the solution to the diffusion equation with a damping term, Appl. Anal. 98 (7) (2019), https://doi.org/10.1080/00036811.2017.1422725. [5] H. Zhan, Z. Feng, Solutions of evolutionary p(x)−Laplacian equation based on the weighted variable exponent space, Z. Angew. Math. Phys. 68 (2017) 1–17. [6] H. Zhan, Z. Feng, Existence of solutions to an evolution p−Laplacian equation with a nonlinear gradient term, Electron. J. Differ. Equ. 311 (2017) 1–15. [7] J.P. Aubin, Un théorˇeme de compacité, C. R. Acad. Sci. 256 (1963) 5042–5044. [8] J. Simon, Compact sets in the space Lp (0, t; B), Ann. Mat. Pura Appl. (4) 146 (1952) 65–96. [9] K. Lee, A. Petrosyan, J.L. Vazquez, Large time geometric properties of solutions of the evolution p−Laplacian equation, J. Differ. Equ. 229 (2006) 389–411. [10] K. Kobayasi, H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Differ. Equ. 252 (2012) 137–167. [11] H. Frid, Y. Li, A boundary value problem for a class of anisotropic degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal. 226 (2017) 975–1008. [12] J. Zhao, Existence and nonexistence of solutions for ut = div(|∇u|p−2 ∇u) + f (∇u, u, x, t), J. Math. Anal. Appl. 172 (1993) 130–146. [13] J. Yin, C. Wang, Evolutionary weighted p−Laplacian with boundary degeneracy, J. Differ. Equ. 237 (2007) 421–445. [14] S. Alkis Tersenov, S. Aris Tersenov, Existence of Lipschitz continuous solutions to the Cauchy-Dirichlet problem for anisotropic parabolic equations, J. Funct. Anal. 272 (2017) 3965–3986.
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25
[15] S. Lian, W. Gao, H. Yuan, C. Cao, Existence of solutions to an initial Dirichlet problem of evolutional p(x)−Laplace equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29 (2012) 377–399. [16] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (1) (2010) 204–228. [17] K. Ho, I. Sim, On degenerate p(x)−Laplacian equations involving critical growth with two parameters, Nonlinear Anal. 132 (2016) 95–114. [18] A.S. Tersenov, The one dimensional parabolic p(x)−Laplace equation, Nonlinear Differ. Equ. Appl. 23 (2016) 1–11. [19] Z. Wu, J. Zhao, J. Yin, H. Li, Nonlinear Diffusion Equations, Word Scientific Publishing, Singapore, 2001. [20] Z. Li, B. Yan, W. Gao, Existence of solutions to a parabolic p(x)-Laplace equation with convection term via Linfinity estimates, Electron. J. Differ. Equ. 46 (2015) 1–21. [21] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl. 361 (2010) 371–391. [22] M.M. Bokalo, O.M. Buhrii, R.A. Mashiyev, Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonlinear Evol. Equ. Appl. 6 (2013) 67–87. [23] P. Kloeden, J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal. 13 (2014) 2543–2557.
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Available online at www.sciencedirect.com
ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde
Well-posedness problem of an anisotropic parabolic equation Huashui Zhan a , Zhaosheng Feng b,∗ a School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China b Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Received 16 October 2018; revised 24 April 2019; accepted 15 August 2019
Abstract In this paper, we are concerned with well-posedness of an anisotropic parabolic equation with the convection term. When some diffusion coefficients are degenerate on the boundary ∂ and the others are positive on , we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation. A new concept, the general characteristic function of the domain , is introduced and applied. The existence and stability of the solutions is established under the given partial boundary value conditions. © 2019 Elsevier Inc. All rights reserved. MSC: 35B35; 35G31; 35K55 Keywords: Anisotropic parabolic equation; Characteristic function; Partial boundary value condition; Stability
1. Introduction As we know, the anisotropic parabolic equation ut =
N N ∂ ∂bi (u, x, t) , (x, t) ∈ QT , ai (x)|uxi |pi −2 uxi + ∂xi ∂xi i=1
* Corresponding author.
E-mail address: [email protected] (Z. Feng). https://doi.org/10.1016/j.jde.2019.08.014 0022-0396/© 2019 Elsevier Inc. All rights reserved.
i=1
(1.1)
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plays an important role in mathematical modelling of various physical processes, such as flows of incompressible turbulent fluids, gases in pipes and filtration in glaciology [1–3]. Here, pi > 1, ai (x) ∈ C(), i ∈ {1, 2, · · · , N }, QT = × (0, T ), and is a bounded domain with a smooth boundary ∂. We consider equation (1.1) under the initial value condition: u(x, 0) = u0 (x), x ∈ ,
(1.2)
and the partial boundary value condition: u(x, t) = 0, (x, t) ∈ 1 × (0, T ),
(1.3)
where 1 is a relatively open subset of ∂ to be determined. Set 2 = ∂ \ 1 , and let p0 = min{p1 , p2 , · · · , pN−1 , pN } > 1, p 0 = max{p1 , p2 , · · · , pN−1 , pN }. Assume that ai (x) > 0, i ∈ {1, 2, · · · , N } when x ∈ , and ai1 (x) ≥ c1 > 0, ai2 (x) ≥ c2 > 0, · · · , aik (x) ≥ ck > 0, x ∈ , aj1 (x) = 0, aj2 (x) = 0, · · · , ajl (x) = 0, x ∈ ∂,
(1.4)
where {i1 , i2 , · · · , ik , j1 , j2 , · · · , jl } is a permutation of {1, 2, · · · , N } and ci s are positive constants. For a smooth bounded domain ⊂ RN , the function φ(x) is said to be a general characteristic function of if it is a nonnegative continuous function and satisfies φ(x) > 0, x ∈ and φ(x) = 0, x ∈ ∂, and |φxir (x)|pir
[φ(x)]pir −1 is bounded when r = 1, 2, · · · , k. The main feature which distinguishes this paper from other related works lies in the fact that we propose an explicit formula of 1 in terms of a general characteristic function of : |φxir (x)|pir 1 = x ∈ ∂ : = 0, r = 1, 2, · · · , k . (1.5) [φ(x)]pir −1 For example, if a1 (x)|x∈∂ = 0, ai (x) > 0, x ∈ , i ≥ 2, we can choose φ(x) = a1 (x), and then (1.5) becomes
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a1x pi 1 = x ∈ ∂ : a1 i = 0, i ≥ 2 . a1
3
(1.6)
Before summarizing our main results, let us briefly introduce the definitions of weak solutions. Definition 1. A function u(x, t) is said to be a weak solution of equation (1.1) with the initial value (1.2), provided that p 0 0 u ∈ L∞ (QT ), ai (x) uxi i ∈ L1 (QT ), ut ∈ Lp 0, T ; W −1,p () ,
where p 0 =
p0 , p 0 −1
and for any function ϕ ∈ C01 (QT ) there holds
N
p −2 < ut , ϕ > + ai (x) uxi i uxi ϕxi + bi (u, x, t) · ϕxi dxdt = 0. i=1 Q
T
Here, (1.2) is true in the sense of |u(x, t) − u0 (x)|dx = 0.
lim
t→0
(1.7)
Definition 2. The function u(x, t) is said to be a weak solution of equation (1.1) with the initialboundary conditions (1.2) and (1.3), if u(x, t) satisfies Definition 1 and the partial boundary value condition (1.3) is satisfied in the sense of trace. Theorem 1.1. Suppose that bi (u, x, t) is a C 1 function on R × × [0, T ] and u0 ∈ L∞ (), |∇u0 |p ∈ L1 (). 0
Then equation (1.1) with the initial value (1.2) has a weak solution. Moreover, when 1 ≤ s ≤ l − p 1−1 and ajs js (x)dx < ∞, then the initial-boundary value problem (1.1)-(1.2)-(1.3) has a weak solution. Theorem 1.2. Let u(x, t) and v(x, t) be two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively, and with the same partial boundary value condition (1.3). Suppose that φ(x) is a general characteristic function of and 1 is given by (1.5). If ⎛ l 1⎜ ⎝ η s=1
⎞ ⎟ ajs (x)|φxjs (x)|pjs dx ⎠
1 pjs
≤ c,
(1.8)
\η
where η = {x ∈ : φ(x) > η}, and 1 p
|bi (u, x, t) − bi (v, x, t)| ≤ cai i |u − v|, i = 1, 2, · · · , N,
(1.9)
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then we have
|u(x, t) − v(x, t)|dx ≤ c
|u0 (x) − v0 (x)|dx.
(1.10)
Here and in the sequel, c may have different values in different formulae. When b = {bi (u, x, t)} is independent of the diffusion coefficients {ai (x)}, i.e. without condition (1.9), we have the following result regarding the stability of weak solutions of equation (1.1). Theorem 1.3. Suppose that is a normal smooth boundary, and u(x, t) and v(x, t) are two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively, and with the same partial boundary value condition (1.3), where 1 is given by (1.5). Then the stability (1.10) of weak solutions of equation (1.1) is true, provided that condition (1.8) holds and ∂bi (s, x, t) ∂bi (s, x, t) ≥ 0 or ≤ 0 for i ≥ 1. ∂s ∂s If the condition ∂bi (s,x,t) ≥ 0 or ∂bi (s,x,t) ≤ 0 in Theorem 1.3 becomes invalid, the stability ∂s ∂s of weak solutions of equation (1.1) under the partial boundary value condition (1.5) can still be achieved. For example and for simplicity, here we only consider the case of l = 1: a1 (x) = 0, x ∈ ∂,
(1.11)
while ai (x) ≥ ci > 0 for i > 1. The discussions for other cases can be processed in an analogous manner. Theorem 1.4. Let be a normal smooth boundary, and u(x, t) and v(x, t) be two solutions of equation (1.1) with the initial values u0 (x) and v0 (x) respectively. The partial boundary value condition (1.3) holds while 1 is given by (1.6). Suppose that ai (x) ≥ ci > 0 for i > 1, and a1 (x) satisfies (1.11) and a1xi pi ≤ c, a1 (x) a (x) 1
⎛ 1⎜ ⎝ η
⎞ ⎟ a1 (x)|a1x1 (x)|p1 dx ⎠
1 p1
≤ c,
(1.12)
η
where c is a constant. If there exists a function g1 (x) such that |b1 (·, x, t)| ≤ cg1 (x),
(1.13)
|bi (u, x, t) − bi (v, x, t))| ≤ c|u − v|, i ≥ 2,
(1.14)
q1 1 (a1 (x))− p1 g1 (x) dx < ∞,
(1.15)
and
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then the stability (1.10) of weak solutions is true, where a1x1 (x) = 1 . η = {x ∈ : a1 (x) > η}, and q1 = p1p−1
5
∂a1 (x) ∂x1 ,
a1xi (x) =
∂a1 (x) ∂xi ,
It is notable that the expression of 1 = φ , as given by (1.5), depends on the choice of the general characteristic function φ(x). For a smooth domain ⊂ RN , since the general characteristic function φ(x) is not unique, if one chooses a different general characteristic function φ1 (x), the corresponding partial boundary φ1 might be different from φ . Thus the optimal partial boundary 0 in (1.3), should satisfy 0 ⊆ {φ }. We wish to make a conjecture that 0 =
φ .
φ
Whether this conjecture is true or false, the proof is a challenging problem worthy of being investigated in the future. The idea and the technique described in this study are conveniently generalized to the other parabolic equations (e.g. [4–18]), even to those with variable exponents and nonlinearity (e.g. [19–23]). The remainder of this paper is structured as follows. In Section 2, we consider the existence of weak solutions of equation (1.1). Sections 3 and 4 are devoted to the stability of weak solutions of equation (1.1) under the proposed partial boundary condition. In Section 5, we briefly give a comment on the construction of the explicit formula of 1 . 2. Existence of weak solutions Let us recall the approximate problem N p −2 0 uεt − εdiv |∇uε |p −2 ∇uε + ai (x) uεxi i uεxi i=1
−
N ∂bi (uε , x, t) i=1
∂xi
xi
(2.1)
= 0, (x, t) ∈ QT ,
uε (x, t) = 0, (x, t) ∈ ∂ × (0, T ),
(2.2)
uε (x, 0) = uε,0 (x), x ∈ ,
(2.3)
p 0 where uε,0 ∈ C0∞ (), |uε,0 |L∞ () ≤ |u0 |L∞ () , and ∇uε,0 is uniformly convergent to 0 |∇u0 (x)|p in L1 (). It is well-known that the above problem has a unique solution uε ∈ 0
1,p 0
Lp (0, T ; W0
()) [19].
− p 1−1 Lemma 2.1. [5] If ai i (x)dx < ∞, and u is a weak solution of equation (1.1) with the initial value (1.2). Then for any given t ∈ [0, T ) there holds
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|uxi |dx ≤ c, i = 1, 2, · · · , N,
and the trace of u on the boundary ∂ can be defined in the traditional way. − p 1−1 In view of condition (1.4), the inequality ai i (x)dx < ∞ given in Lemma 2.1 can be − p 1−1 replaced by ajs js (x)dx < ∞ for 1 ≤ s ≤ l. Proof of Theorem 1.1. By the maximum principle [19] we know that uε L∞ (QT ) c.
(2.4)
We multiply both sides of equation (2.1) by uε and integrate it over Qt = × [0, t] (t < T ). Since t
1 uε uεt dxdt = 2
0
t
∂u2ε dxdt ∂t
0
1 = 2
u2ε (x, t)dx
1 − 2
u2ε (x, 0)dx
and N ∂bi (uε , x, t) − uε dx ∂x i i=1 N ∂uε bi (uε , x, t) dx = ∂xi i=1 uε N uε N ∂ ∂ bi (s, x, t)dsdx − bi (s, x, t)dsdx = ∂xi ∂xi i=1
0
uε N ∂ bi (s, x, t)dsdx = ∂xi i=1
i=1 0
0
≤c, we have 1 2
u2ε (x, t)dx + ε
0
|∇uε |p dxdt + Qt
N i=1 Q t
ai (x)|uεxi |pi dxdt c, ∀t < T .
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This implies that ai (x)|uεxi |pi dxdt ≤ c
(2.5)
QT
and
0
|∇uε |p ≤ c.
ε
(2.6)
QT 0
1,p 0
For any v ∈ Lp (0, T ; W0
()) and v
1,p 0
0
Lp (0,T ;W0
have
())
= 1, by Young’s inequality, we
pi −2 ai (x)|uεxi | uεxi vxi dxdt QT ai (x) |uεxi |pi + |vxi |pi dxdt ≤c QT
≤c
0 ai (x) |uεxi |pi + |vxi |p + 1 dxdt
QT
ai (x)|uεxi |pi dxdt + c
≤c QT
≤c, and ⎡ ⎢ | < uεt , v > | ≤ c ⎣ε
0
|∇uε |p dxdt +
N i=1 Q
QT
⎤ ⎥ ai (x)|uεxi |pi dxdt + 1⎦
T
≤ c. Then, we obtain uεt
0
0
Lp (0,T ;W −1,p ()
≤ c.
(2.7)
For any ϕ ∈ C01 () and 0 ≤ ϕ ≤ 1, it is easy to see that (ϕuε )t
0
0
Lp (0,T ;W −1,p ())
≤ c.
(2.8)
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For a fixed s such that s > 0
N 2
0
+ 1, it has H0s () → W 1,p (), which implies that
W −1,p () → H −s (). As a result, we have (ϕuε )t
0
≤c
(2.9)
≤ c.
(2.10)
Lp (0,T ;H −s ())
and ϕuε 1,p0
Note that W0
0
1,p0
Lp (0,T ;W0
())
() → Lp0 () → H −s (), one can employ Aubin’s compactness theorem 0
[7,8] to derive that ϕuε → ϕu strongly in Lp (0, T ; Lp0 ()). Thus, ϕuε → ϕu a.e. in QT . In particular, due to the arbitrariness of ϕ, we have uε → u a.e. in QT . − → In view of (2.4)-(2.7), there exists a function u and an n−dimensional vector function ζ = (ζ1 , · · · , ζn ) such that p 0 −2
ε|∇uε |
∇uε 0 in L
p0 p 0 −1
(QT ),
pi
u ∈ L∞ (QT ), |ζi | ∈ L pi −1 (QT ), uε u, weakly-star in L∞ (QT ), bi (uε , x, t) → bi (u, x, t) a.e. in QT , p
i uεxi uxi in Lloc (QT ),
and pi p −2 ai (x) uεxi i uεxi ζi in L pi −1 (QT ).
By a similar argument to the usual evolutionary p−Laplacian [19], we can obtain (1.7) and −−−−−−−−→ − → p−2 ζ · ∇ϕdxdt a |∇u| ∇u · ∇ϕdxdt = QT
QT
for any given function ϕ ∈ C01 (QT ), where −−−−−−−−→ a |∇u|p−2 ∇u = ai (x)|uxi |pi −2 uxi . The desired result follows Lemma 2.1 immediately.
2
(2.11)
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3. Stability under partial boundary condition For small η > 0, let s Sη (s) =
hη (τ )dτ, hη (s) = 0
2 |s| . 1− η η +
(3.1)
Clearly, hη (s) ∈ C(R), and hη (s) ≥ 0, | shη (s) |≤ 1, | Sη (s) |≤ 1, lim Sη (s) = sgns. η→0
If we denote Hη (s) =
s 0
(3.2)
Sη (s)ds, then
lim Sη (s) = sgn(s), lim Hη (s) = |s|, s ∈ (−∞, +∞),
(3.3)
lim hη (s)s = 0.
(3.4)
η→0
η→0
and η→0
Proof of Theorem 1.2. Let u(x, t) and v(x, t) be two weak solutions of equation (1.1) with the initial values u(x, 0) and v(x, 0) respectively, and with the partial homogeneous boundary value condition: u(x, t) = v(x, t) = 0, (x, t) ∈ 1 × [0, T ).
(3.5)
Let η = {x ∈ : φ(x) > η} and φη (x) =
1, φ(x) η ,
if x ∈ η , if x ∈ \ η .
φx (x)
Obviously, φηxi = iη when x ∈ \ η . Otherwise, it is identically zero. Choosing χ[τ,s] φη Sη (u − v) as the test function, we deduce that s φη Sη (u − v)
∂(u − v) dxdt ∂t
τ
+
N s i=1 τ
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi (uxi − vxi )hη (u − v)φη (x)dxdt
(3.6)
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10
+
k s
air (x) |uxir |pir −2 uxir − |vxir |pir −2 vxir Sη (u − v)φηxir dxdt
r=1 τ
+
l s
ajr (x) |uxjr |pjr −2 uxjr − |vxjr |pjr −2 vxjr Sη (u − v)φηxjr dxdt
r=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt
i=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
i=1 τ
= 0.
(3.7)
For the first term on the left hand side of (3.7), following [4, Lemma 3.1] we find s φη (x)Sη (u − v)
lim
η→0
τ
(3.8)
|u − v|(x, s)dx −
=
∂(u − v) dxdt ∂t
|u − v|(x, τ )dx.
For the second term on the left hand side of (3.7), it is easy to see that
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi (uxi − vxi )hη (u − v)φη (x)dx ≥ 0.
(3.9)
To evaluate the third term on the left hand side of (3.7), in consideration of (1.3) and (1.6), by a straightforward calculation we obtain ai (x) |ux |pir −2 ux − |vx |pir −2 vx Sη (u − v)φηx dx r ir ir ir ir ir air (x) |uxir |pir −2 uxir − |vxir |pir −2 vxir Sη (u − v)φηxir dx = \η air (x) |uxir |pir −1 + |vxir |pir −1 |Sη (u − v)φηxir |dx ≤ \η
≤ \η
! pir −1 pir 1 air (x)φ(x) |uxir |pir −1 + |vxir |pir −1 η
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·(air (x)) pir ⎛
dx ⎞
⎜ ≤⎝
1 1 pir [φ(x)]xir Sη (u − v) η pir −1 [φ(x)] pir
11
⎟ 1 φ(x)air (x) |uxir |pir + |vxir |pir dx ⎠ η
1 qir
\η
⎛ ⎜ ·⎝
⎞
|pir
\η
⎛ ⎜ ≤ c⎝
|[φ(x)]xir 1 ⎟ dx ⎠ air (x)|Sη (u − v)|pir η [φ(x)]pir −1 ⎞
⎟ air (x) |uxir |pir + |vxir |pir dx ⎠
1 pir
1 qir
\η
⎛ ⎜1 ·⎝ η
⎞
|Sη (u − v)|pir \η
where qir = Denote
|[φ(x)]xir
|pir
[φ(x)]pir −1
⎟ dx ⎠
1 pir
(3.10)
,
pir pir −1 .
η1 = {x ∈ \ η : dist(x, 2 ) > dist(x, 1 )}, η2 = {x ∈ \ η : dist(x, 2 ) ≤ dist(x, 1 )}. Then we have 1 η
|Sη (u − v)|pir | \η
1 ≤ η
|Sη (u − v)|pir
η1
1 + η
|Sη (u − v)|pir η2
|[φ(x)]xir |pir
dx
|[φ(x)]xir |pir
dx
[φ(x)]pir −1
[φ(x)]pir −1
|[φ(x)]xir |pir [φ(x)]pir −1
dx.
By making use of u(x, t) = v(x, t) = 0, (x, t) ∈ 1 × (0, T ), we have
(3.11)
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1 η→0 η
|Sη (u − v)|pir
lim
η1
=
sign(u − v)
|[φ(x)]xir |pir [φ(x)]pir −1
|[φ(x)]xir |pir
1
[φ(x)]pir −1
dx (3.12)
d
=0. Moreover, by using the identity |[φ(x)]xir |pir
[φ(x)]pir (x) −1
= 0, x ∈ 2 ,
we derive that 1 η→∞ η
|Sη (u − v)|pir
lim
η2
1 ≤ lim η→∞ η = 2
η2
|[φ(x)]xir |pir [φ(x)]pir −1
|(φ(x))xir |pir [φ(x)]pir −1
|[φ(x)]xir |pir [φ(x)]pir −1
dx
dx (3.13)
d
=0. From (3.10)-(3.13), we obtain pir −2 pir −2 lim air (x) |uxir | uxir − |vxir | vxir Sη (u − v)φηxir dx = 0. η→0
(3.14)
To evaluate the fourth term on the left hand side of (3.7), by a direct calculation we deduce that aj (x) |ux |pjr −2 ux − |vx |pjr −2 vx Sη (u − v)φηx dx r jr jr jr jr jr pjr −2 pjr −2 ajr (x) |uxjr | uxjr − |vxjr | vxjr Sη (u − v)φηxjr dx = \η ajr (x) |uxjr |pjr −1 + |vxjr |pjr −1 [φ(x)]xjr Sη (u − v)|dx ≤ \η
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⎛ ⎜ ≤ c⎝
⎞
⎟ ajr (x) |uxjr |pjr + |vxjr |pjr dx ⎠
13
1 qjr
\η
⎛ 1⎜ · ⎝ η
⎞ ⎟ ajr (x)|[φ(x)]xi |pjr dx ⎠
1 pjr
.
\η
Using condition (1.8), we have
pjr −2 pjr −2 lim ajr (x) |uxjr | uxjr − |vxjr | vxjr Sη (u − v)φηxjr dx = 0. η→0
(3.15)
To evaluate the fifth term in (3.7), in view of condition (1.9), it follows Hölder’s inequality that
≤c
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx 1 pi a (u − v)x φη (u − v)hη (u − v) dx i i
⎛ ⎞1 ⎛ ⎞1 pi qi p p q i i i ≤c ⎝ ai (x)(|uxi | + |vxi | )dx ⎠ ⎝ |(u − v)hη (u − v)| dx ⎠ ,
where qi =
pi pi −1 .
By (3.2), we have
s [bi (u, x, t) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt = 0.
lim
η→0 τ
(3.16)
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For the sixth term in (3.7), using condition (1.9) again leads to N lim [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx η→0 i=1 N [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx = lim η→0 i=1 \ η ≤ lim
η→0
≤ lim
η→0
N 1
1 p
ai i |φxi (x)||Sη (u − v)(u − v)|dx
η
i=1
k 1
η→0
1 p
air ir |φxir (x)||Sη (u − v)(u − v)|dx
η
r=1
+ lim
(3.17)
\η
\η
l 1 r=1
1 p
ajrjr |φxjr (x)||Sη (u − v)(u − v)|dx.
η \η
By the partial homogeneous boundary condition (1.3), we have
lim
k 1
η→0
r=1
1 p
air ir |φxir (x)||Sη (u − v)(u − v)|dx
η \η
⎞
⎛ k ⎜1 ≤ lim ⎝ η→0 η r=1
⎡
⎢1 ·⎣ η
|φxir
\η
(x)|pir
|φ(x)|pir −1
air |φ(x)|
pir −1 pir
⎟ |u − v|pir dx ⎠
r ppi−1 ir
1 pir
⎤ pir −1 |Sη (u − v)|
pir −1 pir
⎥ dx ⎦
\η
⎛ ≤ c lim
η→0
k r=1
⎜1 ⎝ η
\η
⎛
⎜ = c⎝
1
⎞ |φxir
(x)|pir
|(φ(x))|pir −1
⎟ |u − v|pir dx ⎠
⎞ |φxir
(x)|pir
|φ(x)|pir −1
⎟ |u − v|pir d⎠
1 pir
1 pir
pir
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⎛
⎞
|φxir
⎜ +c ⎝
2
(x)|pir
|φ(x)|pir −1
⎟ |u − v|pir d⎠
15
1 pir
= 0. In view of condition (1.8), we get
lim
l 1
η→0
s=1
1 p
ajsjs |φxjs (x)||Sη (u − v)(u − v)|dx
η \η
⎛ 1⎜ ≤ lim ⎝ η→0 η
⎞
pjs
ajs (x)|φxjs (x)|
⎟ dx ⎠
1 pjs
⎝
\η
dx ⎠
pjs
(3.18)
⎞ pjs −1
η→0
|Sη (u − v)(u − v)|
pjs pjs −1
⎛ ≤c lim ⎝
⎞ pjs −1
⎛
|Sη (u − v)(u − v)|
pjs pjs −1
dx ⎠
pjs
⎛
≤c ⎝
⎞ pjs −1 pjs
|u − v|dx ⎠
.
In accordance with (3.17)-(3.18), we derive that s [bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
lim
η→0 τ
(3.19)
⎞ pi −1 ⎛ s pi ⎠ ⎝ |u − v|dxdt . ≤c τ
Combining (3.8)-(3.9), (3.14)-(3.16) with (3.19), and letting η → 0 in (3.7) leads to
|u(x, s) − v(x, s)|dx
⎞ pi −1 ⎛ s pi N ⎠ ⎝ |u(x, τ ) − v(x, τ )|dx + c |u − v|dxdt . (3.20)
i=1
τ
Let κ(s) = |u(x, s) − v(x, s)|dx. Without s loss of generality, we assume that there exists τ ∈ [0, T ) and κ(τ ) > 0. For any s > τ and τ k(t)dt > 0, we denote that τ0 τ0 = max{t ∈ [τ, s], κ(t) > 0} and
k(t)dt = c1 . τ
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16
Then, τ < τ0 ≤ s, and s
τ0 k(t)dt ≥
τ
k(t)dt = c1 . τ
Since u, v ∈ L∞ (QT ), there exist constants Ci > 0 such that c
s
k(t)dt
s
τ
τ
pip−1 i
k(t)dt
≤
c
s τ
k(t)dt c1
pip−1 i
≤ Ci = Ci (c, c1 , T , q), i = 1, 2, · · · , N.
(3.21)
By (3.20) and (3.21), we have " κ(s) − κ(τ ) ≤
N
# s Ci + c
i=1
k(t)dt. τ
Using Gronwall’s inequality, we get
|u(x, s) − v(x, s)| dx c
|u(x, τ ) − v(x, τ )| dx.
Due to the arbitrariness of τ , we obtain
|u(x, s) − v(x, s)| dx c
|u0 (x) − v0 (x)| dx.
2
Proof of Theorem 1.3. From the proof of Theorem 1.2, we see that (3.8)-(3.9), and (3.14) hold too. Let us evaluate the last two formulas involving the convection terms on the left hand side of (3.7). We start with ≤c
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx 1 −1 pi a (u − v)x a pi φη (u − v)hη (u − v) dx i i i
⎞1 ⎛ ⎞1 ⎛ pi qi qi −1 pi pi pi ⎠ ⎠ ⎝ ⎝ ai (x) |uxi | + |vxi | dx , ≤c ai (u − v)hη (u − v) dx
where
(3.22)
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17
⎞1 qi qi −1 pi ⎝ a (u − v)hη (u − v) dx ⎠ i ⎛
⎛
=⎝
−1 pi −1
ai
⎞1
(3.23)
qi
|(u − v)hη (u − v)|qi dx ⎠ .
If {x ∈ : |u − v| = 0} is a measure zero set, by |(u − v)hη (u − v)| ≤ c and
−1 pi −1
ai
(x)dx ≤ c,
we get qi −1 pi a (u − v)hη (u − v) dx i η→0 lim
−1 pi −1
≤c
ai
(x)dx
(3.24)
{:|u−v|=0}
=0. If {x ∈ : |u − v| = 0} has a positive measure, by |(u − v)hη (u − v)| ≤ c and
−1 pi −1
ai
(x)dx ≤ c,
it follows Lebesgue’s dominated convergence theorem that qi −1 pi a (u − v)hη (u − v) dx = 0. i η→0 lim
(3.25)
By (3.22)-(3.25), we derive that lim [bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dx = 0. η→0
Meanwhile, if
∂bi (s,x,t) ∂s
≥ 0 for i ≥ 1, we denote D1η = {x ∈ \ η : φxi (x) < 0}.
To evaluate the last term on the left hand side of (3.7), we re-write it as
(3.26)
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18
−
[bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx
[bi (u, x, t) − bi (v, x, t)]φηxi sign(u − v)dx
=−
(3.27)
[bi (u, x, t) − bi (v, x, t)]φηxi [Sη (u − v) − sign(u − v)]dx.
−
Note that lim D1η ⊆ 1 .
η→0
We then have − lim
[bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx
η→0
1 η→0 η
= − lim ≤ lim
η→0
≤c
c η
bi (ζ, x, t)(u − v)(φ(x))xi sign(u − v)dx
|u − v|[−(φ(x))xi ]dx
(3.28)
D1η
|u − v|d
1
=0, where bi (ζ, x, t) = ∂bi (u,x,t) |u=ζ . ∂u If ∂bi (s,x,t) ≤ 0 for i ≥ 1, we denote ∂s D1η = {x ∈ \ η : φxi (x) > 0}. Proceeding in a similar way, we can achieve (3.28) too. Let ρ(x) = dist(x, ∂). Since is a smooth domain, there exists a finite open cover of ∂: ∂ ⊂
$
Uα .
α
Denote αη = {x ∈ \ η : x ∈ Uα } and
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α = αη
19
∂.
Using the flattening technique and the L’Hospital rule leads to lim [bi (u, x, t) − bi (v, x, t)]φηxi [Sη (u − v) − sign(u − v)]dx η→0 [bi (u, x, t) − bi (v, x, t)]φηx [Sη (u − v) − sign(u − v)] dx lim ≤c i α
≤c
α
=c
α
1 + η
η→0 αη
lim
η→0 α
η |u − v||Sη (u − v) − sign(u − v)|dρ 0
%
lim
η→0 α
η
1 d η
|Sη (u − v) − sign(u − v)||u − v| (3.29)
⎤
|u − v||hη (u − v)|dρ ⎦ d
0
=c
α
=c
α
η
1 lim η→0 η lim
|u − v||hη (u − v)|dρd α 0
η→0 α
|u − v||hη1 (u − v)|d
=0, where η1 ∈ [0, η]. By virtue of (3.27)-(3.29), we obtain lim − [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx = 0. η→0
The rest is similar to the one as we show in the proof of Theorem 1.2, and thus we arrive at (1.10). 2 4. Proof of Theorem 1.4 Let η = {x ∈ : a1 (x) > η}, and φη (x) =
1, 1 η a1 (x),
if x ∈ η , if x ∈ \ η .
(4.1)
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20
Choosing χ[τ,s] φη Sη (u − v) as the test function, we have s φη Sη (u − v)
∂(u − v) dxdt ∂t
τ
+
N s
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi )(uxi − vxi hη (u − v)φη (x)dxdt
i=1 τ
s
a1 (x) |ux1 |p1 −2 ux1 − |vx1 |p1 −2 vx1 Sη (u − v)φηx1 dxdt
+ τ
+
N s
ai (x) |uxi |pi −2 uxi − |vxi |pi −2 vxi Sη (u − v)φηxi dxdt
(4.2)
i=2 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)](u − v)xi φη hη (u − v)dxdt
i=1 τ
+
N s
[bi (u, x, t)) − bi (v, x, t)]φηxi Sη (u − v)dxdt
i=1 τ
=0. As we show in the preceding section, for the first two terms of the left hand side of (4.2), formulas (3.8) and (3.9) still hold. To evaluate the third term of the left hand side of (4.2), by (1.12) we have p1 −2 p1 −2 lim a1 (x)(|ux1 | ux1 − |vx1 | vx1 )Sη (u − v)φηx1 dx η→0 1 a1 (x) |ux1 |p1 −1 + |vx1 |p1 −1 |(a1 (x))x1 Sη (u − v)|dx ≤ lim η→0 η \η
⎛ ⎜ ≤ lim ⎝
η→0
⎞1
a1 (x) |ux1 |
p1
\η
= 0. Meanwhile, we can deduce that
+ |vx1 |
p1
⎟ dx ⎠
q1
⎛ 1⎜ ⎝ η
\η
⎞ ⎟ a1 (x)|(a1 (x))x1 |p1 dx ⎠
(4.3) 1 p1
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21
N pi −2 pi −2 ai (x) |uxi | uxi − |vxi | vxi Sη (u − v)φηxi dx i=2 \ η ≤
N c i=2
≤c
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
η \η
N c i=2
η
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
\η
→0, as η → 0. This is due to the partial boundary value condition (1.3) with the formula (1.6):
c η
|uxi |pi −1 + |vxi |pi −1 (a1 (x))xi Sη (u − v) dx
\η
1 ≤ η
pi −1 pi
a1
pi −1
|uxi |
pi −1
+ |vxi |
(a1 (x))x Sη (u − v) i pi −1 pi
dx
a1
⎛
⎞ pi −1
⎜1 ≤c ⎝ η
a1 (x) |uxi | + |vxi | pi
pi
⎟ dx ⎠
pi
\η
⎛ ⎜1 ·⎝ η
⎜1 ≤c ⎝ η
|a1xi |pi p −1
\η
⎛
⎞1
a1 i
|a1xi |pi a1 i
⎛
⎜ →c ⎝
1
⎟ |Sη (u − v)| dx ⎠
|pi
p −1 a1 i
pi
pi
⎞1 |a1xi
pi
⎞1 p −1
\η
⎟ |Sη (u − v)| dx ⎠ pi
⎟ |sign(u − v)|pi d⎠
pi
⎛ ⎞1 pi p i ⎜ |a1xi | ⎟ pi +c⎝ |sign(u − v)| d⎠ p −1 a1 i 2
=0. For the fifth term of the left hand side of (4.2), it follows Hölder’s inequality that N [bi (u, x, t)) − bi (v, x, t)](u − v)x φη hη (u − v)dx i i=1
(4.4)
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22
≤ [b1 (u, x, t)) − b1 (v, x, t)](u − v)x1 φη hη (u − v)dx N [bi (u, x, t)) − bi (v, x, t)](u − v)x φη hη (u − v)dx + i i=2 ≤ c |g1 (x)(u − v)x1 φη hη (u − v)|dx
+
N
|(u − v)xi φη (u − v)hη (u − v)|dx.
(4.5)
i=2
Using (1.15), we have |g1 (x)(u − v)x1 φη hη (u − v)|dx
⎛ ⎞1 ⎛ ⎞1 p1 q1 1 − p1 p1 q1 p ⎝ ⎠ ⎠ ⎝ 1 ≤c a1 (x) |ux1 | + |vx1 | dx |(a1 (x)) g1 (x)(u − v)hη (u − v)| dx
→0, as η → 0. (4.6) Let us recall (1.14): |bi (u, x, t) − bi (v, x, t)| ≤ c|u − v|, i ≥ 2, and thus get N
|(u − v)xi φη (u − v)hη (u − v)|dx
i=2
⎞1 ⎛ ⎞1 ⎛ pi qi N ⎝ ≤c |uxi |pi + |vxi |pi dx ⎠ ⎝ |(u − v)hη (u − v)|qi dx ⎠ i=2
→0, as η → 0. For the last term of the left hand side of (4.2), we have N [bi (u, x, t) − bi (v, x, t)]φηx Sη (u − v)dx lim i η→0 i=1
(4.7)
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23
= lim [bi (u, x, t) − bi (v, x, t)]φηxi Sη (u − v)dx η→0 i=1 \ η N
≤ lim
η→0
N c i=1
\η
⎛ ≤ lim
η→0
N i=1
|(a1 (x))xi ||Sη (u − v)(u − v)|dx
η
1⎜ ⎝ η
⎞1 ⎛ ⎟ |(a1 (x))xi |pi dx ⎠
pi
⎜ ⎝
\η
⎞ p1 −1
|Sη (u − v)(u − v)|
p1 p1 −1
⎟ dx ⎠
p1
\η
→ 0, as η → 0.
(4.8)
Now, we let η → 0 in (4.2). By (4.3)-(4.8), we have
|u(x, s) − v(x, s)|dx
|u(x, τ ) − v(x, τ )|dx.
Due to the arbitrariness of τ , we obtain |u(x, s) − v(x, s)|dx |u0 (x) − v0 (x)|dx.
5. Explicit formula of 1 Let us recall the explicit expression of 1 as constructed in (1.5), i.e. 1 = x ∈ ∂ :
|φxir (x)|pir
[φ(x)]pir −1
= 0, r = 1, 2, · · · , k ,
where φ(x) ∈ C 1 () is a nonnegative function satisfying φ(x) > 0, x ∈ and φ(x) = 0, x ∈ ∂,
(5.1)
such that |φxir (x)|pir
[φ(x)]pir −1 is bounded for every r = 1, 2, · · · , k. Since the boundary ∂ is a smooth (N − 1)-dimensional manifold, the distance ρ(x) = dist(x, ∂) is smooth when x is near the boundary ∂. If we choose φ = ρ α (x), only when α > p 0 , then |φxir (x)|pir
[φ(x)]pir −1
= 0, r = 1, 2, · · · , k.
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24
If pir = p for all 1 ≤ r ≤ k, we choose φ = ρ α (x) with α = p, then |φxir (x)|pir
[φ(x)]pir −1
=
|αρ α−1 ρxir |p = |αρxir |p ρ α(p−1)
is bounded. In fact, there always exists a nonnegative φ(x) ∈ C 1 () satisfying (5.1) such that is bounded. To see this, without loss of generality, we can assume that i1 ≤ i2 ≤ nonnegative φ(x) ∈ C 1 () satisfies (5.1) but φ
pik
|φxi (x)|pir r
[φ(x)]pir −1
|φxi (x)|pir r
[φ(x)]pir −1 · · · ≤ ik . If the
is unbounded, then the function φ1 =
satisfies (5.1) and pi
|φxirk (x)|pir
[φ(x)]pik (pir −1)
= 0, 1 ≤ r < k,
and pi
|φxikk (x)|pir [φ(x)]pik (pik − 1) is bounded. References [1] S.V. Antontsev, J.I. Diaz, S. Shmarev, Energy Methods for Free Boundary Problems, Applications to Nonlinear PDES and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 48, Birkhäuser, Boston, MA, 2002. [2] J.I. Diaz, J. Padial, Uniqueness and existence of a solution in BVt (q) space to a doubly nonlinear parabolic problem, Publ. Mat. 40 (1996) 527–560. [3] J.I. Diaz, F. Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (1994) 1085–1111. [4] H. Zhan, The uniqueness of the solution to the diffusion equation with a damping term, Appl. Anal. 98 (7) (2019), https://doi.org/10.1080/00036811.2017.1422725. [5] H. Zhan, Z. Feng, Solutions of evolutionary p(x)−Laplacian equation based on the weighted variable exponent space, Z. Angew. Math. Phys. 68 (2017) 1–17. [6] H. Zhan, Z. Feng, Existence of solutions to an evolution p−Laplacian equation with a nonlinear gradient term, Electron. J. Differ. Equ. 311 (2017) 1–15. [7] J.P. Aubin, Un théorˇeme de compacité, C. R. Acad. Sci. 256 (1963) 5042–5044. [8] J. Simon, Compact sets in the space Lp (0, t; B), Ann. Mat. Pura Appl. (4) 146 (1952) 65–96. [9] K. Lee, A. Petrosyan, J.L. Vazquez, Large time geometric properties of solutions of the evolution p−Laplacian equation, J. Differ. Equ. 229 (2006) 389–411. [10] K. Kobayasi, H. Ohwa, Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle, J. Differ. Equ. 252 (2012) 137–167. [11] H. Frid, Y. Li, A boundary value problem for a class of anisotropic degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal. 226 (2017) 975–1008. [12] J. Zhao, Existence and nonexistence of solutions for ut = div(|∇u|p−2 ∇u) + f (∇u, u, x, t), J. Math. Anal. Appl. 172 (1993) 130–146. [13] J. Yin, C. Wang, Evolutionary weighted p−Laplacian with boundary degeneracy, J. Differ. Equ. 237 (2007) 421–445. [14] S. Alkis Tersenov, S. Aris Tersenov, Existence of Lipschitz continuous solutions to the Cauchy-Dirichlet problem for anisotropic parabolic equations, J. Funct. Anal. 272 (2017) 3965–3986.
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25
[15] S. Lian, W. Gao, H. Yuan, C. Cao, Existence of solutions to an initial Dirichlet problem of evolutional p(x)−Laplace equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29 (2012) 377–399. [16] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (1) (2010) 204–228. [17] K. Ho, I. Sim, On degenerate p(x)−Laplacian equations involving critical growth with two parameters, Nonlinear Anal. 132 (2016) 95–114. [18] A.S. Tersenov, The one dimensional parabolic p(x)−Laplace equation, Nonlinear Differ. Equ. Appl. 23 (2016) 1–11. [19] Z. Wu, J. Zhao, J. Yin, H. Li, Nonlinear Diffusion Equations, Word Scientific Publishing, Singapore, 2001. [20] Z. Li, B. Yan, W. Gao, Existence of solutions to a parabolic p(x)-Laplace equation with convection term via Linfinity estimates, Electron. J. Differ. Equ. 46 (2015) 1–21. [21] S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl. 361 (2010) 371–391. [22] M.M. Bokalo, O.M. Buhrii, R.A. Mashiyev, Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonlinear Evol. Equ. Appl. 6 (2013) 67–87. [23] P. Kloeden, J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal. 13 (2014) 2543–2557.