The well-posedness of an anisotropic parabolic equation based on the partial boundary value condition

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R E S E A R C H

Open Access

The well-posedness of an anisotropic

parabolic equation based on the partial

boundary value condition

Huashui Zhan

*

*_{Correspondence:}

[email protected] School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, P.R. China

Abstract

Consider the anisotropic parabolic equation with the variable exponent

ut= N

i=1

(

ai(x)|uxi|

pi(x)–2_u

xi

)

xi,

withai(x),pi(x)∈C1(),pi(x) > 1,ai(x)≥0. If some of{ai(x)}are degenerate on the

boundary, a partial boundary value condition is imposed, the stability of weak solutions can be proved based on the partial boundary value condition.

MSC: 35K55; 35K92; 35K85; 35R35

Keywords: anisotropic parabolic equation; partial boundary value condition; the well-posedness

1 Introduction The equation

ut=div

a(x)|∇u|p(x)–_∇_u_, ₍_x_,_t₎_∈_Q

T, (.)

comes from the so-called electrorheological ﬂuids theory (see [, ]), where⊂RN _{is a}

bounded domain with smooth boundary∂,p(x) >  is a measurable function. Ifa(x)≡, there are many related papers to study equation (.) with the usual initial-boundary value conditions

u(x, ) =u(x), x∈, (.)

u(x,t) = , (x,t)∈∂×(,T), (.)

one can see [–] and the references therein.

If a(x) >  whenx∈ buta(x)|x∈∂= , then the stability of weak solutions can be

proved without the boundary value condition (.), provided that the diﬀusion coeﬃcient

a(x) satisﬁes some other restrictions. One can see our previous works [–].

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In this paper, we will consider an anisotropic parabolic equation of the type

ut= N

i=

ai(x)|uxi|pi(x)–uxi

xi, (x,t)∈QT. (.)

We denote that

p+_i =max x∈

pi(x), p–i =min x∈

pi(x)

for anyi∈ {, , . . . ,N}and denote that

p=min

p–_,p–_, . . . ,p_N–_–,p–_N, p> ,

p=maxp+_,p_+, . . . ,p+_N_–,p+_N.

Ifai(x)≡, the existence of a weak solution was proved in []. Also, one can refer to the

excellent papers [–].

LetI={i,i, . . . ,ik} ⊂ {, , . . . ,N},J={j,j, . . . ,jl} ⊂ {, , . . . ,N},k+l=N,I∩J=∅.

Not only we assume thatai(x)∈C(), and whenx∈,ai(x) > , but we also assume that

ai(x)≥c> , ai(x)≥c> , . . . , aik(x)≥ck> , x∈, (.)

aj(x) = , aj(x) = , . . . , ajl(x) = , x∈∂. (.)

Besides the initial value condition (.), instead of the usual boundary value condition (.), by assumptions (.)-(.), only a partial boundary value condition

u(x,t) = , (x,t)∈×(,T) (.)

should be imposed. To see that, let us give a simple example to show whatis. LetN= ,

p(x) =p(x)≡p(x), the domainbe a square,

=(x,x) :  <x< ,  <x< 

.

Consider the equation

ut=

∂ ∂x

a(x)|ux|

p(x)–_u

x

+ ∂ ∂x

|ux|

p(x)–_u

x

. (.)

Then we conjecture that

=

(x,x) :  <x< ,x= 

∪(x,x) :  <x< ,x= 

. (.)

This conjecture was proved in [] recently.

However, in general, it is diﬃcult to depict out the geometric character of. We have

tried to depict outby the Fichera function in [], but it seems not so successful. In this

short paper,

=

x∈∂:|( l

j=aj(x))x_ir|pir(x)

[l_j_=aj(x)]pir(x)–

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we will study the well-posedness of the equation basing on the partial boundary value condition (.). Also, we denote that

=∂\.

Deﬁnition . A functionu(x,t) is said to be a weak solution of equation (.) with the

initial value condition (.) if

u∈L∞(QT),

∂u

∂t ∈L

₍_Q

T), ai(x)|uxi|pi(x)∈L

,T;L(), (.)

and for any functionϕ∈C(QT),ϕ∈L∞(QT) andϕxi∈L

_(,_T_;_W,pi(x)

loc ()) such that

QT

∂u

∂t(ϕϕ) + N

i=

ai(x)|uxi|pi(x)–uxi(ϕϕ)xi

dx dt= . (.)

The initial value condition (.) is satisﬁed in the sense of

lim t→

u(x,t) –u(x)dx= . (.)

Besides, if the partial boundary value condition (.) is satisﬁed in the sense of the trace, then we say that there is a weak solution of the initial-boundary value problem (.)-(.)-(.).

In this paper, we ﬁrst study the existence of the weak solution.

Theorem . If p> ,ai(x)satisﬁes conditions(.), (.),

u∈L∞(), uxi∈L

pi(x)_(), _(.)

then there is a solution of equation(.)with the initial value(.).Moreover,if for every

≤r≤l, a–/(_j_r pjr(x)–)(x)dx<∞, then the initial-boundary value problem(.)-(.)

-(.)has a solution.

Here,Lpi(x)_{() is the variable exponent space, its deﬁnition is given in Section .} Secondly, we will study the stability of weak solutions to the initial-boundary value prob-lem (.)-(.)-(.).

Theorem . If u and v are two solutions of equation(.)with the same partial boundary

value condition(.)and with the initial values u(x),v(x),respectively,if l> and for

every≤r≤l,

n

\n

ajr(x)

_l

s=

ajs(x)

x_jr

p_jr(x)

dx

 p+ jr

≤c, (.)

then

u(x,t) –v(x,t)dx≤

u(x) –v(x)dx. (.)

Here,n={x∈:

l

s=ajs(x) >



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Ifl=  in Theorem ., without loss of generality, we may assume that

a(x) = , x∈∂, (.)

whileai(x) >  fori> . Then we have the following.

Theorem . If(.)is true and ai(x) > for i> ,u and v are two solutions of equation

(.)with the same partial boundary value condition(.)and with the initial values u(x),

v(x),respectively,if for every i≥

n

axi(x)

pi(x)

dx

 p+_i

≤c (.)

and

n

\n

a(x)ax(x)

p(x)

dx

 p+_

≤c, (.)

then the stability of weak solutions(.)is true. Here, ax(x) =

∂a(x)

∂x ,axi(x) =

∂a(x)

∂xi as

usual,n={x∈:a(x) >_n}.

At the end of the introduction, we would like to suggest that there are many papers devoted to the anisotropic elliptic equations, for examples, one can see [–] and the references therein. For example, Fu and Shan studied the problem of removable isolated singularities for elliptic equations with variable exponents in []. They gave a suﬃcient condition for removability of the isolated singular point for the equations inW,p(x)_().

Cencelj and Repov˘s studied the perturbation by a critical term and a superlinear subcrit-ical nonlinearity of a quasilinear elliptic equation containing a singular potential in []. By means of variational arguments and a version of the concentration-compactness prin-ciple in the singular case, they proved the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem. Konaté and Ouaro studied nonlinear anisotropic problems with bounded Radon diffuse measure and variable exponent in []. They proved the existence and uniqueness of an entropy solution. By the way, the definition of weak solutions and the method used in [] are different from the ones in this paper. Moreover, only the case when the domain is the

n-dimensional unit cube is considered in [], and the diﬀusion coeﬃcientai(x) =ai(xi) is

restricted only dependent on the single variablexi.

2 The existence

We ﬁrstly give some basic concepts about the exponent variable spaces. .Lp(x)() space.

Lp(x)() =

u:uis a measurable real-valued function,

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The spaceLp(x)_{() is equipped with the following Luxemburg norm:}

|u|_Lp(x)() =inf

λ>  :

u(_λx)p(x)dx≤ .

The space (Lp(x)_(),_{| · |}

Lp(x)₍₎) is a separable, uniformly convex Banach space. .W,p(x)_{() space.}

W,p(x)() =u∈Lp(x)() :|∇u| ∈Lp(x)(),

endowed with the following norm:

|u|_W,p(x)=|u|_Lp(x)₍₎+|∇u|_Lp(x)₍₎, ∀u∈W,p(x)().

We useW_,p(x)() to denote the closure ofC_∞() inW,p(x)_.

Lemma . ([–]) The spaces (Lp(x)_(),_{| · |}

Lp(x)₍₎), (W,p(x)(),| · |_W,p(x)₍₎) and

W_,p(x)()are reﬂexive Banach spaces.

Lemma .([]) If,for any given i∈ {, , . . . ,N},a–

 pi(x)–

i (x)dx<∞,then

|uxi|dx≤c. (.)

By this lemma, one can see that if ai(x) satisﬁes (.), (.) and if for every ≤r≤l,

a

–/(p_jr(x)–)

jr (x)dx<∞, then (.) is satisﬁed. Thus, we can deﬁne the trace ofuon the boundary∂.

Proof of Theorem. Consider the partially regularized equation

ut= N

i=

ai(x)|uxi|pi(x)–uxi

xi+εu, (x,t)∈QT (.)

with the initial boundary conditions

u(x, ) =uε(x), x∈, (.)

u(x,t) = , (x,t)∈∂×(,T). (.)

Here, we letuε(x)∈C∞() and strongly convergent tou(x) inW,p



 ().

Since ai(x) satisﬁes (.) and (.), similar to the proof of the usual evolutionary p

-Laplacian equation, we can prove that there is a solutionuε∈L(,T;W_,p()) of the

initial-boundary value problem (.)-(.), which satisﬁes

uεL∞(QT)≤c, uεtL₍_Q

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Multiplying (.) byuεand integrating it overQT, we have

 

u_εdx+

n

i=

QT

ai(x)|uεxi|pi(x)dx dt+ε

QT

|∇uε|dx dt

= 

u_(x)dx, (.)

then

ε

QT

|∇uε|dx≤c (.)

and

N

i=

QT

ai(x)|uεxi|pi(x)dx dt≤c. (.)

Hence, by (.), (.), (.), there exists a functionuand ann-dimensional vector−→ζ =

(ζ, . . . ,ζn) satisfying that−→ζ = (ζ, . . . ,ζn)

u∈L∞(QT),

∂u

∂t ∈L

₍_Q

T), ζi∈L

,T;L pi

(x) pi(x)–₍₎_,

anduε→ua.e.∈QT,

uεu, weakly star inL∞(QT),

uε→u, inL

,T;Lr_loc(),

∂uε

∂t

∂u

∂t inL

₍_Q

T),

ε∇uε, inL(QT),

ai(x)|uεxi|pi(x)–uεxi ζi inL

_,_T_;_L_pipi(x()–x) ₍₎_.

Here,r< Np

N–p.

Now, similar to the general evolutionaryp-Laplacian equation, we are able to prove that (the details are omitted here)

lim t→

u(x,t) –u(x)dx= 

and

N

i=

QT

ai(x)|uxi|pi(x)–uxiϕxidx dt=

N

i=

QT

ζi(x)ϕxidx dt (.)

for any functionϕ∈C

(QT). By a process of the limit [], we can show that (.) is also

true for anyϕ=ϕϕ, whereϕ∈C(QT),ϕ∈L∞ andϕxi∈L(,T;W

,pi(x)

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usatisfies equation (.) with the initial value (.) in the sense of Definition .. At last, according to Lemma ., the partial boundary value condition (.) is satisfied in the sense

of trace. Theorem . is proved.

3 The stability

Lemma .([–])

(i) p(x)-Hölder’s inequality.Letq(x)andq(x)be real functions with_q_₍_x₎+_q_₍_x₎= 

andq(x) > .Then the conjugate space ofLq(x)()isLq(x)().And,for any

u∈Lq(x)₍₎_and_v∈_Lq(x)₍₎_,_{we have}

uv dx≤|u|_Lq_(x)₍₎|v|_Lq_(x)().

(ii)

|u|_Lp(x)₍₎= , then

|u|p(x)dx= ,

|u|_Lp(x)() > , then|u|p –

Lp(x)≤

|u|p(x)dx≤ |u|p+ Lp(x),

|u|_Lp(x)() < , then|u|p +

Lp(x)≤

|u|p(x)dx≤ |u|p– Lp(x).

Now, we will prove Theorem .. For any given positive integern, letgn(s) be an odd

function, and

gn(s) =

⎧ ⎨ ⎩

, s>_n,

n_s_e–ns_, __≤_s_≤ 

n.

Clearly,

lim

n→gn(s) =sign(s), s∈(–∞, +∞). (.)

Proof of Theorem. Letuandvbe two weak solutions of equation (.) with the initial valuesu(x, ),v(x, ), respectively.

Letn={x∈:

l

r=ajr(x) >



n}, and

φn(x) =

⎧ ⎨ ⎩

, ifx∈n,

nl_r_=ajr(x), ifx∈\n.

(.)

Obviously,φnxi=n( l

r=ajr(x))xiwhenx∈\n, in other places, it is identical to zero. We can chooseϕ=χ[τ,s]φn,ϕ=gn(u–v),ϕ=χ[τ,s]φngn(u–v) as the test function, then

s

τ

φngn(u–v)

∂(u–v) ∂t dx dt

+

N

i=

s

τ

ai(x)

|uxi|pi(x)–uxi–|vxi|

pi(x)–_v

xi

(uxi–vxi)g

n(u–v)φn(x)dx dt

+

k

r=

s

τ

air(x)

|ux_ir|pir(x)–ux_ir –|vx_ir|pir(x)–vx_ir

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+

l

r=l

s

τ

ajr(x)

|ux_jr|pjr(x)–ux_jr–|vx_jr|pjr(x)–vx_jr

gn(u–v)φnx_jrdx dt

= . (.)

In the ﬁrst place, as usual, we have

ai(x)

pi(x)–_v

xi

(uxi–vxi)g

n(u–v)φn(x)dx≥, (.)

and sinceut∈L(QT), using the Lebesgue dominated theorem, we have

lim η→

s

τ

φn(x)gn(u–v)

∂(u–v) ∂t dx dt

=

|u–v|(x,s)dx–

|u–v|(x,τ)dx. (.)

In the second place, we deal with the third term on the left-hand side of (.). For sim-plicity, in what follows, we denotel_r_=ajr(x) as

l j=aj(x),

air(x)

|ux_ir|pir(x)–ux_ir–|vx_ir|pir(x)–vx_ir

gn(u–v)φnx_irdx

=

\n

air(x)

|ux_ir|pir(x)–ux_ir–|vx_ir|pir(x)–vx_ir

≤

\n

air(x)

|ux_ir|pir(x)–+|vx_ir|pir(x)–gn(u–v)φnx_irdx

≤

\n

nair(x)

l

j=

aj(x)

pir(x)– pir(x)

|ux_ir|pir(x)–+|vx_ir|pir(x)–

·air(x) 

pir(x)_g

n(u–v) n

 pir(x)

[l_j_=aj(x)]x_ir

[l_j_=aj(x)]

pir(x)– pir(x)

dx

≤

\n

n l

j=

aj(x)air(x)

|ux_ir|pir(x)+|vx_ir|pir(x)

dx  q ir ·

\n

nair(x)gn(u–v)

p_ir(x)|(

l

[l_j_=aj(x)]pir(x)– dx  p ir ≤c

\n

air(x)

|ux_ir|pir(x)+|vx_ir|pir(x)

dx  q ir · n

\n

gn(u–v) p_ir(x)|(

l

[l_j_=aj(x)]pir(x)– dx

 p

ir_. _(.)

Here,p

ir=p

+

irorp

–

iraccording to (ii) of Lemma ..qjr(x) =

p_jr(x)

p_jr(x)–,qir has a similar sense. If we denote that

n=

x∈\n:dist(x,) >dist(x,)

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n=

x∈\n:dist(x,)≤dist(x,)

,

then

n

\n

l

[l_j_=aj(x)]pir(x)– dx

≤n

n

gn(u–v)pir(x)

|(l_j_=aj(x))x_ir|pir(x)

[l_j_=aj(x)]pir(x)– dx

+n

n

l

[l_j_=aj(x)]pir(x)–

dx. (.)

Since

u=v= , x∈,

by the deﬁnition of the trace, we have

lim n→∞n

n

l

[l_j_=aj(x)]pir(x)– dx

=



sign(u–v)|( l

[l_j_=aj(x)]pir(x)–

d= . (.)

Moreover, since

|(l_j_=aj(x))x_ir|pir(x)

[l_j_=aj(x)]pir(x)–

= , x∈,

lim n→∞n

n

l

[l_j_=aj(x)]pir(x)– dx

≤ lim n→∞n

n |(l

[l_j_=aj(x)]pir(x)– dx

=



|(l_j_=aj(x))x_ir|pir(x)

[l_j_=aj(x)]pir(x)–

d= .

(.)

By (.)-(.), we conclude that

lim n→

air(x)

= . (.)

In the third place, we deal with the last term on the left-hand side of (.)

ajr(x)

gn(u–v)φnx_jrdx

=

\n

ajr(x)

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≤

\n

ajr(x)

|ux_jr|pjr(x)–+|vx_jr|pjr(x)–

l

j=

aj(x)

x_jr

gn(u–v)|dx

≤c

\n

ajr(x)|ux_jr|pjr(x)+|vx_jr|pjr(x)dx

 q+ jr

×n

\n

ajr(x)

_l

j=

aj(x)

xi

p_jr(x)

dx

 p+ jr

. (.)

Here,qjr(x) =

p_jr(x)

p_jr(x)–,q +

jr=maxx∈qjr(x). By assumption (.)

n

\n

ajr(x)

_l

j=

aj(x)

x_jr

p_jr(x)

dx

 p+ jr

≤c.

Then

lim n→∞

ajr(x)

= . (.)

Now, letn→ ∞in (.). Then

u(x,s) –v(x,s)dx≤

u(x,τ) –v(x,τ)dx, (.)

by the arbitrariness ofτ, we have

u(x) –v(x)dx.

4 The case ofl= 1

Proof of Theorem. Letuandvbe two weak solutions of equation (.) with the initial valuesu(x, ),v(x, ), respectively. Sincea(x) satisﬁes (.) and fori≥,ai(x) > ,x∈,

we can letn={x∈:a(x) >_n}and

φn(x) =

⎧ ⎨ ⎩

, ifx∈n,

na(x), ifx∈\n.

(.)

Obviously,φnxi=naxiwhenx∈\n, in other places, it is identical to zero. We can chooseχ[τ,s]φngn(u–v) as the test function, then

s

τ

φngn(u–v)

∂(u–v) ∂t dx dt

+

N

i=

s

τ

ai(x)

pi(x)–_v

xi

(uxi–vxi)g

n(u–v)φn(x)dx dt

+ s

τ

a(x)

|ux|

p(x)–_u

x–|vx|

p(x)–_v

x

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+ N i= s τ ai(x)

pi(x)–_v

xi

gn(u–v)φnxidx dt

= . (.)

Certainly, we have

ai(x)

pi(x)–_v

xi

(uxi–vxi)g

n(u–v)φn(x)dx≥ (.)

and

lim η→

s

τ

φn(x)φngn(u–v)

∂(u–v) ∂t dx dt

=

|u–v|(x,s)dx–

|u–v|(x,τ)dx. (.)

Now, we deal with the third term on the left-hand side of (.).

a(x)

|ux|

p(x)–_u

x–|vx|

p(x)–_v

x

gn(u–v)φnxdx

=

\n

a(x)

|ux|

p(x)–_u

x–|vx|

p(x)–_v

x

gn(u–v)φnxdx

≤

\n

a(x)

|ux|

p(x)–₊|_v

x|

p(x)–_g

n(u–v)φnxdx

≤n

\n

a(x)

|ux|

p(x)–₊|_v

x|

p(x)–_g

n(u–v)(ax(x)dx

≤cn

\n

a(x)

|ux|

p(x)₊|_v

x|

p(x)_dx 

q+_

×

\n

a(x)gn(u–v) p(x)

ax(x)

p(x)

dx

 p+_

≤cn–

 p+ 

–_q+ 

n

\n

a(x)

|ux|

p(x)₊|_v

x|

p(x)_dx 

q+_

×

\n

gn(u–v) p(x)

ax(x)

p(x)

dx

 p+_

. (.)

If we denote that

dn= sup x∈\n

dist(x,∂) (.)

and

n=

x∈\n:dist(x,) >dn

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then

n

\n

air(x)gn(u–v)

p_ir(x)

_l

j=

aj(x)

x_ir

p_ir(x)

dx  p+ ir ≤n n

air(x)gn(u–v)

p_ir(x)

_l

j=

aj(x)

x_ir

p_ir(x)

dx  p+ ir +n n

air(x)gn(u–v)

p_ir(x)

_l

j=

aj(x)

x_ir

p_ir(x)

dx  p+ ir ≤cn n

gn(u–v) p_ir(x)

dx  p+ ir +n n

air(x) _l j=

aj(x)

x_ir

p_ir(x)

dx

 p+ ir

. (.)

By the deﬁnition of the trace, we have

lim n→∞n

n

gn(u–v)pirdx= . (.)

Moreover, sinceai(x)∈C() andl> , by (.), we always have

_l

j=

aj(x)

x_ir ∂ = . (.)

Thus, by the fact thatai(x)∈C(), we get

lim n→∞n

n

air(x) _l j=

aj(x)

x_ir p_ir dx  p+ ir

≤cmax x∈ _l j=

aj(x)

x_ir

= . (.)

By (.)-(.), we conclude that

lim n→

air(x)

= . (.)

In the third place, we deal with the last term on the left-hand side of (.)

ajr(x)

=

\n

ajr(x)

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≤

\n

ajr(x)

|ux_jr|pjr(x)–+|vx_jr|pjr(x)–

l

j=

aj(x)

x_jr

gn(u–v)|dx

≤c

\n

ajr(x)

|ux_jr|pjr(x)+|vx_jr|pjr(x)

dx

 q+ jr

×n

\n

ajr(x)

_l

j=

aj(x)

xi

p_jr(x)

dx

 p+ jr

. (.)

Here,qjr(x) =

p_jr(x)

p_jr(x)–,q+jr=maxx∈qjr(x). By assumption (.),

n

\n

ajr(x)

_l

j=

aj(x)

x_jr

p_jr(x)

dx

 p+ jr

≤c.

Then

lim n→∞

ajr(x)

= . (.)

Now, letn→ ∞in (.). Then

u(x,τ) –v(x,τ)dx, (.)

by the arbitrariness ofτ, we have

u(x) –v(x)dx.

5 Conclusion

The anisotropic parabolic equation is considered in this paper. If in some directions the diﬀusion coeﬃcients are degenerate on the boundary, while in other directions they are not degenerate, how to give a suitable partial boundary value condition to match the equa-tion was studied by the author in []. If a partial boundary value condiequa-tion is imposed, only when the domain is anN-dimensional cube, the stability of weak solutions is proved []. This short paper solves the problem when the domain is a usual bounded domain, gives a complete supplement of the paper [].

Acknowledgements

The paper is supported by NSF of Fujian Province (Grant No. 2015J1092), supported by SF of Xiamen University of Technology (Grant No. XYK201448), China.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

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