**R E S E A R C H**

**Open Access**

## Symmetry of solutions to parabolic

## Monge-Ampère equations

### Limei Dai

**_{Correspondence:}

[email protected] School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China

**Abstract**

In this paper, we study the parabolic Monge-Ampère equation

–u*t*det

*(*

*D*2

*u*

*)*

=*f*(t,u) in

### ×(0,

*T].*

Using the method of moving planes, we show that any parabolically convex solution is symmetric with respect to some hyperplane. We also give a counterexample in

R*n*_{×}_{(0,}_{T] and an example in a cylinder to illustrate the results.}

**MSC:** 35K96; 35B06

**Keywords:** parabolic Monge-Ampère equations; symmetry; method of moving

planes

**1 Introduction**

The Monge-Ampère equation has been of much importance in geometry, optics, stochas-tic theory, mass transfer problem, mathemastochas-tical economics and mathemastochas-tical ﬁnance theory. In optics, the reﬂector antenna system satisﬁes a partial diﬀerential equation of Monge-Ampère type. In [, ], Wang showed that the reﬂector antenna design problem was equivalent to an optimal transfer problem. An optimal transportation problem can be interpreted as providing a weak or generalized solution to the Monge-Ampère mapping problem []. More applications of the Monge-Ampère equation and the optimal trans-portation can be found in [, ]. In the meantime, the Monge-Ampère equation turned out to be the prototype for a class of questions arising in diﬀerential geometry.

For the study of elliptic Monge-Ampère equations, we can refer to the classical
pa-pers [–] and the study of parabolic Monge-Ampère equations; see the references [–
]*etc.*The parabolic Monge-Ampère equation –*ut*det(*D**u*) =*f* was ﬁrst introduced by

Krylov [] together with the other parabolic versions of elliptic Monge-Ampère
equa-tions; see [] for a complete description and related results. It is also relevant in the
study of deformation of surfaces by Gauss-Kronecker curvature [, ] and in a
maxi-mum principle for parabolic equations []. Tso [] pointed out that the parabolic
equa-tion –*ut*det(*D**u*) =*f* is the most appropriate parabolic version of the elliptic

Monge-Ampère equationdet(*D*_{u}_{) =}_{f}_{in the proof of Aleksandrov-Bakelman maximum principle}
of second-order parabolic equations. In this paper, we study the symmetry of solutions to
the parabolic Monge-Ampère equation

–*ut*det

*D**u*=*f*(*t*,*u*), (*x*,*t*)∈*Q*, (.)

*u*= , (*x*,*t*)∈*SQ*, (.)

*u*=*u*(*x*), (*x*,*t*)∈*BQ*, (.)

where*D*_{u}_{is the Hessian matrix of}_{u}_{in}_{x}_{,}_{Q}_{=}_{}_{×}_{(,}_{T}_{],}_{}_{is a bounded and convex}
open subset in R*n*_{,}_{SQ}_{=}_{∂}_{×}_{(,}_{T}_{) denotes the side of}_{Q}_{,} _{BQ}_{=}_{}_{× {}_{}_{}} _{denotes the}

bottom of*Q*, and*∂pQ*=*SQ*∪*BQ*denotes the parabolic boundary of*Q*,*f* and*u*are given

functions.

There is vast literature on symmetry and monotonicity of positive solutions of elliptic
equations. In , Gidas*et al.*[] ﬁrst studied the symmetry of elliptic equations, and
they proved that if=R*n*_{or}_{}_{is a smooth bounded domain in}_{R}*n*_{, convex in}_{x}_{and}

symmetric with respect to the hyperplane{*x*∈R*n*_{:}_{x}

= }, then any positive solution of the Dirichlet problem

*u*+*f*(*u*) = , *x*∈,

*u*= , *x*∈*∂*

satisﬁes the following symmetry and monotonicity properties:

*u*(–*x*,*x*, . . . ,*xn*) =*u*(*x*,*x*, . . . ,*xn*), (.)

*ux*(*x*,*x*, . . . ,*xn*) < (*x*> ). (.)

The basic technique they applied is the method of moving planes ﬁrst introduced by
Alexandrov [] and then developed by Serrin []. Later the symmetry results of elliptic
equations have been generalized and extended by many authors. Especially, Li []
con-sidered fully nonlinear elliptic equations on smooth domains, and Berestycki and
Niren-berg [] found a way to deal with general equations with nonsmooth domains using the
maximum principles on domains with small measure. Recently, Zhang and Wang []
in-vestigated the symmetry of the elliptic Monge-Ampère equationdet(*D**u*) =*e*–*u*and they
got the following results.

Letbe a bounded convex domain inR*n*with smooth boundary and symmetric with
respect to the hyperplane{*x*∈R*n*_{:}_{x}

= }, then each solution of the Dirichlet problem

det*D**u*=*e*–*u*, *x*∈,

*u*= , *x*∈*∂*

has the above symmetry and monotonicity properties (.) and (.). Extensions in various directions including degenerate problems [] or elliptic systems of equations [] were studied by many authors.

For the symmetry results of parabolic equations on bounded and unbounded domains,
the reader can be referred to [, , ] and the references therein. In particular, when
*Q*=×*J*,*J*= (,*T*], Gidas*et al.*[] studied parabolic equations –*ut*+*u*+*f*(*t*,*r*,*u*) =

and –*ut*+*F*(*t*,*x*,*u*,*Du*,*D**u*) = , and they proved that parabolic equations possessed the

ﬁrst studied the asymptotic symmetry results for classical, bounded, positive solutions of the problem

*ut*–*u*=*f*(*t*,*u*), (*x*,*t*)∈×*J*, (.)

*u*= , (*x*,*t*)∈*∂*×*J*. (.)

The symmetry of general positive solutions of parabolic equations was investigated in [,
, ] and the references therein. A typical theorem of*J*=Ris as follows.

Letbe convex and symmetric in*x*. If*u*is a bounded positive solution of (.) and
(.) with*J*=Rsatisfying

inf

*t*∈R*u*(*x*,*t*) > (*x*∈,*t*∈*J*),

then*u*has the symmetry and monotonicity properties for each*t*∈R:
*u*–*x*,*x*,*t*=*ux*,*x*,*t* *x*=*x*,*x*∈,*t*∈R,

*ux*(*x*,*t*) < (*x*∈,*x*> ,*t*∈R).

The result of*J*= (,∞) is as follows.

Assume that*u*is a bounded positive solution of (.) and (.) with*J*= (,∞) such that
for some sequence*tn*→ ∞,

lim inf

*n*→∞ *u*(*x*,*tn*) > (*x*∈).

Then*u*is asymptotically symmetric in the sense that

lim

*t*→∞

*u*–*x*,*x*,*t*–*ux*,*x*,*t*= (*x*∈),

lim sup

*t*→∞

*ux*(*x*,*t*)≤ (*x*∈,*x*> ).

In this paper, using the method of moving planes, we obtain the same symmetry of solu-tions to problem (.), (.) and (.) as elliptic equasolu-tions.

**2 Maximum principles**

In this section, we prove some maximum principles. Letbe a bounded domain inR*n*_{, let}

*aij*_{(}_{x}_{,}_{t}_{),}_{b}_{(}_{x}_{,}_{t}_{),}_{c}_{(}_{x}_{,}_{t}_{) be continuous functions in}_{Q}_{,}_{Q}_{=}_{}_{×}_{(,}_{T}_{]. Suppose that}_{b}_{(}_{x}_{,}_{t}_{) < ,}

*c*(*x*,*t*) is bounded and there exist positive constants*λ*andsuch that

*λ*|*ξ*|≤*aij*(*x*,*t*)*ξiξj*≤|*ξ*|, ∀*ξ*∈R*n*.
Here and in the sequel, we always denote

*Di*=

*∂*
*∂xi*

, *Dij*=

*∂*

*∂xi∂xj*

.

We use the standard notation*C**k*,*k*_{(}_{Q}_{) to denote the class of functions}_{u}_{such that the}

derivatives*Di*
*xD*

*j*

**Theorem .** *Letλ*(*x*,*t*)*be a bounded continuous function on Q*,*and let the positive *
*func-tionϕ*∈*C*,_{(}_{Q}_{)}_{satisfy}

*b*(*x*,*t*)*ϕt*+*aij*(*x*,*t*)*Dijϕ*–*λ*(*x*,*t*)*ϕ*≤. (.)

*Suppose that u*∈*C*,_{(}_{Q}_{)}_{∩}* _{C}*

_{(}

_{Q}_{)}

_{satisﬁes}*b*(*x*,*t*)*ut*+*aij*(*x*,*t*)*Diju*–*c*(*x*,*t*)*u*≤, (*x*,*t*)∈*Q*, (.)

*u*≥, (*x*,*t*)∈*∂pQ*. (.)

*If*

*c*(*x*,*t*) >*λ*(*x*,*t*), (*x*,*t*)∈*Q*, (.)

*then u*≥*in Q*.

*Proof* We argue by contradiction. Suppose there exists (*x*,*t*)∈*Q*such that*u*(*x*,*t*) < . Let

*v*(*x*,*t*) = *u*(*x*,*t*)

*ϕ*(*x*,*t*), (*x*,*t*)∈*Q*.

Then*v*(*x*,*t*) < . Set*v*(*x*,*t*) =min* _{Q}v*(

*x*,

*t*), then

*x*∈and

*v*(

*x*,

*t*) < . Since

*v*(·,

*t*) at-tains its minimum at

*x*, we have

*Dv*(

*x*,

*t*) = ,

*D*

*v*(

*x*,

*t*)≥. In addition, we have

*vt*(

*x*,

*t*)≤. A direct calculation gives

*vt*=

*utϕ*–*uϕt*

*ϕ* ,

*Dijv*=

*ϕDiju*–

*u*

*ϕ**Dijϕ*–

*ϕDivDjϕ*–

*ϕDjvDiϕ*.

Taking into account*u*(*x*,*t*) < , we have at (*x*,*t*),

≤*ϕaijDijv*=*aijDiju*–

*aij _{D}*

*ijϕ*

*ϕ* *u*

≤*aijDiju*+

*u*

*ϕ*(*bϕt*–*λϕ*)

≤*aijDiju*+

*b*

*ϕutϕ*–*λu*

=*aijDiju*+*but*–*λu*

< *aijDiju*+*but*–*cu*

≤.

This is a contradiction and thus completes the proof of Theorem ..

**Corollary .** *Suppose that* *is unbounded*, *Q*=×(,*T*]. *Besides the conditions of*
*Theorem*.,*we assume*

lim inf

|*x*|→∞*u*(*x*,*t*)≥. (.)

*Then u*≥*in Q*.

*Proof* Still consider*v*(*x*,*t*) in the proof of Theorem .. Condition (.) shows that the
minimum of*v*(*x*,*t*) cannot be achieved at inﬁnity. The rest of the proof is the same as the

proof of Theorem ..

Ifis a narrow region with width*l*,

=*x*∈R*n*| <*x*<*l*,

then we have the following narrow region principle.

**Corollary .**(Narrow region principle) *Suppose that u*∈*C*,_{(}_{Q}_{)}_{∩}* _{C}*

_{(}

_{Q}_{)}

_{satisﬁes}_{(.)}

*and*(.).

*Let the width l ofbe suﬃciently small*.

*If on∂pQ*,

*u*≥,

*then we have u*≥

*in Q*.*Ifis unbounded*,*and*lim inf_{|}*x*|→∞*u*(*x*,*t*)≥,*then the conclusion is also true*.

*Proof* Let <*ε*<*l*,

*ϕ*(*x*,*t*) =*t*+sin*x*+*ε*

*l* .

Then*ϕ*is positive and

*ϕt*= ,

*aij _{D}*

*ijϕ*= –

*l*

*a*_{ϕ}_{.}

Choose*λ*(*x*,*t*) = –*λ*/*l*. In virtue of the boundedness of*c*(*x*,*t*), when*l*is suﬃciently small,
we have*c*(*x*,*t*) >*λ*(*x*,*t*), and thus

*bϕt*+*aijDijϕ*–*λϕ*

=*b*–

*l*

*a**ϕ*–

–*λ*

*l*

*ϕ*

=*b*–

*l*

*a**ϕ*+*λ*
*l**ϕ*

≤*b*< .

**3 Main results**

In this section, we prove that the solutions of (.), (.) and (.) are symmetric by the method of moving planes.

**Deﬁnition .** A function*u*(*x*,*t*) :*Q*→Ris called parabolically convex if it is continuous,
convex in*x*and decreasing in*t*.

Suppose that the following conditions hold.
(A) *fu*(*t*,*u*)/*f*(*t*,*u*)is bounded in[,*T*]×R.

(B) *∂u*/*∂x*< and

*u*(*x*)≤*uxλ*, *x*∈*λ*, (.)

where*xλ*_{= (}_{λ}_{–}_{x}

,*x*, . . . ,*xn*),*λ*=∩ {*x*∈:*x*≤*λ*}(*λ*< ).

**Theorem .** *Let* *be a strictly convex domain in* R*n* *and symmetric with respect to*
*the plane*{*x*∈:*x*= },*Q*=×(,*T*].*Assume that conditions*(A)*and*(B)*hold and*
*u*∈*C*,_{(}_{Q}_{)}_{∩}* _{C}*

_{(}

_{Q}_{)}

_{is any parabolically convex solution of}_{(.), (.)}

_{and}_{(.).}

_{Then}*u*(

*x*,

*x*,

*t*) =

*u*(–

*x*,

*x*,

*t*),

*where*(

*x*,

*t*) = (

*x*,

*x*,

*t*)∈R

*n*+,

*and when x*≥,

*∂u*(

*x*,

*t*)/

*∂x*≤.

*Proof*Let in

*λ*

_{×}

_{(,}

_{T}_{],}

_{u}λ_{(}

_{x}_{,}

_{t}_{) =}

_{u}_{(}

_{x}λ_{,}

_{t}_{), that is,}

*uλ*(*x*,*x*, . . . ,*xn*,*t*) =*u*(*λ*–*x*,*x*, . . . ,*xn*,*t*), (*x*,*t*)∈*λ*×(,*T*].

Then

*D**uλ*(*x*,*x*, . . . ,*xn*,*t*) =*PTD**u*(*λ*–*x*,*x*, . . . ,*xn*,*t*)*P*,

where*P*=diag(–, , . . . , ). Therefore,

–*uλ*

*t* det

*D**uλ*_{= –}_{u}

*t*(*λ*–*x*,*x*, . . . ,*xn*,*t*)det

*D**u*(*λ*–*x*,*x*, . . . ,*xn*,*t*)

=*ft*,*u*(*λ*–*x*,*x*, . . . ,*xn*,*t*)

=*ft*,*uλ*. (.)

We rewrite (.) in the form

log–*uλ _{t}*+logdet

*D*

*uλ*=log

*ft*,

*uλ*. (.)

On the other hand, from (.), we have

log(–*ut*) +log

det*D**u*=log*f*(*t*,*u*). (.)

According to (.) and (.), we have

log(–*ut*) –log

Therefore

*d*
*ds*log

–*sut*– ( –*s*)*uλt*

*ds*+

*d*
*ds*log det

*sD**u*+ ( –*s*)*D**uλds*

=

*d*
*ds*log*f*

*t*,*su*+ ( –*s*)*uλ _{ds}*

_{.}

As a result, we have

*b*(*x*,*t*)*u*–*uλ _{t}*+

*aij*(

*x*,

*t*)

*u*–

*uλ*–

_{ij}*c*(

*x*,

*t*)

*u*–

*uλ*= , (

*x*,

*t*)∈

*λ*×(,

*T*], (.)

where

*b*(*x*,*t*) =

*ds*
*sut*+ ( –*s*)*uλt*

,

*aij*(*x*,*t*) =

*g _{s}ijds*,

*c*(*x*,*t*) =

*fu*

*f*

*t*,*su*+ ( –*s*)*uλds*,

*gsij*is the inverse matrix of*sD**u*+ ( –*s*)*D**uλ*. Then*b*(*x*,*t*) < ,*c*(*x*,*t*) is bounded and by the

*a priori*estimate [] we know there exist positive constants*λ*andsuch that

*λ*|*ξ*|≤*aijξiξj*≤|*ξ*|, ∀*ξ*∈R*n*.

Let

*wλ*_{=}_{u}_{–}_{u}λ_{,}

then from (.),

*b*(*x*,*t*)*wλ _{t}* +

*aij*(

*x*,

*t*)

*wλ*–

_{ij}*c*(

*x*,

*t*)

*wλ*= , (

*x*,

*t*)∈

*λ*×(,

*T*]. (.)

Clearly,

*wλ*(*x*,*t*) = , *x*∈*∂λ*∩ {*x*=*λ*}, <*t*≤*T*. (.)

Because the image of*∂*∩*∂λ*about the plane{*x*=*λ*}lies in, according to the
maxi-mum principle of parabolic Monge-Ampère equations,

*uλ*(*x*,*t*)≤, ∀*x*∈*∂*∩*∂λ*.

Thus

On the other hand, from (.),

*wλ*(*x*, ) =*u*(*x*) –*uxλ*≥, *x*∈*λ*. (.)

From Corollary ., when the width of*λ*_{is suﬃciently small,}_{w}λ_{(}_{x}_{,}_{t}_{)}_{≥}_{, (}_{x}_{,}_{t}_{)}_{∈}_{}λ_{×}

(,*T*].

Now we start to move the plane to its right limit. Deﬁne

=sup*λ*< |*wλ*(*x*,*t*)≥,*x*∈*λ*, <*t*≤*T*.

We claim that

= .

Otherwise, we will show that the plane can be further moved to the right by a small dis-tance, and this would contradict with the deﬁnition of.

In fact, if< , then the image of*∂*∩*∂*_{under the reﬂection about}_{{}_{x}_{=}_{}_{}}_{lies}

inside. According to the strong maximum principle of parabolic Monge-Ampère
equa-tions, for*x*∈,*u*_{< . Therefore, for}_{x}_{∈}_{∂}_{∩}_{∂}_{, we have}_{w}_{> . On the other hand,}

by the deﬁnition of, we have for*x*∈_{,}_{w}_{≥}_{. So, from the strong maximum principle}

[] of linear parabolic equations and (.), we have for (*x*,*t*)∈×(,*T*],

*w*(*x*,*t*) > . (.)

Let*d*be the maximum width of narrow regions so that we can apply the narrow region
principle. Choose a small positive constant*δ*such that+*δ*< ,*δ*≤*d*/ –. We consider
the function*w*+*δ*_{(}_{x}_{,}_{t}_{) on the narrow region}

+*δ*×(,*T*] =

+*δ*∩ *x*>–
*d*

×(,*T*].

Then*w*+*δ*_{(}_{x}_{,}_{t}_{) satisﬁes}

*b*(*x*,*t*)*w _{t}*+

*δ*+

*aij*(

*x*,

*t*)

*Dijw*+

*δ*–

*c*(

*x*,

*t*)

*w*+

*δ*= , (

*x*,

*t*)∈+

*δ*×(,

*T*]. (.)

Now we prove the boundary condition

*w*+*δ*(*x*,*t*)≥, (*x*,*t*)∈*∂p*

+*δ*×(,*T*]. (.)

Similar to boundary conditions (.), (.) and (.), boundary condition (.) is satisﬁed
for*x*∈*∂*+*δ*_{∩}_{∂}_{,}_{x}_{∈}* _{∂}*+

*δ*

_{∩ {}

_{x}_{=}

_{}_{+}

_{δ}_{}}

_{and for}

_{t}_{= . In order to prove (.) is}

satisﬁed for*x*∈*∂*+*δ*∩ {*x*=–*d*/}, we apply the continuity argument. By (.) and
the fact that (–*d*/,*x*, . . . ,*xn*) is inside, there exists a positive constant*c*such that

*w*

–*d*

,*x*, . . . ,*xn*,*t*

Because*wλ*_{is continuous in}_{λ}_{, then for small}_{δ}_{, we still have}

*w*+*δ*

–*d*

,*x*, . . . ,*xn*,*t*

≥.

Therefore boundary condition (.) holds for small*δ*. From Corollary ., we have

*w*+*δ*(*x*,*t*)≥, *x*∈+*δ*, <*t*≤*T*. (.)

Combining (.) and the fact that*wλ*_{is continuous for}_{λ}_{, we know that}* _{w}*+

*δ*

_{(}

_{x}_{,}

_{t}_{)}

_{≥}

_{ for}

*x*∈_{when}_{δ}_{is small. Then from (.), we know that}

*w*+*δ*(*x*,*t*)≥, *x*∈+*δ*, <*t*≤*T*.

This contradicts with the deﬁnition of, and so= .

As a result,*w*_{(}_{x}_{,}_{t}_{)}_{≥}_{ for}_{x}_{∈}* _{}*

_{, which means that as}

_{x}_{< ,}

*u*(*x*,*x*, . . . ,*xn*,*t*)≥*u*(–*x*,*x*, . . . ,*xn*,*t*).

Since is symmetric about the plane{*x*= }, then for*x* ≥,*u*(–*x*,*x*, . . . ,*xn*,*t*) also

satisﬁes (.). Thus we can move the plane from the right towards the left and get the reverse inequality. Therefore

*∂u*(*x*,*t*)/*∂x*≤, *x*≥,

*u*(*x*,*x*, . . . ,*xn*,*t*) =*u*(–*x*,*x*, . . . ,*xn*,*t*). (.)

Equation (.) means that *u* is symmetric about the plane {*x* = }. Theorem . is

proved.

If we put the*x*axis in any direction, from Theorem ., we have the following.

**Corollary .** *Ifis a ball*,*Q*=×(,*T*],*then any parabolically convex solution u*∈
*C*,_{(}_{Q}_{)}_{of}_{(.), (.)}_{and}_{(.)}_{is radially symmetric about the origin}_{.}

**Remark .** Solutions of (.) inR*n*×(,*T*] may not be radially symmetric. For example,

–*ut*det

*D**u*=*e*–*u*, (*x*,*t*)∈R*n*×(,*T*] (.)

has a non-radially symmetric solution. In fact, we know that*f*(*x*) = log( +*e*√*x*_{) –}√_{}_{x}_{–}

log (*x*> ) satisﬁes*f*=*e*–*f* inR, and*f*(*x*) =*f*(–*x*),*x*< . Deﬁne

*u*(*x*,*t*) =log(*T*–*t*) +*f*(*x*) +*f*(*x*) +· · ·+*f*(*xn*),

We conclude this paper with a brief examination of Theorem .. Let*B*=*B*() be the
unit ball inR*n*_{, and let radially symmetric function}_{u}_{(}_{x}_{) =}_{u}_{(}_{r}_{),}_{r}_{=}_{|}_{x}_{|}_{satisfy}

*u*(*r*)(*u*_{}(*r*))*n*–*u*_{}(*r*)

*rn*– = –, <*r*< , (.)

*u*() =*u*_{}() = . (.)

**Example .** Let*u*satisfy (.) and (.). Then any solution of

–*ut*det

*D**u*= , (*x*,*t*)∈*B*×(,*T*], (.)

*u*= , (*x*,*t*)∈*∂B*×(,*T*), (.)

*u*=*u*, (*x*,*t*)∈*B*× {} (.)

is of the form

*u*= –(*n*+ )*t*+ *n*+_{u}

(*r*), (.)

where*r*=|*x*|.

*Proof* According to Corollary ., the solution is symmetric. Let

*u*(*x*,*t*) =*u*(*r*,*t*), *r*=|*x*|.

Then

*ui*=

*∂u*(*r*,*t*)

*∂r*
*xi*

*r*,

*uij*=

*∂*_{u}_{(}_{r}_{,}_{t}_{)}

*∂r*
*xixj*

*r* +

*∂u*(*r*,*t*)

*∂r*

*δij*

*r* –
*xixj*

*r*

,

det*D**u*=

*∂u*/*∂r*
*r*

*n*–

*∂**u*

*∂r*.

Therefore (.) is

–*∂u*

*∂t*

*∂u*/*∂r*
*r*

*n*–* _{∂}*

_{u}*∂r* = . (.)

We seek the solution of the form

*u*(*r*,*t*) =*T*(*t*)*u*(*r*).

Then

–*u*(*r*)*T*(*t*)(*u*

(*r*)*T*(*t*))*n*–
*rn*– *u*

(*r*)*T*(*t*) = .

That is,

*u*(*r*)(*u*_{}(*r*))*n*–* _{u}*
(

*r*)

*rn*– = –

Therefore

*T*(*t*)*T*(*t*)*n*= . (.)

By (.), we know that

*T*() = . (.)

From (.) and (.), we have

*T*(*t*) =(*n*+ )*t*+

*n*+_{.}

As a result,

*u*(*r*,*t*) = –(*n*+ )*t*+ *n*+ _{u}_{(}_{r}_{).}

From the maximum principle, we know that the solution of (.)-(.) is unique. Thus any solution of (.), (.) and (.) is of the form of (.).

**Competing interests**

The author declares that they have no competing interests.

**Acknowledgements**

The research was supported by NNSFC (11201343), Shandong Province Young and Middle-Aged Scientists Research Awards Fund (BS2011SF025), Shandong Province Science and Technology Development Project (2011YD16002).

Received: 3 April 2013 Accepted: 5 August 2013 Published: 20 August 2013
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doi:10.1186/1687-2770-2013-185

**Cite this article as:**Dai:Symmetry of solutions to parabolic Monge-Ampère equations.*Boundary Value Problems*2013