Existence of solution to parabolic equations with mixed boundary condition on non-cylindrical domain

Existence of solution to parabolic equations with mixed boundary condition on non-cylindrical domain

Tujin Kim111The author’s research was supported by TWAS, UNESCO and AMSS in Chinese Academy of Sciences. Daomin Cao 222Supported by Chinese Science Fund for Creative Research Group(10721101) Institute of Mathematics, Academy of Sciences, Pyongyang, Democratic People’s Republic of Korea33footnotemark: 3 Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China
Abstract

In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak solution to the problem. In the case of the linear equation the parts for every type of boundary condition are any open subsets of the boundary being nonempty the part for Dirichlet condition at any time. Due to this it is difficult to reduce the problem to one on a cylindrical domain by diffeomorphism of the domain. By a transformation of unknown function and penalty method we connect the problem to a monotone operator equation for functions defined on the non-cylindrical domain. In this way a semilinear problem is considered when the part of boundary for Dirichlet condition is cylindrical.

keywords:
Parabolic equation, Non-cylindrical domain, Mixed boundary condition, Existence
MSC: 35K20, 35K55, 35A15

1 Introduction

There are vast literature for parabolic differential equations on non-cylindrical domain and various methods have been used to study them. In bhl () the energy inequality for a linear equation with homogeneous Dirichlet boundary condition is proved, thus unique existence of solution is studied. For domains expanded along time existence and uniqueness of solution to initial boundary value problem of the linear(cf. e (), ku () and kp ()), semilinear(cf. krr ()) and nonlinear(cf. ku ()) equations with homogeneous Dirichlet boundary condition are studied. For such domains and boundary conditions krr () also deals with attractor; and bps () considers unique existence of solution to a linear Schrödinger-type equation. In bg () dealing with the Dirichlet problem, they assume only Hölder continuity on time-regularity of the boundary. In ls () semigroup theory is improved and the obtained result is applied to the initial boundary value problem of a linear parabolic equation with inhomogeneous Dirichlet condition on non-cylindrical domain. There are some literatures for unique existence of initial boundary value problems of linear equations relying on the method of potentials (see c () and references therein). Domains in l1 () and l2 (), where existence, uniqueness and regularity are studied, are more general, that is, ”initial” condition is given on a hypersurface in spatial-temporal space instead of the plane . In lms1 () optimal regularity of solution to a special kind of 1-dimensional problem is considered. Neumann problem of heat equations (cf. hl ()), parabolic equation with Robin type boundary condition (cf. ks ()) in non-cylindrical domains and behavior of solutions to the initial-boundary value problems of nonlinear equations (cf. k () and t ()) are studied. In lcm () and mlm () optimal control and controllability of parabolic equation with homogenous Dirichlet condition on non cylindrical domain, respectively are studied.

Also, there are many literatures for the initial boundary value problems with mixed boundary conditions.
Under certain assumptions the non-cylindrical domains are transformed to a cylindrical one. The initial boundary value problems of linear parabolic equations with mixed time dependent lateral boundary condition on cylindrical domains are studied (cf. bfo1 (), bfo2 (), sa ()). The boundary conditions on the lateral surfaces in bfo1 () may be either two of the following classical ones: Dirichlet, Neumann and Robin, but one part for a kind of boundary condition is a connected and relatively open subset of the lateral surface and the boundary of the part is tangent to the plane . In bfo2 () the first part is concerned with a classical problem on cylindrical domains and the result is applied to the problem with zero initial condition and lateral mixed boundary conditions on a cylindrical domain, where the non-cylindrical surface for boundary condition is transformed to a cylindrical one by a diffeomorphism. The lateral boundary surface of the cylindrical domain in bfo2 () is also divided into two parts, and one part for a kind of boundary condition is connected and relatively open subset of the boundary surface and at each point is transverse to the hyperplane . Developing a method in abstract evolution equations, sa () is concerned with linear parabolic problems on cylindrical domains with mixed variable inhomogeneous Dirichlet and Neumann conditions. But, here change along time of distance of the sections of part of boundary for Dirichlet condition must be dominated by a Lipschitz continuous function in time . In sa () as application of the result, unique existence of solution to a linear parabolic problem with homogeneous Dirichlet boundary condition on non-cylindrical domains is considered.

Section 4 in l3 () deals with the linear parabolic problem on non-cylindrical domain with Robin and Dirichlet boundary conditions, where the surface for Dirichlet condition is cylindrical type. lvw () and lwv () study existence, uniqueness and regularity of solutions to the initial boundary value problems of linear and semilinear parabolic equations on non-cylindrical domains, which is related to the the combustion phenomena. The domains in lvw () and lwv () are bordered with a part of cylindrical type surface where homogeneous Neumann condition is given, non-cylindrical hypersurfaces where Dirichlet boundary one is given and planes . Thus, by change of spatial independent variable they transform the problems to classical problems on cylindrical domains where Dirichlet and Neumann conditions, respectively, are given on cylindrical surfaces. In lt () some differential inclusions are studied and the result is applied to the following problem

 ut−Δu∈F(u)inΩ,−∂∂nu∈β(u)onγ,
 u=0onΓ−γ,u(x,0)=ξinΩ0,

where is a non-cylindrical domain in spatial-temporal space, is its lateral surface, is a part of a cylindrical surface, and is a proper lower semicontinuous convex function from to with .

On the other hand, in s () a time-dependent Navier-Stokes problem on a non-cylindrical domain with a mixed boundary condition is considered. In s () the part of boundary where homogeneous Dirichlet condition is given is cylindrical type and the boundary condition on the other part of boundary is such a special one that guarantee existence of solution to an elliptic operator equation obtained by penalty method.

In this paper we are concerned with linear and semilinear parabolic equations on non-cylindrical domains with mixed boundary conditions which may include inhomogeneous Dirichlet, Neumann and Robin conditions together. In the case of linear equation the parts for every type of boundary condition are any open subsets of the boundary being nonempty the part for Dirichlet condition at any time. This rises difficulty in reducing the problem to one on cylindrical domains in bfo1 (), bfo2 () and sa () or one in lvw (), lwv (), l3 () and lt ().

Our idea is to use a transformation of unknown function by which the problem is connected to a monotone operator equation for functions defined on the non-cylindrical domain. In this way we can also consider semilinear equation when the part of boundary for Dirichlet condition is cylindrical.

This paper is composed of 5 sections. In Section 2 notation, the problem, the definition of weak solution and the main result are stated. In Section 3 by a change of unknown function an equivalent problem is derived. Section 4 is devoted to an auxiliary penalized problem. In Section 5 the proof of the main result is completed.

2 Problem and main result

Let be bounded connected domains of and be open subsets of such that and . Let be outward normal unit vector on the boundary and be outward normal unit vector on for fixed .
Let and . Let .

For function defined on define by

 β(y)=(∫T0∥y∥2Ω(t)dt)12

whenever the integral make sense. Let

 D(Q)={φ:φ∈C∞(¯Q),φ|Σ0=0},V(Q)={the completion of D(Q) under the normβ(y)},W(Q)={the completion of D(Q) in the spaceH1(Q)}

and be duality product between and . By the condition , is a norm in .

We use the following

Assumption 2.1

The hypersurface belongs to for and to for and for any there exist a diffeomorphisms on in the class which maps onto , and is in for , where is the unit operator.

Remark 2.1

Let and be any functions such that and Jacobian , where and . Then , where satisfies Assumption 2.1.

We are concerned with the following initial boundary value problem

 ∂y∂t−N∑i,j=1∂∂xi(aij(x,t)∂y∂xj)+N∑i=1bi(x,t)∂y∂xi+c(x,t,y)=g(x,t), (2.1)
 y|Σ0=¯y|Σ0,(k(x,t)y+N∑i,j=1aij(x,t)ni∂y∂xj)∣∣Σ1=f(x,t), (2.2)
 y(x,0)=y0(x)∈L2(Ω(0)), (2.3)

where and are functions satisfying the following conditions

is Lipschitz continuous with respect to uniformly for  and measurable with respect to for fixed and and

Remark 2.2

On a part of where we have Neumann condition.

When , in view of (2.2) we have

 ∫Q∂y∂tudxdt =(y(x,T),u(x,T))Ω(T)−(y0,u(x,0))Ω(0) (2.4) +∫Σ1yucos(^ν,t)dσ−∫Qy∂u∂tdxdt,
 −∫QN∑i,j=1∂∂xi(aij(x,t)∂y∂xj)udxdt =N∑i,j=1∫Qaij(x,t)∂y∂xj∂u∂xidxdt (2.5) +∫Σ1k(x,t)yudσ−∫Σ1f(x,t)udσ,

where is the angle between and the positive direction of -axis. Also, if , then (cf. Lemma 2.2, ch. 2 in ggz ()).

In view of (2.4) and (2.5), we introduce the following

Definition 2.1

A function is called a solution to (2.1)-(2.3) if satisfies the following

 y−¯y∈V(Q), −∫Qy∂v∂tdxdt+N∑i,j=1∫Qaij(x,t)∂y∂xj∂v∂xidxdt+N∑i=1∫Qbi(x,t)∂y∂xivdxdt +∫Qc(x,t,y)vdxdt+∫Σ1[yvcos(^ν,t)+k(x,t)yv]dσ =(y0,v(x,0))Ω(0)+∫Qg(x,t)vdxdt+∫Σ1f(x,t)vdσ ∀v∈W(Q)withv(x,T)=0.

Our main result of this paper is the following

Theorem 2.1

Suppose that conditions hold and that either is linear with respect to or and is invariant under the diffeomorphism in Assumption 2.1. Then there exists a solution to problem (2.1)-(2.3) provided that is valid.

3 Transformation of unknown function

For the sake of simplicity, we will use the same constants in the estimates unless confusion will be caused.

It is known that there exists a function such that

 ψ(x)>0∀x∈Ω(0),ψ|∂Ω(0)=0% and|∇ψ|>0∀x∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Ω(0)∖ω0,

where (cf. Lemma 1.1 in cik ()).

Lemma 3.1

There exists a function such that

 φ(x,t)>0onQ,φ(x,t)=0onΣand−∂φ(x,t)∂n>η>0onΣ,

where is the space of functions which are twice continuously differentiable with respect to and continuously differentiable with respect to on .

Proof Take , where is the diffeomorphism from onto , which is the inverse of the one given in Assumption 2.1. Then the conclusion follows from the properties of function and Assumption 2.1.

Let us make a change by

 u=ek1t+k2φ(x,t)y, (3.1)

where and are constants to be determined later. Let and (2.2) is satisfied. Then,

 ek1t+k2φ(x,t)∂y∂t=∂u∂t−(k1+k2∂φ∂t)u, −ek1t+k2φ(x,t)N∑i,j=1∂∂xi(aij(x,t)∂y∂xj)= −N∑i,j=1∂∂xi(aij∂u∂xj)+2∑ijaijk2∂φ∂xj∂u∂xi +k2u[∑ij∂aij∂xi∂φ∂xj−k2∑ijaij∂φ∂xi∂φ∂xj+∑ijaij∂2φ∂xi∂xj], ek1t+k2φ(x,t)N∑i=1bi(x,t)∂y∂xi=N∑i=1[bi(x,t)∂u∂xi−bi(x,t)k2∂φ∂xiu].

Also, we have that

 N∑i,j=1aij(x,t)ni∂u∂xj∣∣Σ1= =(N∑i,j=1aij(x,t)ni∂y∂xjek1t+k2φ(x,t)+N∑i,j=1aij(x,t)nik2∂φ(x,t)∂xju)∣∣Σ1 =(N∑i,j=1aij(x,t)ni∂y∂xjek1t+k2φ(x,t)+k2∂φ(x,t)∂nN∑i,j=1aij(x,t)ninju)∣∣Σ1 =(f(x,t)ek1t−k(x,t)u+k2∂φ(x,t)∂nN∑i,j=1aij(x,t)ninju)∣∣Σ1

where the fact that and its corollary have been used.

Taking into account these facts, we have

 ∂u∂t−N∑i,j=1∂∂xi(aij(x,t)∂u∂xj)+N∑i=1Bi(x,t)∂u∂xi+C(x,t,u)=G(x,t), (3.2)
 u|Σ0=¯u|Σ0,(K(x,t)u+N∑i,j=1aij(x,t)ni∂u∂xj)∣∣Σ1=F(x,t), (3.3)
 u(x,0)=u0≡y0(x)ek2φ(x,0)∈L2(Ω(0)), (3.4)

where

 Bi(x,t)=bi(x,t)+2∑jaijk2∂φ∂xj, (3.5) C(x,t,u)=ek1t+k2φ(x,t)c(x,t,e−k1t−k2φ(x,t)u)−(k1+k2∂φ∂t)u +k2u[∑ij∂aij∂xi∂φ∂xj−k2∑ijaij∂φ∂xi∂φ∂xj+∑ijaij∂2φ∂xi∂xj−∑ibi∂φ∂xi], G(x,t)=ek1t+k2φ(x,t)g, F(x,t)=ek1tf, ¯u=ek1t+k2φ(x,t)¯y,

and

 K(x,t)=k(x,t)−k2∂φ∂n∑i,jaijninj. (3.6)

Now, we take

which is possible by Lemma 3.1 and . Functions and , respectively, satisfy the conditions for and in and .

Lemma 3.2

In the sense of Definition 2.1 existence of solution to problems (2.1)-(2.3) and (3.2)-(3.4) are equivalent

Proof First, let us prove that if is a solution in the sense of Definition 2.1 to problem (2.1)-(2.3), then is a solution in the sense of Definition 2.1 to problem (3.2)-(3.4).

For , put . Then

 −∫Qy∂v∂tdxdt=−∫Que−k1t−k2φ∂ek1t+k2φ¯v∂tdxdt =−∫Qu∂¯v∂tdxdt−∫Q(k1+k2∂φ∂t)uvdxdt, N∑i,j=1∫Qaij(x,t)∂y∂xj∂v∂xidxdt= =N∑i,j=1∫Qaij(x,t)∂(e−k1t−k2φu)∂xj∂(ek1t+k2φ¯v)∂xidxdt =N∑i,j=1∫Qaij[∂u∂xje−k1t−k2φ−uk2e−k1−k2φ∂φ∂xj]× ×[ek1t+k2φ∂¯v∂xi+ek1t+k2φk2∂φ∂xi¯v]dxdt =N∑i,j=1∫Qaij[∂u∂xj∂¯v∂xi+k2∂φ∂xi∂u∂xj¯v−k22∂φ∂xj∂φ∂xiu¯v−k2∂φ∂xju∂¯v∂xi]dxdt≡I.

Integrating by parts in the last term above, we have

 I =N∑i,j=1∫Qaij(x,t)∂y∂xj∂v∂xidxdt=N∑i,j=1∫Qaij∂u∂xj∂¯v∂xidxdt +N∑i,j=1∫Q2k2aij∂φ∂xi∂u∂xj¯vdxdt−∫Σ1aijk2∂φ∂nninju¯vdσ +N∑i,j=1∫Qk2[∂aij∂xi∂φ∂xju¯v−k2aij∂φ∂xj∂φ∂xiu¯v+aij∂2φ∂xi∂xju¯v]dxdt.

Also,

 ∑i∫Qbi∂y∂xivdxdt=∑i∫Qbi∂u∂xi¯vdxdt−∑i∫Qbik2∂φ∂xiu¯vdxdt, ∫Qc(x,t,y)vdxdt=∫Qc(x,t,e−k1−k2φu)ek1t+k2φ¯vdxdt.

The facts and are equivalent, and so from above we can see that is a solution to (3.2)-(3.4) in the sense of Definition 2.1. In the same way we can see that if is a solution to problem (3.2)-(3.4) in the sense of Definition 2.1, then is a solution in the sense of Definition 2.1 to (2.1)-(2.3).

Therefore, in what follows we will consider the existence of a solution to problem (3.2)-(3.4). To this end, in the next section we will consider an auxiliary problem.

4 An auxiliary problem

The main purpose in this section is to find a function satisfying the following

 um−¯u∈W(Q), (4.1) ∫Q1m∂um∂t∂v∂tdxdt−∫Qum∂v∂tdxdt+∫QN∑i,j=1aij(x,t)∂um∂xj∂v∂xidxdt +∫QN∑i=1Bi(x,t)∂um∂xivdxdt+∫QC(x,t,um)vdxdt +∫Σ1[umvcos(^ν,t)+K(x,t)umv]dσ+(um(x,T),v(x,T))Ω(T) =(u0,v(x,0))Ω(0)+∫QG(x,t)vdxdt+∫Σ1F(x,t)vdσ ∀v∈W(Q),

where are positive integers.

We have the following result.

Theorem 4.1

Let in (3.1) be as (3.7). Then, for some in (3.1), which is taken before (4.7), there exists a unique solution to problem (4.1).

Proof Set , define an operator and an element , respectively, by

 ∀w,v∈W(Q); ⟨Amw,v⟩=∫Q1m∂u∂t∂v∂tdxdt−∫Qu∂v∂tdxdt+N∑i,j=1∫Qaij(x,t)∂u∂xj∂v∂xidxdt +N∑i=1∫QBi(x,t)∂u∂xivdxdt+∫QC(x,t,u)vdxdt +∫Σ1[uvcos(^ν,t)+K(x,t)uv]dσ+(u(x,T),v(x,T))Ω(T)

and

 ⟨L,v⟩=(u0,v(x,0))Ω(0)+∫QG(x,t)vdxdt+∫Σ1F(x,t)vdσ.

Now, let us consider problem of finding such that

 Amw=L. (4.2)

By the conditions and , operator is Lipschitz continuous. For any letting , we have that

 ⟨Amw1−Amw2,w⟩= (4.3) ∫Q1m∂w∂t∂w∂tdxdt−∫Qw∂w∂tdxdt+N∑i,j=1∫Qaij(x,t)∂w∂xj∂w∂xidxdt +N∑i=1∫QBi(x,t)∂w∂xiwdxdt+∫Q[C(x,t,w1+¯u)−C(x,t,w2+¯u)]wdxdt

On the other hand, by integrating by parts we get

 −∫Qw∂w∂tdxdt=12[|w(0)|2Ω(0)−|w(T)|2Ω(T)−∫Σ1w2cos(^ν,t)dσ]. (4.4)

From (4.3) and (4.4) we conclude that for any

 ⟨Amw1−Amw2,w⟩=∫Q1m∂w∂t∂w∂tdxdt+∫QN∑i,j=1aij(x,t)∂w∂xj∂w∂xidxdt (4.5) +∫QN∑i=1Bi(x,t)∂w∂xiwdxdt+∫Q[C(x,t,w1+¯u)−C(x,t,w2+¯u)]wdxdt +∫Σ1[12w2cos(^ν,t)+K(x,t)w2]dσ+12|w(0)|2Ω(0)+12|w(x,T)|2Ω(T).

It follows from (3.6) and the choice of mentioned above that

 ∫Σ1[12w2cos(^ν,t)+K(x,t)w2]dσ≥0. (4.6)

Note that in (3.5) and in (3.6) are independent of . Therefore, taking in the expression of in (3.5) a negative number small enough independently of , we have

 ∫QN∑i,j=1aij(x,t)∂w∂xj∂w∂xidxdt+∫QN∑i=1Bi(x,t)∂w∂xiwdxdt (4.7) +∫Q[C(x,t,w1+¯u)−C(x,t,w2+¯u)]wdxdt≥ρ2∫T0∥w∥2Ω(t)dt

By (4.5)-(4.7) we have

 ⟨Amw1−Amw2,w1−w2⟩≥α∥w∥2H1(Q),∃α>0,∀w1,w2∈W(Q)

(Note that depends on .) Now, by the theory of monotone operator, there exists a unique solution to problem (4.2) (cf. Theorem 2.2, ch. 3 in ggz ()). Thus, is the solution asserted in the theorem.

5 Proof of Theorem 2.1

Let be as in the proof of Theorem 4.1, and as in (3.7). When is the solution to (4.1) asserted in Theorem 4.1, putting , by (4.2), (4.3), (4.5)-(4.7) we have that

 +|wm(x,0)|2Ω(0)+|wm(x,T)|2Ω(T) (5.1) ≤c[