- Open Access
Asymptotics of solutions of second order parabolic equations near conical points and edges
Boundary Value Problems volume 2014, Article number: 252 (2014)
The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain , where K is an infinite cone in , . They obtain the asymptotics of the Green function near the vertex () and edge (), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied.
The present paper is concerned with an initial-boundary value problem for a second order parabolic equation in a n-dimensional domain with a -dimensional edge M, . In particular, we are interested in the asymptotics of the solution near the edge. The largest part of the paper deals with the problem
in the domain . Here
is an infinite cone (angle if ), Ω is a subdomain of the unit sphere with boundary ∂ Ω, and
is a linear second order differential operator with variable coefficients.
Initial-boundary value problems for parabolic equations in domains with angular or conical points and edges were studied in a number of papers. Most of these papers deal with the heat equation. Concerning the heat equation in domains with angular or conical points, we mention the papers by Grisvard , Kozlov and Maz’ya , de Coster and Nicaise , where the asymptotics of the solutions near the singular boundary points was studied. For domains with edges, Solonnikov ,  and Nazarov  estimated the Green function and proved the existence of solutions of the Dirichlet and Neumann problems for the heat equation in weighted Sobolev spaces. Kozlov and Rossmann ,  and Kweon  investigated the asymptotics of solutions of the Dirichlet problem for the heat equation near an edge.
A theory for general parabolic problems with time-independent coefficients in domains with conical points was developed in papers by Kozlov –. This theory includes the asymptotics of solutions in weighed Sobolev spaces and a description of the singularities of the Green function near the conical points. The goal of the present paper is to extend these results to the case of time-dependent coefficients and to domains with edges. Moreover, we consider solutions in weighted Sobolev spaces with arbitrary . However, we restrict ourselves to second order parabolic equations, and we consider only the first terms in the asymptotics. In our previous paper , we obtained point estimates for the Green function. These estimates together with results of the theory of elliptic boundary value problems are used in the present paper in order to describe the behavior of solutions near the edge.
We outline the main results of the present paper. For an arbitrary point , we put and . The same notation is used for multi-indices . We assume that are real-valued functions and that
where ϵ is a small positive number. Besides this assumption, we impose some smoothness conditions on the coefficients and (see (21) and (22)). The condition (3) ensures in particular that the difference of the operators and
is small in the operator norm . Here is the Sobolev space on with the norm
For , this space is denoted by .
In Sections 2 and 3, we deal with the asymptotics of the Green function near the edge M of . In the case of constant coefficients , the asymptotics can easily be obtained by means of the asymptotics of the Green function for the heat equation which is given in , . If the coefficients are variable, then the terms in the asymptotics contain the eigenvalues and eigenfunctions of the following operator pencil :
for , , , where . Analogous estimates are valid for the derivatives of G (cf. Theorem 3.1). Using this result, we show in Section 3 (see Theorem 3.2) that admits the decomposition
for and . Here, μ is a certain number greater than . The coefficient in (6) satisfies the estimate
for . Moreover, admits the decomposition
where is the function (37) and
In Section 4, we apply the results of the foregoing section in order to describe the behavior of the solutions of the problem (1) near the edge M. By Theorem 4.2, the following result holds. Suppose that , where
Then the solution of the problem (1) admits the decomposition
ℰ is the extension operator introduced in Section 4.2, and . Note that the function belongs to the anisotropic Sobolev-Slobodetskiĭ space , where s is the function .
Section 5 in closing deals with the initial-boundary value problem in a bounded domain with an edge. Under some smoothness conditions on the coefficients of the differential operator, we obtain the same decomposition of the weak solution near an edge point as in the case of the previously considered domain (see Theorem 5.1 at the end of the paper).
2 Green function of parabolic equations with constant coefficients
In this section, we assume that
where are real numbers, for all i, j. Here A denotes the square matrix with the elements , is the nabla operator and its transposed, i.e., is the row vector with the components .
There exists a coordinate transformation with a constant square matrix S such that the problem
takes the form
in the new coordinates ξ, where is a certain cone in with vertex at the origin. This coordinate transformation can be constructed as follows. Let be the matrix with the elements , , the matrix with the elements , , and B the matrix with the elements , , . Furthermore, let and be the nabla operators in the coordinates and , respectively. Then the operator can be written as
There exists an invertible matrix U such that (the identity matrix). This is true for the matrix , where Λ is the diagonal matrix of the (positive) eigenvalues of the matrix and the rows of V are the orthonormalized eigenvectors of . For the coordinates , , we have , and, consequently,
Obviously, the transformation maps onto the set , where is a cone in . Since is a symmetric matrix with only positive eigenvalues, there exists an invertible matrix W such that . For and , we get
We denote the Green function of the problem (8) by . This means that
Then the function
is the Green function of the problem (7).
In order to describe the asymptotics of near the edge, i.e., for small , we introduce the following notation. We denote by the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on the subdomain of the unit sphere with Dirichlet boundary condition, and by an orthonormalized (in ) system of eigenfunction to the eigenvalues . Furthermore, let
be the solutions of the quadratic equation . Then the functions
are special solutions of the equation in . We also introduce the functions
which are special solutions of the heat equation for and .
Suppose that μ is a real number satisfying the inequalities and for all j. By , Theorem 2.1], the Green function admits the decomposition
where is the fundamental solution of the heat equation in and is the Green function of the Dirichlet problem for the heat equation in . The remainder in (11) satisfies the estimate
for , , . Here, for , while is an arbitrarily small positive real number if .
Using the decomposition (11), we obtain an analogous decomposition for the Green function . For this, we introduce the functions
where denotes the quadratic form
Note that the form is independent of the coordinate transformation since and . Since U and W are invertible matrices, there exists a positive constant κ such that
We furthermore note that both and are solutions of the equation
which have the form
This means in particular that are eigenvalues and are eigenfunctions of the pencil which is defined as
Suppose thatandfor all j. Then the Green functionadmits the decomposition
for, , , , and. Here, for, whileis an arbitrarily small positive real number if.
where , , , and , and
The right-hand side of (16) is equal to
We derive another formula for the coefficient in Theorem 2.1. If , then
for . Let denote the weighted Sobolev space (closure of ) with the norm
(cf., Theorem 5.1]). Let
In the integral
We set , , , and we substitute in the integral on the right-hand side. Then we obtain
Multiplying the last equality by , we arrive at the formula
3 Green function of parabolic equations with variable coefficients
in , i.e., the solution of the problem
In this section, we will employ an estimate for the Green function which was proved in . For this end, we assume in the following that the coefficients of L satisfy some additional smoothness conditions. To be more precise, we suppose that
with some for and that
3.1 Estimates of Green function
Furthermore, let the pencil be defined by (5), and let be its eigenvalues, where
The following estimate for the Green function is proved in , Theorem 4.4].
for, , , , , . Here, denotes the same nonnegative number as in Theorem 2.1.
3.2 Asymptotics of Green function
Analogously to the matrix U in Section 2, let be a matrix such that
where is the matrix with the elements , . Under our assumptions on the coefficients, we may assume that the elements of U are -functions. Then the numbers are eigenvalues of the Beltrami operator −δ (with Dirichlet boundary conditions) on the subdomain of the unit sphere, where . As in Section 2, we denote by an orthonormalized system of eigenfunctions corresponding to the eigenvalues . Moreover, we set
Then the functions
are special solutions of the equation for . By (20), we have
for , . Suppose that p and β are such that
where . The coefficients satisfy the equality (cf. (19))
In the following lemma, we give an estimate of these functions.
Suppose that, where C is the same constant as in Theorem 3.1. Then the function (26) satisfies the estimate
for, , , and. Here, κ is a certain positive number, λ is an arbitrary number less than, andis the same nonnegative number as in Theorem 2.1.
We define for . Then
We estimate the first integral on the right-hand side of the last equality using the decomposition (25). Theorem 3.1 yields
for , , and , where ε is an arbitrarily small positive number. Furthermore,
The number λ can be chosen such that . Consequently,
The same estimate holds for
Thus, we obtain
Since for and is exponentially decaying for large , we get
where is the intersection of K with the sphere and n is the normal vector to this sphere. By Theorem 3.1, the integrand on the right-hand side of the last equality has the upper bound
This proves the lemma. □
For the estimation of the remainder , we need the following lemma.
Suppose thatis a solution of the problemin K, on ∂K, wherefor a certain integer. Then
with a constantindependent of ε.
First note that the eigenvalues and of the pencil have no generalized eigenfunctions (see, e.g., , Section 2.2]). Let be the Green function of the Dirichlet problem for the operator in the cone K, for smooth ζ vanishing in a neighborhood of . Then
By , Theorem 2.2], the function g satisfies the estimates
Moreover, in the case , the estimates for and for are valid. For arbitrary integer ν, let for , else. Furthermore, let
Then it follows from the above estimates for and from , Lemmas 3.5.1 and 3.5.4] that
where c is independent of μ, ν, and f (cf., Lemma 3.5.6]). Consequently,
By Hölder’s inequality,
Thus, we obtain
The last inequality together with the estimate
Now we are able to prove the main result of this section.
Suppose that and
where C is the same constant as in Theorem 3.1. Then the Green functionadmits the decomposition
whereis defined by (24), and
for, , , , . Here, is the same constant as in Theorem 2.1. The coefficientssatisfy the estimate (27).
Let ζ be a smooth function on the interval , for and for . Furthermore, let for and . It follows from the equality
for , where
Here, denotes the commutator of and χ. Furthermore, for . We estimate the -norm of the function for . By , Theorem 4.1],
Here, the constant c is independent of , y, t, τ. Indeed, by Lemma 3.2, we have
with a constant c independent of , y, t, τ. Furthermore, under the condition (3), the inequality
which implies (32) if ϵ is sufficiently small. Next, we estimate the -norm of . By Theorem 3.1,
for , where . Here, λ can be chosen such that . Therefore,
Analogously, we obtain
Since vanishes for and , we obtain the estimate
by means of Theorem 3.1. Using Lemma 3.1, we get the same estimate for the -norm of the functions . Consequently, (32) implies
for . We prove an analogous estimate for the - and t-derivatives of . Obviously,
for , where f is the same function as above. Since, moreover, for and , we get
for . The -norms of can be estimated in the same way as f. This together with (33) leads to the estimate
for . Analogously, the inequality
holds for and . Applying the estimate
for , (cf., Lemma 1.2.3]), we get
for , , , . This proves (31). □
where C is the same constant as in Theorem 3.1.
3.3 Asymptotics of the coefficients
Let be the Green function of the first boundary value problem for the operator
with constant coefficients depending on the parameters and t. This means that
We write the operator in the form
where and denote the nabla operators in the - and -variables, respectively. As in Section 2, let and be square and continuously differentiable (with respect to and t) matrices such that and . By Theorem 2.1, the function admits the decomposition
if and for all j. Here,
(cf. (13)), where is the coefficients matrix of the operator , and
is a quadratic form with respect to and satisfying the inequality (14). We define , i.e.,
for , , .
The coefficientsin Theorem 3.2admit the decomposition
where satisfies the estimate
For shortness, we write instead of in the proof of this theorem. Since for , , we have
for , , , . This means that the function satisfies the equation
On the other hand, it follows from (26) that
Here denotes the formally adjoint differential operator to L. Consequently,
for and . Furthermore, for , . This follows from the representation