- Open Access
Asymptotics of solutions of second order parabolic equations near conical points and edges
© Kozlov and Rossmann; licensee Springer. 2014
Received: 27 August 2014
Accepted: 21 November 2014
Published: 11 December 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.
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.
For , this space is denoted by .
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
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 .
is the Green function of the problem (7).
which are special solutions of the heat equation for and .
for , , . Here, for , while is an arbitrarily small positive real number if .
for, , , , and. Here, for, whileis an arbitrarily small positive real number if.
3 Green function of parabolic equations with variable coefficients
3.1 Estimates of Green function
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
In the following lemma, we give an estimate of these functions.
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.
This proves the lemma. □
For the estimation of the remainder , we need the following lemma.
with a constantindependent of ε.
Now we are able to prove the main result of this section.
for, , , , . Here, is the same constant as in Theorem 2.1. The coefficientssatisfy the estimate (27).
for , , , . This proves (31). □
where C is the same constant as in Theorem 3.1.
3.3 Asymptotics of the coefficients
for , , .