Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

Abstract

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by ~freezing" the coefficients at the singular point. The main point of the paper is in proving that the re.solvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.

Solutions of the heat equation in domains with singularities

“…The graphs of roots of the transcendental equation D,,(a) = 0 obtained by numerical methods are given in [12]. The results presented below analytically justify the distribution of roots obtained numerically.…”

“…The proof of this theorem coincides word for word with that of Theorem 1 in [11], where the multidi, mensional case is treated. …”

“…Then by virtue of (4) ]lull,+~,~ ~ cllfll,,~, Theorems 6 and 7 can be proved in the same way as Theorems 2 and 3 in [11]. The derivation of (7) uses Theorem 5 as well as the results of w concerning the spectrum of the operator pencil (5).…”

“…Then it follows from (28) that cl and c5 are linear functions of c2 and c,, that is, the eigenspace is two-dimensional. In the domain p < -1, from (12) we find that the spectrum of (5) has the form (37 I. If P0 6.…”

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

“…In [AB96], [AB98] the first boundary problem is studied for the heat equation in a bounded plane domain with cuspidal points at the boundary at which the tangent coincides with a characteristic t = c, where c is a constant. The paper [AT12] contributed to the study of the first boundary problem for the 1D heat equation in a bounded plane domain by evaluating the first term of the asymptotic of a solution at the characteristic point.…”