Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities
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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 resolvent 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.
KeywordsAsymptotic Behavior Singular Point Asymptotic Expansion Parabolic Equation Dirichlet Problem
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