Regularity of solutions of initial–boundary value problems for parabolic equations in domains with conical points

Abstract

The purpose of this paper is to establish the well-posedness and the regularity of solutions of the initialboundary value problems for general higher order parabolic equations in infinite cylinders with the bases containing conical points.

On Parabolic Equations with Mixed Dirichlet–Robin Type Boundary Conditions in a Non-rectangular Domain

“…The functional framework dealt with by the author is different from that used in our study. Smoothness and asymptotic behavior near conical points of the solutions are established for such kind of problems in cylinders with conical points in [2] and [3].…”

On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points

“…In this paper, by reducing some conditions on eigenvalues of the spectrum problem, we obtained generally asymptotic expansions of solutions of the second boundary value problem for Schrödinger systems in a neighborhood of conical points (to compare see [3], [4] for example). Results are obtained for the second initial boundary value problem in infinite cylinders with the coefficients depending on both the time and spatial variables, while previous results were given for the first initial boundary value problem [2], [4] or in a finite cylinder [2], [15], [16] or for coefficients independent of the time variable [9], [15], [16], [18] or for other kinds of systems [3], [4], [9], [18].…”

“…In the previous papers [1], [2], [3], [4], to consider asymptotic behavior of solutions of boundary value problems for non-stationary systems, the authors just studied the case when the associated spectrum problem has simple eigenvalues. And now, by weakening restrictions on eigenvalues of the spectrum problem (extending them to semisimple eigenvalues having invariant multiplicity), we obtain generally asymptotic formulas of solutions as linear combinations of special singular vector functions and regular vector functions.…”

The Dirichlet–Cauchy problem for nonlinear hyperbolic equations in a domain with edges

“…In [14,7,8] linear parabolic/hyperbolic equations in domains with conical points were investigated in weighted Sobolev spaces in which the unique solvability of the IBVPs, the regularity and the asymptotic of the solution near conical points were obtained. For this issue, we mention some of papers which considered linear nonstationary equations in domains with conical points or with edges.…”