具有退化的2b-抛物型方程的多点时间问题

I. Pukalskyy, B. Yashan
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引用次数: 0

摘要

近几十年来,偏微分方程的非局部条件问题引起了人们的特别关注。对这些问题的兴趣是由于边值问题一般处理的需要和它们丰富的实际应用(扩散、振荡、土壤中盐和水分的输送、等离子体物理、数学生物学等过程)。研究了一类具有退化的非一致2b-抛物型方程的多点时间问题。2b阶抛物方程的系数允许在一些点的时间和空间变量中存在任意阶的幂奇点。研究了具有光滑系数的辅助问题的解。利用先验估计,建立了在特殊Hölder空间中求解问题及其导数的不等式。利用Archel和Riess定理,将收敛序列与辅助问题的紧致解序列区分开来,该紧致解序列的极限值是给定问题的解。在Hölder空间中建立了具有幂律权值的2b-抛物型方程多点时间问题解的估计。幂权重的阶数由抛物方程的高项组系数的简并阶数和低项组系数的幂特征决定。在方程右侧有一定的限制条件下,得到给定问题解的积分像。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A MULTIPOINT IN-TIME PROBLEM FOR THE 2b-PARABOLIC EQUATION WITH DEGENERATION
In recent decades, special attention has been paid to problems with nonlocal conditions for partial differential equations. Such interest in such problems is due to both the needs of the general therapy of boundary value problems and their rich practical application (the process of diffusion, oscillations, salt and moisture transport in soils, plasma physics, mathematical biology, etc.). A multipoint in-time problem for a nonuniformly 2b-parabolic equation with degeneracy is studied. The coefficients of the parabolic equation of order 2b allow for power singularities of arbitrary order both in the time and spatial variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions of auxiliary problems, the limiting value of which will be the solution of the given problem. Estimates of the solution of the multipoint time problem for the 2b-parabolic equation are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of degeneracy of the coefficients of the groups of higher terms and the power features of the coefficients of the lower terms of the parabolic equation. With certain restrictions on the right-hand side of the equation, an integral image of the solution to the given problem is obtained.
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