Analytic methods in the theory of differential and pseudo-differential equations of parabolic type
type: Item
instance of: scholarly article
title: Analytic methods in the theory of differential and pseudo-differential equations of parabolic type
zbMATH Open document ID: 1062.35003
author: Anatoly N. Kochubei
author: S. D. Eĭdel'man
author: Stepan Ivasyshen
published in: Operator Theory: Advances and Applications
publication date: 2004-11-23T00:00:00Z
review text: This monograph is a thorough exposition of analytic methods for constructing, investigating and using fundamental solutions of the Cauchy problem, \(L_N(t,x,\partial_t,\partial_x)u(t,x)=f(t,x)\) for \((t,x)\in Q\subset\mathbb R^{n+1}\) with coefficients depending on all variables, for generalized classes of linear parabolic equations in the sense of Petrovsky. The linear differential expression \(L_N\) is a scalar if \(N=1\) and an \(N\times N\) matrix if \(N>1\). The four classes are: \textbf{E}\(_1\): \(\vec{2b}\)-parabolic partial differential equations (PDE) (parabolic equations of a quasi-homogeneous structure), in which the weights of the time variable with respect to the spatial variables \(x_j\) are \(1/(2b_j)\). \textbf{E}\(_2\): Degenerate PDEs of Kolmogorov's structure which generalize classical Kolmogorov equations of diffusion with inertia. The weights of the time variable are \(1/(2b)\) with respect to the main spatial variables and \(1+1/(2b)\) and \(2+1/(2b)\) with respect to the two groups of spatial variables corresponding to the degeneration. \textbf{E}\(_3\): Pseudodifferential equations with nonsmooth quasi-homogeneous symbols. \textbf{E}\(_4\): Fractional diffusion equations, with weight of the time variable \(\alpha/2\). As a first step, a fundamental solution of the Cauchy problem is constructed through various modifications of the classical Levi method as the sum of the main term, called parametrix, and an integral with the parametrix as kernel and an unknown density which is determined from an iteratively solved integral equation. Then this solution is studied and used for finding classes of existence, uniqueness and correctness. The simultaneous treatment of substantially different generalizations of the classical Petrovs\-ky's definition of parabolic equations makes it possible to present an interesting comparative analysis of the methods and results. For the classes \textbf{E}\(_1\), \textbf{E}\(_2\) and \textbf{E}\(_4\), the fundamental…
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35K30
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35S10
zbMATH DE Number: 2117880
zbMATH Keywords: parabolic equation in the sense of Petrovsky
zbMATH Keywords: Cauchy problem
zbMATH Keywords: degenerate PDEs of Kolmogorov's structure
zbMATH Keywords: Kolmogorov equations
zbMATH Keywords: pseudodifferential equations with nonsmooth quasihomogeneous symbols
zbMATH Keywords: fractional diffusion equations
zbMATH Keywords: fundamental solution
zbMATH Keywords: parametrix
zbMATH Keywords: integral equation
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: correctness
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2117880