Analytic methods in the theory of differential and pseudo-differential equations of parabolic type
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 solution decays exponentially as the spatial variables grow, and the existence, uniqueness and correctness classes for the Cauchy problem consist of rapidly increasing functions. For the class \textbf{E}\(_3\), fundamental solutions of the Cauchy problem have only power-like decay, and this result cannot be improved. Fundamental solutions of the Cauchy problem for fractional diffusion equations require qualitatively different estimates for the parametrix.
parabolic equation in the sense of Petrovsky
degenerate PDEs of Kolmogorov's structure
pseudodifferential equations with nonsmooth quasihomogeneous symbols
scientific article; zbMATH DE number 2117880