Abstract

The work proposes a method for analyzing the Cauchy problem for a wide class of linear evolution partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integrodifferential equation, which can be considered as an ordinary differential equation in an appropriate Banach space. The latter space is chosen so that the contraction mapping principle can be applied. To derive the corresponding estimates for the operators generated by the transformed equation, we impose the condition that the inverse Fourier transform of the coefficients has compact support in the space variable and the spaces of coefficients of the original equation are determined from the Paley–Wiener Fourier transform theorems. In this case, the apparatus of the Bochner integral theory in pseudonormed spaces, as well as the theory of countably normed spaces and Sobolev spaces, is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of the Fourier transform of finite sums for the operator exponential.

中文翻译：

---

偏微分方程的傅立叶变换方法。Part 2. 线性方程 Cauchy 问题解的存在唯一性

摘要

这项工作提出了一种分析具有可变系数的线性演化偏微分方程的柯西问题的方法。通过应用（逆）傅里叶变换，原始方程简化为积分微分方程，可以将其视为适当 Banach 空间中的常微分方程。选择后一个空间以便可以应用收缩映射原理。为了推导出由变换方程生成的算子的相应估计，我们施加了这样的条件，即系数的傅里叶逆变换在空间变量中具有紧凑的支持，并且原始方程的系数空间由 Paley–Wiener Fourier 确定变换定理。在这种情况下，使用伪范数空间中 Bochner 积分理论的装置，以及可数范数空间和 Sobolev 空间的理论。函数类被区分，其中解决方案的存在性和唯一性被证明。对于具有分离变量的系数的方程，精确解以指数运算符的有限和的傅里叶变换的形式获得。