Heat Equation with Memory in Anisotropic and Non-Homogeneous Media

• Published in: Acta mathematica Sinica. English series Vol. 27; no. 2; pp. 219 - 254
• Main Authors: Yong, Jiongmin, Zhang, Xu
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01-02-2011; Springer Nature B.V
• Subjects: Analysis; Anisotropy; Conduction; Heat; Heat conduction; Heat equations; Kernels; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Media; Propagation; Studies; Temperature; 傅里叶定律; 双曲线方程; 各向异性; 存储器; 热传导方程; 热方程; 空间相关; 非均匀介质; Heat equation with memory; well-posedness; 80A05; propagation speed; 35L05; 35K05; anisotropic and non-homogeneous media; 45K05
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-010-0077-1

A modified Fourier＇s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.