Abstract
In this chapter, we discuss fundamental properties of the nonlinear heat equation ∂ t u = Δ u associated with the nonlinear Laplacian Δ defined in Chap. 11. In particular, we establish the existence and the regularity of global solutions to the heat equation. Coupled with the Bochner inequalities in the previous chapter, the analysis of heat flow leads to various analytic and geometric applications as we will see in the following chapters. We remark that, due to the nonlinearity, there is no heat kernel.
Although it is nonlinear, our Laplacian is locally uniformly elliptic by virtue of the smoothness and the strong convexity of Finsler structures. Therefore one can analyze the heat equation by using well-established techniques in partial differential equations. We remark that analytic arguments in the proof of the regularity will be only sketched since they are beyond the scope of this book.
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Ohta, Si. (2021). Nonlinear Heat Flow. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_13
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DOI: https://doi.org/10.1007/978-3-030-80650-7_13
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