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Various Properties of Solutions of the Heat Equation

  • Mi-Ho Giga
  • Yoshikazu Giga
  • Jürgen Saal
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 79)

Abstract

Here we establish the tools used in Chapter 1 in order to analyze the asymptotic behavior of solutions for the heat equation. We start by deriving L p -L q estimates for solutions and their derivatives and the uniqueness theorem for weak solutions. For this purpose, we prepare the Young inequality for convolution, which has a wide range of applications. Furthermore, algebraic and commutativity properties, in particular concerning differentiation of convolutions, are stated. These properties turn out to be helpful in the proof of smoothness for t > 0 for the solution of the heat equation in Chapter 1. Next, we consider the continuity of the solution at time t= 0, in the case that the initial value is continuous. Continuity is proved by a fairly general method that applies to a large class of equations.

Keywords

Integral Equation Weak Solution Fundamental Solution Heat Equation Integral Sign 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoMeguro-kuJapan
  2. 2.Technische Universität Darmstadt Center of Smart InterfacesDarmstadtGermany

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