Abstract
This chapter explains how one can justify theoretically the formal manipulations of Chap. 1 We introduce precisely what is the concept of weak solutions and justify the use of integrations by parts. We show that the concept of solutions in the sense of distributions has good stability properties and that intuitive conservation laws can be entirely justified. We also introduce simple examples of semilinear equations that can be treated easily based on the notion of weak solutions.
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Notes
- 1.
\(\mathcal{D}\) is the vector space of \(C^{\infty }\) functions with compact supports.
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Perthame, B. (2015). Weak Solutions of Parabolic Equations in Whole Space. In: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-19500-1_3
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DOI: https://doi.org/10.1007/978-3-319-19500-1_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19499-8
Online ISBN: 978-3-319-19500-1
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