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Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 258))

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Abstract

In this chapter we begin to study nonlinear partial differential equations. The results which we obtain here all follow from the maximum principles which were obtained in Chapters 8 and 9. We shall show how they apply to nonlinear elliptic and parabolic partial differential equations. As a first application, we will use the strong maximum principles to prove comparison theorems ; i.e., pointwise inequalities between different solutions. These say, roughly, that if u and v are two solutions, and if uv on ∂D, it follows that u d v on D. Such theorems can be quite useful in obtaining qualitative information about solutions. For example, comparison theorems are often used to obtain information about the asymptotic behavior of solutions of parabolic equations as t→ + ∞. As a second application of the maximum principle, we shall show how it can be used to prove existence theorems. This is the method of “upper” and “lower” solutions, the solution being the limit of a monotone iteration scheme, where the monotonicity is a consequence of the maximum principle.

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© 1994 Springer Science+Business Media New York

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Smoller, J. (1994). Comparison Theorems and Monotonicity Methods. In: Shock Waves and Reaction—Diffusion Equations. Grundlehren der mathematischen Wissenschaften, vol 258. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0873-0_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0873-0_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6929-8

  • Online ISBN: 978-1-4612-0873-0

  • eBook Packages: Springer Book Archive

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