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Nonlinear Parabolic Equations

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Partial Differential Equations III

Part of the book series: Applied Mathematical Sciences ((AMS,volume 117))

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Abstract

We begin this chapter with some general results on the existence and regularity of solutions to semilinear parabolic PDE, first treating the pure initial-value problem in § 1, for PDE of the form

$$\frac{\partial u} {\partial t} = Lu + F(t,x,u,\nabla u),\quad u(0) = f,$$
(0.1)

where u is defined on [0, T) × M, and M has no boundary. Some of the results established in § 1 will be useful in the next chapter, on nonlinear, hyperbolic equations. We also give a precursor to results on the global existence of weak solutions, which will be examined further in Chap. 17, in the context of the Navier–Stokes equations for fluids.

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Michael E. Taylor

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Taylor, M.E. (2011). Nonlinear Parabolic Equations. In: Partial Differential Equations III. Applied Mathematical Sciences, vol 117. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7049-7_3

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