Abstract
We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these problems. As a result, we obtain an explicit formula for the Young measure even without the non-degeneracy assumption used in the previous works. The main new idea is an application of the Radon–Nikodym theorem to decompose the Young measure.
Highlights
We are interested in the limiting behavior of the following problems: for the reaction-diffusion system
By a small abuse of notation, we extend functions Si by a constant value to the whole of R
For monotone F the problem is fairly classical and has been studied for a great variety of reaction-diffusion systems, with more than two components [5, 6, 14, 29] or reaction-diffusion equation coupled with an ODE [21]
Summary
We are interested in the limiting behavior (as ε → 0) of the following problems: for the reaction-diffusion system. Let S1(λ) ≤ S2(λ) ≤ S3(λ) be the solutions of equation F (Si (λ)) = λ (see Figure 1) For monotone F the problem is fairly classical and has been studied for a great variety of reaction-diffusion systems, with more than two components [5, 6, 14, 29] or reaction-diffusion equation coupled with an ODE [21]. -called nonstandard analysis was used to study the limit problem in the space of grid functions [7, 8]. It is known [33, 35] that both systems exhibit the following surprising phenomenon: as ε → 0, F (uε) → v and vε → v converge strongly without any known a priori estimates allowing to conclude so. The non-degeneracy condition excludes piecewise affine functions used in more explicit computations as in [26]
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