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Well-posedness for mutational equations under a general type of dissipativity conditions

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Abstract

This paper is concerned with mutational analysis found by Aubin and developed by Lorenz. To extend their results so that they can be applied to quasi-linear evolution equations initiated by Kato, we focus on a mutational framework where for each r > 0 there exists M ≥ 1 such that d(ϑ(t, x), ϑ(t, y)) ≤ Md(x, y) for t ∈ [0, 1] and x, y ∈ D r (φ), where ϑ is a transition and Dr(φ) is the revel set of a proper lower semicontinuous functional φ. The setting that the constant M may be larger than 1 plays an important role in applying to quasi-linear evolution equations. In that case, it is difficult to estimate the distance between two approximate solutions to mutational equations. Our strategy is to construct a family of metrics depending on both time and state, with respect to which transitions are contractive in some sense.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Engineering, Chuo University, Tokyo, 112-8551, Japan

    Yoshikazu Kobayashi

  2. Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka, 422-8529, Japan

    Naoki Tanaka

Authors
  1. Yoshikazu Kobayashi
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  2. Naoki Tanaka
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Corresponding author

Correspondence to Yoshikazu Kobayashi.

Additional information

Partially supported by JSPS KAKENHI Grant Numbers 25400145 and 16K05212.

Partially supported by JSPS KAKENHI Grant Numbers 25400134 and 16K05199. Received May 11, 2016 and in revised form December 9, 2016

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Kobayashi, Y., Tanaka, N. Well-posedness for mutational equations under a general type of dissipativity conditions. Isr. J. Math. 225, 1–33 (2018). https://doi.org/10.1007/s11856-018-1660-x

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  • DOI: https://doi.org/10.1007/s11856-018-1660-x

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