# 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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