Well-posedness for mutational equations under a general type of dissipativity conditions
- 33 Downloads
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.
Unable to display preview. Download preview PDF.
- T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; Dedicated to Konrad Jörgens), Lecture Notes in Mathematics, Vol. 448, Springer, Berlin, 1975, pp. 25–70.Google Scholar
- T. Lorenz, Mutational Analysis, A Joint Framework for Cauchy Problems in and beyond Vector Spaces, Lecture Notes in Mathematics, Vol. 1996, Springer-Verlag, Berlin, 2010.Google Scholar