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- W44018047 abstract "It is well-known that the semigroup S on D(A) generated by a nonlinear (possible multivalued) operator A (via (2.8) or (2.13) may not be differentiable (see Crandall and Liggett [4]). However, in continuous case this semigroup is differentiable everywhere on [0,+~) (Theorem 2.3 here, of R.H. Martin Jr.). The most general under which an operator of dissipative type generates a semigroup is the condition (2.9) introduced by Kobayashi [8] and extended by the author (see (2.19)) to non-autonomous case. This is strictly more general than the standard condition (2.10). In applications to PDE the range is most useful. However, the tangential is still important in PDE, since e.g. in continuous case it is equivalent to Nagumo (2.14) (which can be applied in PDE). Moreover, this (2.9) has a unifying effect in the theory of non-linear semigroups. Indeed, first of all from (2.9) (Theorem 2.1) we can easily derive the exponential formula of Crandall and Liggett ((2.13) in Theorem 2.2 her). This is a known fact. Another important fact is that Theorem 2.3 (i.e. the differentiability of S in continuous case) can also be easily derived from Theorem 2.1. This is important since the original proof of Theorem 2.3 is more difficult than the proof of Theorem 2.1 itself. Such aspects are presented in Section 2. Finally we note that (2.9) can also be used in the theory of m-dissipativity of the sum of two mdissipative operators [8, 13]." @default.
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- W44018047 date "1984-01-01" @default.
- W44018047 modified "2023-09-24" @default.
- W44018047 title "Some problems on non-linear semigroups and the blow-up of integral solutions" @default.
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- W44018047 doi "https://doi.org/10.1007/bfb0072775" @default.
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