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- W1504030762 abstract "The following well-known perturbation theorem is of fundamental importance in semigroup theory: Let $A$ be $m$-dissipative (i.e., $A$ generates a $(C_{0})$ contraction semigroup). If $P_{1}$ is dissipative and $A$ bounded with relative $A$ bound less than one, and if $P_{2}$ is bounded, then $A+P_{1}+P_{2}$generates a $(C_{0})$ semigroup. This result is generalized to allow $A,P_{1},P_{2}$. all to depend on a real parameter $t$. Thus in many cases, establishing the well-posedness of the Cauchy problem for $$u'(t) = (A(t) + P_{1} (t) + P_{2} (t))u (t),,,,,,({}^{'}=d/dt})$$ is reduced to proving the well-posedness of the Cauehy problem for $u'(t)=A(t)u(t)$. Applications are given to temporally inhomogeneous scattering, theory and to second order evolution equations of the form $$u''(t) + B(t)u'(t) + C(t)u(t) = 0;$$; here both $B(t)$ and $C(t)$ can be unbounded. Some Concrete examples are given, including mixed problems for $$u_{tt} + alpha u_{xx} + beta u_{x} + gamma u_{tx} + delta u_{t} + varepsilon u + phi$$ with time dependent boundary conditions; here all the coefficients are smooth functions on $$left{{(t,x): - infty < t < infty ,,0 leqslant x leqslant 1} right}$$, $alpha$ is positive and $gamma$ is sufficiently small." @default.
- W1504030762 created "2016-06-24" @default.
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- W1504030762 date "1974-06-01" @default.
- W1504030762 modified "2023-10-18" @default.
- W1504030762 title "A perturbation theorem for evolution equations and some applications" @default.
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- W1504030762 doi "https://doi.org/10.1215/ijm/1256051221" @default.
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