Abstract
Let B be a reflexive Banach space, let A be a closed operator in B with dense domain of definition D(A). Let all λ, Re λ ≥ 0, be in the resolvent set of -A and let \(\left\| {{{\left( {\lambda + A} \right)}^{ - 1}}} \right\| \leqslant \frac{{M'}}{{\left| \lambda \right| + 1}}\) for all λ Re λ ≧ 0. This means that-A generates an analytic semigroup e-tA (cf. (I.2)). We consider here the nonlinear initial value problem
in B. Usually then M is a locally Hölder 1 or Lipschitz continuous mapping from D(A1-ρ) into B for some ρ, 0 < ρ < 1. A1-ρe-tAx is continuous for t > 0, x ∈ B, 1 ≧ ρ ≧0 and fulfills the estimate \(\left\| {{A^{1 - \rho }}{e^{ - tA}}x} \right\| \leqslant c\left( {\delta ,M'} \right){e^{ - \delta t}}/{t^{1 - p}},\delta < 1/M',O \leqslant t \leqslant T\)
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© 1985 Springer Fachmedien Wiesbaden
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von Wahl, W. (1985). Local Solutions of First Order Semilinear Evolution Equations. In: The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects of Mathematics / Aspekte der Mathematik, vol E 8. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-13911-9_2
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DOI: https://doi.org/10.1007/978-3-663-13911-9_2
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-08915-3
Online ISBN: 978-3-663-13911-9
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