Local Solutions of First Order Semilinear Evolution Equations

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

$$\left\{ {\begin{array}{*{20}{c}} {u' + Au + M\left( u \right) = O,} \\ {u\left( O \right) = \varphi \in D\left( A \right)} \\ \end{array} } \right.$$

(II. 1.1)

in B. Usually then M is a locally Hölder ¹ or Lipschitz continuous mapping from D(A^1-ρ) into B for some ρ, 0 < ρ < 1. A^1-ρe^-tAx is continuous for t > 0, x ∈ B, 1 ≧ ρ ≧0 and fulfills the estimate $\left\| {{A^{1 - \rho }}{e^{ - tA}}x} \right\| \leqslant c\left( {\delta ,M'} \right){e^{ - \delta t}}/{t^{1 - p}},\delta < 1/M',O \leqslant t \leqslant T$