Determining Nodes for Semilinear Parabolic Evolution Equations in Banach Spaces

Abstract

We are concerned with the determination of the asymptotic behavior of mild solutions to the abstract initial value problem for semilinear parabolic evolution equations in $L_{p}$ by the asymptotic behavior of these mild solutions on a finite set. More precisely, if the asymptotic behavior of the mild solution is known on an suitable finite set which is called determining nodes, then the asymptotic behavior of the mild solution itself is entirely determined. Not only the asymptotic equivalence but also the rate of monomial or exponential convergence can be clarified. We prove the above properties by the theory of analytic semigroups on Banach spaces.