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.

中文翻译：

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Banach空间中半线性抛物线发展方程的确定节点

我们关注的是，在有限集上，这些温和解的渐近性决定了\（L_ {p} \）中半线性抛物线演化方程的抽象初值问题的温和解的渐近性。更准确地说，如果在称为确定节点的合适有限集上知道了温和溶液的渐近行为，那么就完全确定了温和溶液本身的渐近行为。不仅可以证明渐近等价，而且可以弄清单项式或指数式收敛的速率。我们通过Banach空间上的解析半群理论证明了上述性质。