Determining nodes for semilinear parabolic evolution equations in Banach spaces. (English) Zbl 1447.35058
Summary: 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.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K58 | Semilinear parabolic equations |
| 35K90 | Abstract parabolic equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
abstract initial value problem; mild solutions; rate of monomial or exponential convergence; analytic semigroups on Banach spaces.References:
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