Nonlinear Evolution Equations with Infinitely Many Derivatives

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• 作者：Humberto Prado ; Enrique G. Reyes
• 关键词：Holomorphic semigroups ; Differential equations with infinitely many derivatives ; Nonlinear parabolic equations
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：10
• 期：7
• 页码：1577-1590
• 全文大小：489 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262
• 卷排序：10

文摘

We study the nonlinear evolutionary euclidean bosonic string equation$$\begin{aligned} u_t = \Delta e^{-c \Delta }\,u + U(t,u) , \quad c > 0 \; \end{aligned}$$on the Euclidean space \({\mathbb {R}}^n\). We interpret the nonlocal operator \(\Delta e^{-c\,\Delta }\) using entire vectors of \(\Delta \) in \(L^2({\mathbb {R}}^n)\). We prove that it generates a bounded holomorphic \(C_0\)-semigroup on \(L^2({\mathbb {R}}^n)\) (so that it also satisfies maximal \(L^p\) regularity) and we show the well-posedness of the corresponding nonlinear Cauchy problem.KeywordsHolomorphic semigroupsDifferential equations with infinitely many derivativesNonlinear parabolic equationsCommunicated by Daniel Aron Alpay.