Abstract
We introduce the class of \(L^{q,\alpha }\)-Hölder functions and study the local and global existence and uniqueness of solution for abstract evolution differential equations assuming that the non-linear term is a \(L^{q,\alpha }\)-Hölder function. In the last section, some examples with applications to partial differential equations are presented.
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References
Hernández, E., Wu, J., Chadha, A.: Existence, uniqueness and approximate controllability of abstract differential equations with state-dependent delay. J. Differ. Equ. 269(10), 8701–8735 (2020)
Hernandez, E., Fernandes, D., Wu, J.: Existence and uniqueness of solutions, well-posedness and global attractor for abstract differential equations with state-dependent delay. J. Differ. Equ. 302(25), 753–806 (2021)
Goldstein, J.: Semigroups of linear operators and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1985)
Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems, PNLDE, vol. 16. Birkhäauser Verlag, Basel (1995)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
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The authors wish to thank the referees and the editor responsible for this paper, for their valuable comments and suggestions.
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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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Hernandez, E., Lisboa, L. & Fernandes, D. Abstract Differential Equations and \(L^{q,\alpha } \)-Hölder Functions. Mediterr. J. Math. 21, 58 (2024). https://doi.org/10.1007/s00009-024-02596-2
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DOI: https://doi.org/10.1007/s00009-024-02596-2
Keywords
- Abstract Cauchy problem
- local and global existenceof solution
- uniqueness of solution
- mild solution
- strong solution
- classical solution
- strict solution
- analytic semigroup