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PreprintCC BY-NC-ND 4.0arXiv

Solutions of Duhamel's formula with a sum of mappings

  • Radosław Czaja
  • Maria Kania
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Abstract

Semilinear evolution equations are solved by means of the variation of constants formula under the action of smoothing linear operators and a finite number of nonlinearities operating between chosen Banach spaces of a given family. Admissible spaces of initial conditions are determined for the existence and uniqueness of a suitable notion of solution. Its maximal time of existence, short time and blow up time profiles, global extendibility and regularity are analyzed. For extrapolated fractional power scale of Banach spaces the fulfillment of the corresponding Cauchy problem is shown. Chosen applications to the modified viscous Cahn-Hilliard equation in R^N, the evolution of cell population of varying genotype and the semilinear Schrödinger equation are presented.

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Cite

BibTeXRIS
BibTeX
@misc{czaja2026solutions,
  title = {{Solutions of Duhamel's formula with a sum of mappings}},
  author = {Radosław Czaja and Maria Kania},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07642},
  eprint = {2609.07642},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07642},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Solutions of Duhamel's formula with a sum of mappings
AU  - Radosław Czaja
AU  - Maria Kania
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07642
UR  - https://arxiv.org/abs/2609.07642
AB  - Semilinear evolution equations are solved by means of the variation of constants formula under the action of smoothing linear operators and a finite number of nonlinearities operating between chosen Banach spaces of a given family. Admissible spaces of initial conditions are determined for the existence and uniqueness of a suitable notion of solution. Its maximal time of existence, short time and blow up time profiles, global extendibility and regularity are analyzed. For extrapolated fractional power scale of Banach spaces the fulfillment of the corresponding Cauchy problem is shown. Chosen applications to the modified viscous Cahn-Hilliard equation in R^N, the evolution of cell population of varying genotype and the semilinear Schrödinger equation are presented.
ER  -