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Blow‐up regions for a class of fractional evolution equations with smoothed quadratic nonlinearities

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  • Diego Chamorro
  • Elena Issoglio

Abstract

We consider an n‐dimensional parabolic‐type PDE with a diffusion given by a fractional Laplace operator and with a quadratic nonlinearity of the “gradient” of the solution, convoluted with a term b$\mathfrak {b}$ which can be singular. Our first result is the well‐posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow‐up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b$\mathfrak {b}$ to ensure blow‐up of the solution in finite time.

Suggested Citation

  • Diego Chamorro & Elena Issoglio, 2022. "Blow‐up regions for a class of fractional evolution equations with smoothed quadratic nonlinearities," Mathematische Nachrichten, Wiley Blackwell, vol. 295(8), pages 1462-1479, August.
    • Handle: RePEc:bla:mathna:v:295:y:2022:i:8:p:1462-1479
    DOI: 10.1002/mana.202000480
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    1. Michael Hinz & Elena Issoglio & Martina Zähle, 2014. "Elementary Pathwise Methods for Nonlinear Parabolic and Transport Type Stochastic Partial Differential Equations with Fractal Noise," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 123-141, Springer.
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