Article Contents

2023, Volume 43, Issue 3&4: 1686-1701. Doi: 10.3934/dcds.2022039

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Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth

Michel Pierre^1, and
Didier Schmitt^2,

1.
Ecole Normale Supérieure de Rennes, and Institut de Recherche Mathématique de Rennes, Campus de Ker Lann, 35170-Bruz, France
2.
Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, BP 70239, 54506 Vandœ uvre-lès-Nancy Cedex, France

Received: September 2021

Revised: February 2022

Early access: April 2022

Published: March 2023

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Abstract

We provide explicit examples of finite time $ L^\infty $-blow up for the solutions of $ 2\times 2 $ reaction-diffusion systems for which three main properties hold: positivity is preserved for all time, the total mass is uniformly controlled and the growth of the nonlinear reaction terms is superquadratic. They are obtained by choosing the space dimension large enough. This is to be compared with recent global existence results of uniformly bounded solutions for the same kind of systems with quadratic or even slightly superquadratic growth depending on the dimension. Such blow up may occur even with homogeneous Neumann boundary conditions. All these $ L^\infty $-blowing up solutions may be extended as weak global solutions. Blow up examples are also provided in space dimensions one, two and three with various growths.

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Mathematics Subject Classification: Primary: 35B44, 35K57, 35K40; Secondary: 35K91.

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References

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