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doi: 10.3934/dcds.2022039
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## Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth

 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

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.

Citation: Michel Pierre, Didier Schmitt. Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022039
##### References:
 [1] J. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500. [2] M. C. Caputo, T. Goudon and A. F. Vasseur, Solutions of the 4-species quadratic reaction-diffusion systems are bounded and $C^\infty$-smooth, in any space dimension, Anal. PDE, 12 (2019), 1773-1804.  doi: 10.2140/apde.2019.12.1773. [3] B. P. Cupps, J. Morgan and B. Q. Tang, Uniform boundedness for reaction-diffusion systems with mass dissipation, SIAM J. Math. Anal., 53 (2021), 323-350.  doi: 10.1137/19M1277977. [4] K. Fellner, J. Morgan and B. Q. Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 635-651.  doi: 10.3934/dcdss.2020334. [5] K. Fellner, J. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Ann. IHP C, Ana. Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003. [6] T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Sup., 43 (2010), 117-142.  doi: 10.24033/asens.2117. [7] Y.I. Kanel', Solvability in the large of a system of reaction-diffusion equations with the balance condition, Differentsialýe Uravneniya, 26 (1990), 448-458. [8] E.-H. Laamri, Global existence of classical solutions for a class of reaction-diffusion systems, Acta Appl. Math., 115 (2011), 153-165.  doi: 10.1007/s10440-011-9613-y. [9] M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168.  doi: 10.1007/s000280300007. [10] M. Pierre, Global existence in reaction-diffusion systems with dissipation of mass : A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4. [11] M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269.  doi: 10.1137/S0036141095295437. [12] M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106.  doi: 10.1137/S0036144599359735. [13] M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction-diffusion systems with quadratic growth, Indiana Univ. Math. J., 68 (2019), 291-322.  doi: 10.1512/iumj.2019.68.7447. [14] P. Souplet, Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth, J. Evol. Equ., 18 (2018), 1713-1720.  doi: 10.1007/s00028-018-0458-y. [15] B. Q. Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Commun. Math. Sci., 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.

show all references

##### References:
 [1] J. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500. [2] M. C. Caputo, T. Goudon and A. F. Vasseur, Solutions of the 4-species quadratic reaction-diffusion systems are bounded and $C^\infty$-smooth, in any space dimension, Anal. PDE, 12 (2019), 1773-1804.  doi: 10.2140/apde.2019.12.1773. [3] B. P. Cupps, J. Morgan and B. Q. Tang, Uniform boundedness for reaction-diffusion systems with mass dissipation, SIAM J. Math. Anal., 53 (2021), 323-350.  doi: 10.1137/19M1277977. [4] K. Fellner, J. Morgan and B. Q. Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 635-651.  doi: 10.3934/dcdss.2020334. [5] K. Fellner, J. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Ann. IHP C, Ana. Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003. [6] T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Sup., 43 (2010), 117-142.  doi: 10.24033/asens.2117. [7] Y.I. Kanel', Solvability in the large of a system of reaction-diffusion equations with the balance condition, Differentsialýe Uravneniya, 26 (1990), 448-458. [8] E.-H. Laamri, Global existence of classical solutions for a class of reaction-diffusion systems, Acta Appl. Math., 115 (2011), 153-165.  doi: 10.1007/s10440-011-9613-y. [9] M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168.  doi: 10.1007/s000280300007. [10] M. Pierre, Global existence in reaction-diffusion systems with dissipation of mass : A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4. [11] M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269.  doi: 10.1137/S0036141095295437. [12] M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106.  doi: 10.1137/S0036144599359735. [13] M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction-diffusion systems with quadratic growth, Indiana Univ. Math. J., 68 (2019), 291-322.  doi: 10.1512/iumj.2019.68.7447. [14] P. Souplet, Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth, J. Evol. Equ., 18 (2018), 1713-1720.  doi: 10.1007/s00028-018-0458-y. [15] B. Q. Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Commun. Math. Sci., 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.
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