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

Abstract

<p style='text-indent:20px;'>We provide explicit examples of finite time <inline-formula><tex-math id="M1">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-blow up for the solutions of <inline-formula><tex-math id="M2">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> reaction-diffusion systems for which three main properties hold: positivity is preserved for all time, the total mass is uniformly controlled and the growth of the nonlinear reaction terms is superquadratic. They are obtained by choosing the space dimension large enough. This is to be compared with recent global existence results of uniformly bounded solutions for the same kind of systems with quadratic or even slightly superquadratic growth depending on the dimension. Such blow up may occur even with homogeneous Neumann boundary conditions. All these <inline-formula><tex-math id="M3">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-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.</p>