Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems

Dirk Horstmann¹

¹ Mathematisches Institut Universität zu Köln Weyertal 86–90 D-50931 Köln, Germany

Colloquium Mathematicum, Tome 87 (2001) no. 1, pp. 113-127.

Résumé

We give a sufficient condition for the existence of a Lyapunov function for the system $$\eqalign{ a_t=\nabla(k(a,c)\nabla a-h(a,c)\nabla c),\quad\ x\in{\mit\Omega} ,\ t>0,\cr \varepsilon c_t=k_c{\mit\Delta} c-f(c)c+g(a,c),\quad\ x\in{\mit\Omega} ,\ t>0,\cr} $$ for ${\mit\Omega} \subset\mathbb R^N$, completed with either $a=c=0$, or $$ \frac{\partial a}{\partial n}=\frac{\partial c}{\partial n} =0,\quad \hbox{or}\quad k(a,c)\frac{\partial a}{\partial n}=h(a,c)\frac{\partial c}{\partial n},\ c=0\quad \hbox{ on } \partial{\mit\Omega} \times\{t>0\}. $$ Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.

Détail

DOI : 10.4064/cm87-1-7

Keywords: sufficient condition existence lyapunov function system eqalign nabla nabla a h nabla quad mit omega varepsilon mit delta c f a quad mit omega mit omega subset mathbb completed either frac partial partial frac partial partial quad hbox quad frac partial partial frac partial partial quad hbox partial mit omega times furthermore study asymptotic behaviour solution uniform p estimates