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Abstract

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.