Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems
Tom 87 / 2001
Colloquium Mathematicum 87 (2001), 113-127
MSC: 35A07, 35B40, 35K40, 35K50, 35K55, 35K57.
DOI: 10.4064/cm87-1-7
Streszczenie
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.
