Lyapunov functions and \(L^p\)-estimates for a class of reaction-diffusion systems. (English) Zbl 0966.35022

Horstmann, Dirk

Colloq. Math. 87, No. 1, 113-127 (2001).

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

MSC:

35B45	A priori estimates in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
35K50	Systems of parabolic equations, boundary value problems (MSC2000)
35K55	Nonlinear parabolic equations
35K57	Reaction-diffusion equations
35A07	Local existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

asymptotic behaviour of solution; uniform \(L^p\)-estimates