Global existence of solutions of an activator-inhibitor system. (English) Zbl 1101.35046

The author considers the generalized Gierer-Meinhardt system \[ \begin{alignedat}{2} \frac{\partial u_j}{\partial t}&= d_j\triangle u_j-a_ju_j+g_j(x,u) &\quad&\text{in }\Omega \times [0,T),\\ \frac{\partial u_j}{\partial \nu }&= 0&&\text{on } \partial \Omega \times [0,T),\\ u_j(x,0)&= \varphi _j(x) &&\text{in }\Omega,\end{alignedat} \] where \(\Omega \) is a smooth bounded domain in \(\mathbb{R}^n\) with \(\nu \) its unit outer normal, \(j=1,2,\) \(u=( u_1,u_2) \) and \[ \begin{aligned} g_1(x,u) &=\rho _1(x,u) \frac{u_1^p}{u_2^q}+\sigma _1(x) , \\ g_2( x,u) &=\rho _2( x,u) \frac{u_1^r}{u_2^s}+\sigma _2( x). \end{aligned} \] Here \(d_j\) and \(a_j\) are positive constants, \(\rho _1\geq 0\), \(\rho _2>0\), \( \sigma _j\geq 0\) are bounded smooth functions and \(p,q,r,s\) are nonnegative constants satisfying \( 0<\frac{p-1}r<\frac q{s+1}. \) When \(p-1<r\), it is showed that there is a unique global solution. Then, the author also concerns with the asymptotic behavior of the global solutions and asymptotic bounds of global solutions are established which yield new a priori estimates of stationary solutions.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations
92C15 Developmental biology, pattern formation
35B45 A priori estimates in context of PDEs
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