## 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.

### MSC:

 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

### Keywords:

asymptotic bound; generalized Gierer-Meinhardt system
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