Convergence to equilibrium for delay-diffusion equations with small delay
The nonlinear reaction-diffusion equation with delay \[ \begin{aligned} u_ t- \Delta u & = f(u(t),\;u(t-\tau)),\quad\text{in }\Omega\times \mathbb{R}^ +,\\ u & = u_ 0\quad\text{in }\Omega\times [-\tau,0],\\ u & = 0\quad\text{or }\partial u/\partial\nu= 0\quad\text{on }\partial\Omega\times \mathbb{R}^ +\end{aligned} \] is considered, where \(f\) is Lipschitz and satisfies certain one-sided growth conditions. First the equation is shown to have a global unique solution. The main result shows that all bounded solutions (defined on \(C^ 0([-\tau,0]; L^ 2(\Omega))\)) tend to the equilibrium solutions, i.e. the solutions of the elliptic boundary value problem \[ -\Delta u= f(u,u)\quad\text{in }\Omega,\quad u=0\quad\text{or }\partial u/\partial\nu= 0\quad\text{on }\partial\Omega. \]
scientific article; zbMATH DE number 168054
scientific article