Uniform long-time behavior of solutions of parabolic equations depending on slow time
The reaction-diffusion system \[ \partial_t=a\Delta u-f(u,\lambda t)-g_0, \quad u| _{\partial\Omega}=0 \] is considered with the initial condition \(u| _{t=\theta}=u_\theta\). Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary, \(u=(u^1(x,t),\dots,u^m(x,t))=u(x,t)\) is a vector-function, \(x\in\Omega\), \(t\geq 0\). \(a\) is a diagonal matrix \(a=\text{ diag }(\nu_1,\dots,\nu_m)\), \(\nu_i\geq\nu_0>0\). The function \(g_0=(g^1,\dots,g^m)\) depends only on \(x\). The nonlinearity \(f(u,\tau)=(f^1(u,\tau),\dots,f^m(u,\tau))\) is potential, that is \(f_i(u,\tau)=\partial F(u,\tau)/\partial u^i\), \(i=1,\dots,m\), where \(F(0,\tau)=0\). It is assumed that \(f\) is continuously differentiable on \(\mathbb{R}^m\times \mathbb{R}\) and that \(f(u,\tau)u\geq -C\). This system generates a semigroup \(\{S_t\}\) in the space \(E=(H_1(\Omega)\cap\{u| _{\partial\Omega}=0\})\times \mathbb{R}\times \mathbb{R}\). Local invariant manifolds of the semigroup \(\{ S_t\}\) and exponential attractors consisting of invariant manifolds are constructed.
scientific article; zbMATH DE number 1247707