Asymptotic behavior of the drift-diffusion semiconductor equations (Q5894917)

The paper under review is concerned with the following system of PDE's in semiconductor theory: \[ \frac{\partial u_i}{\partial t} - \mu_i\nabla \cdot(\nabla u_i+q_i u_i\nabla\psi) = (c_1u_1+c_2u_2)(1-u_1u_2)+g\quad\text{in}\quad G, \] \[ -\Delta\psi= u_2-u_1+f \quad \text{in}\quad G \] \[ u_i=\bar{u}_i, \;\; \psi=\bar{\psi} \quad\text{on}\quad \Gamma_D, \quad \frac{\partial u_i}{\partial\nu} = \frac{\partial\psi}{\partial\nu} = 0 \quad \text{on}\quad \Gamma_N, \] \[ u_i(\cdot,0) = u_{0i}\quad \text{in} \quad G \] \((i=1,2)\). Here the unknown functions \(u_1,u_2,\psi\) denote the electron density, the hole density and the electrostatic potential, respectively, \(\mu_1,\mu_2=\text{const} >0\) (mobilities of electrons resp. holes), \(c_1, c_2\) are given constants and \(f,g,\bar{u}_1, \bar{u}_2, \bar{\psi}\), \(u_{01}, u_{02}\) are given functions. \(G\) is a bounded domain in \(\mathbb{R}^n\) (\(n\leq 3\)), and \(\partial\Omega=\Gamma_D\cup \Gamma_N\) (disjoint). The authors make use of the existence and uniqueness of a global weak solution to the above system. The main results of the paper are as follows: 1. The dynamical system has a compact, connected, maximal attractor which absorbs \(L^\infty\)-bounded sets. 2. The associated semigroup \(S(t)\) is Fréchet differentiable with respect to the norm in \(L^2\times L^2\). 3. The attractor of the dynamical system has finite Hausdorff dimension.