On the basic equations for carrier transport in semiconductors (Q1101609)

The authors study the system \[ \partial u_ i/\partial t = D_ i \text{div(grad} u_ i+q_ iu_ i \text{grad} v)-R(u_ 1,u_ 2),\quad i=1,2\text{ in } {\mathbb{R}}_+\times G, \] \[ -\text{div}(a \text{grad} v) = f+u_ 1-u_ 2\text{ in } {\mathbb{R}}_+\times G,\quad u_ 1=U_ i,\quad v=V,\quad i=1,2,\text{ on } {\mathbb{R}}_+\times \Gamma_ D, \] \[ \nu (\text{grad} u_ i+q_ iu_ i \text{grad} v)=0,\quad i=1,2,\text{ on } {\mathbb{R}}_+\times \Gamma_ N, \] \[ a(\partial v/\partial \nu)+bv=g\quad on\quad {\mathbb{R}}_+\times \Gamma_ N,\quad u_ i(0,x)=u_ i^{(0)}(x)\text{ in } G, \] where G is an open subset of \({\mathbb{R}}^ N\) (N\(\leq 3)\), whose boundary is constituted of two disjoint parts \(\Gamma_ D\) and \(\Gamma_ N\), \(D_ 1\), \(D_ 2\), \(g_ 1\) and a are strictly positive constants, \(q_ 2=-q_ 1\) and \({\mathbb{R}}: {\mathbb{R}}^ 2\to {\mathbb{R}}\) is a given nonlinearity. Results on existence, uniqueness and asymptotic behavior of the solution \((u_ 1,u_ 2,v)\) as \(t\to +\infty\) were already proved by the first author [Z. Angew. Math. Mech. 65, 101-108 (1985; Zbl 0579.35016)]. The purpose of the paper is to improve those results. In particular, some regularity assumptions are weakened. The open set G, for instance, is supposed to be only a bounded Lipschitzian domain.