On boundedness and stability of solutions to quasilinear parabolic equations with nonlocal nonlinearities (Q1178479)

The author investigates the phenomena which involve that the diffusion of charged particles in an electric field often leads to a reaction diffusion system coupled with a nonlinear second-order elliptic equation: \[ \partial U/\partial t=LU+F(x,t,U,v), \qquad 0=\Delta v+g(t,x,U,v), \] here \(U=(u_ 1,\dots,u_ N)\), \(x\in\Omega\), \(t>0\). The author restricts himself to the case of a single component, \(N=1\), although generalization of the results to weakly coupled systems is possible. A modified method of Lyapunov functions is used.