Abstract: The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are R -vectorvalued functions of time, while they are vectors of R in elliptic problems. Under natural assumptions, firstand second-order sufficient op...

Single-step discretization methods are considered for equations of the form u, + Au = f(t, u), where A is a linear positive definite operator in a Hubert space H. It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data v are only in H, but also that, in contrast to the situation in t...

We consider the equation ut = ∆u + f(u) on RN . Under suitable conditions on f and the initial value u0 = u(·, 0), we show that as t→∞ the solution u(·, t) approaches a planar propagating terrace, or a stacked family of planar traveling fronts. Using this result, we show the asymptotic one-dimensional symmetry of u(·, t) as well as its quasiconvergence in Lloc(R ).

We study the regularity and fractal dimension estimates of global attractors for a class of semilinear degenerate parabolic equations in bounded domains.

The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rm -vector-valued functions of the time, while the are vectors of Rm in elliptic problems. Under natural assumptions, firstand second-order sufficient optima...

The \closure problem" is a well known and widely discussed problem in transport theory. From the full transport equation one can derive a (innnite) sequence of hyperbolic subsystems for the moments. The question arises how to close the system for the rst n moments. In case of Boltzmann equations it can be answered in the theory of extended thermodynamics. Here we consider transport equations wh...

We consider a reaction-diffusion-advection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competit...