Global nonnegative solutions of a nonlinear fourth-order parabolic equation or quantum systems

The existence of nonnegative weak solutions globally in time of nonlinear fourth-order parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on bounded interval subject to initial and Dirichlet and Neumann boundary conditions. Further, the initial datum is assumed only to be nonnegative and to satisfy weak integrability condition. The main difficulty of the existence proof is to ensure that the solutions stay nonnegative and exist globally in time. The rst property is obtained by an exponential transformation of variables. Moreover, entropy-type estimates allow for the proof of the second property. Results concerning the regularity and long-time behavior are given. Finally, numerical experiments underlining the preservation of positivity are presented.