Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems (Q2706211)

The problem of existence of the nonnegative (positive) solutions for the nonlinear fourth-order equation in one-space dimension, arising in study of interface fluctuations in spin systems and in quantum superconductor modeling is studied. As a starting point, known results concerning only nonnegative solutions locally in time in case of periodic boundary conditions are taken into account. In the paper the problem of existence of nonnegative solutions globally in time for general initial data is solved. This problem is considered on a bounded interval subject to initial as well as Dirichlet and Neumann boundary conditions. The property that the solutions stay nonnegative is obtained by an exponential transformation of variables. For the proof of existence of the solutions globally in time the entropy-type estimates are used. The regularity and long-time behaviour of solutions under consideration are demonstrated. The numerical experiments underlining the preservation of positivity are also presented.