A Nonlinear Fourth-order Parabolic Equation with Nonhomogeneous Boundary Conditions

Abstract. A nonlinear fourth-order parabolic equation with nonhomogeneous Dirichlet–Neumann boundary conditions in one space dimension is analyzed. This equation appears, for instance, in quantum semiconductor modeling. The existence and uniqueness of strictly positive classical solutions to the stationary problem are shown. Furthermore, the existence of global nonnegative weak solutions to the transient problem is proved. The proof is based on an exponential transformation of variables and new “entropy” estimates. Moreover, it is proved by the entropy–entropy production method that the transient solution converges exponentially fast to its steady state in the L1 norm as time goes to infinity, under the condition that the logarithm of the steady state is concave. Numerical examples show that this condition seems to be purely technical.