A family of fourth-order q-logarithmic equations

摘要

We prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth-order evolution equations, parametrized by a real parameter q is an element of (0,1], which includes the well known thin-film (q = 1/2) and the Derrida-Lebowitz-Speer-Spohn (DLSS) equation (q = 1), subject to periodic boundary conditions in one spatial dimension. In contrast to the gradient flow approach in [25], our method relies on dissipation property of the corresponding entropy functionals (Tsallis entropies) resulting in required a priori estimates, and extends the existence result from [25] to a wider range of the family members, namely to 0 < q < 1/2. Generalized Beckner-type functional inequalities yield an exponential decay rate of (relative) entropies, which in further implies the exponential stability in the L-1-norm of the constant steady state. Finally, we provide illustrative numerical examples supporting the analytical results. (C) 2016 Elsevier Inc. All rights reserved.