A family of fourth-order q-logarithmic equations

Abstract

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∈(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 L1-norm of the constant steady state. Finally, we provide illustrative numerical examples supporting the analytical results.