A family of fourth-order q-logarithmic equations

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