On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

Abstract

In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function Ξ(z) = e |z| p − 1 and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up at finite time if the initial value is regular. In case of singular initial data, the local-in-time and global-in time existence are derived. In addition, the regularity of the mild solution is also investigated. For the IVP associated with the second one, some modifications on the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.