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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2022-08, Vol.152 (4), p.989-1031
• Main Authors: Tuan Nguyen, Anh, Caraballo, Tomás, Tuan, Nguyen Huy
• Format: Article
• Language: English
• Subjects: Biharmonic equations; Boundary value problems; Formulas (mathematics); Fourier transforms; Nonlinearity; Operators (mathematics); Orlicz space; Partial differential equations; Phase transitions
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/prm.2021.44

In this study, we investigate the intial value problem (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 $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ 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 in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to 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.