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

Abstract

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.

Novel spectral schemes to fractional problems with nonsmooth solutions

“…Through numerous efforts, numerical methods have been developed for the solution of FIVP and FPDP. Some of these contributions can be found in [36][37][38].…”

On Caputo fractional elliptic equation with nonlocal condition

“…The reason they are of interest comes from the model sticking to memory eects. We temporarily list some articles about Caputo or Riemann-Liouville [1,2,3,18,10,12,14,9,11,15,20,13,19] and some other Email address: tiennv55@fe.edu.vn (Van Tien Nguyen) derivatives, see [21, 22? , 23, 24, 25].…”

On a Fractional Parabolic Equation with Regularized Hyper-Bessel Operator and Exponential Nonlinearities

“…It should be noticed that parabolic equations with these polynomial source have been studied almost completely, to our awareness. Indeed, we desire to mention great works [14,15,23,24] as proof of the great interest among mathematicians around the world on this subject. However, as derived by many authors [35][36][37][38] that approximation to infinity behavior of q is more appriciate in some specific cases.…”

“…The second and also the most difficult problem for us as mentioned above, the fast growth of the nonlinearity J. In order to overcome this issue, previous work [23,24] made smallness assumption on the initial data function. It seems to be a efficient method.…”