On initial and terminal value problems for fractional nonclassical diffusion equations

Abstract

In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case 0 < α ≤ 1 and show the ill-posedness in the sense of Hadamard in the case α > 1. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in L q norm is first established.

Note on a time fractional diffusion equation with time dependent variables coefficients

“…We obtain the existence and uniqueness of the global solution for our problem. Our main techniques are based on some previous techniques as in [7,8].…”

“…The regularity estimates for the mild solution are established in some various spaces. To overcome these diculties, we learned a very interesting technique in recent articles [7,8].…”

On a time fractional diffusion with nonlocal in time conditions

“…By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order. Let us refer to many papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

“…Our new techniques in this section are based on applying some Sobolev embedding. Based on the ideas of [7], we develop some similar techniques to deal with the L p case. Let us emphasize that a regularized problem is not considered in [36].…”

Some well‐posed results on the time‐fractional Rayleigh–Stokes problem with polynomial and gradient nonlinearities

“…To provide an overview of the fractional‐time Rayleigh–Stokes problem, we consider some interesting studies closely related to our work as follows. The initial value problem and the terminal value problem for diffusion equations with non‐integer derivative were considered by Tuan et al and Xu et al in their studies 15,16 . Caraballo et al investigate the stochastic‐Stokes problem with the fractional Caputo derivative and finite/infinite delay.…”