On initial and terminal value problems for fractional nonclassical diffusion equations

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 0>\alpha \le 1 and show the ill-posedness in the sense of Hadamard in the case α > 1 \alpha > 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 L^q norm is first established.