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On initial and terminal value problems for fractional nonclassical diffusion equations. (English) Zbl 1456.35222

Summary: 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 \(0<\alpha\leq 1\) and show the ill-posedness in the sense of Hadamard in the case \(\alpha>1\). 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 \(L^q\) norm is first established.

MSC:

35R11 Fractional partial differential equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
26A33 Fractional derivatives and integrals
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35R25 Ill-posed problems for PDEs
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