On initial and terminal value problems for fractional nonclassical diffusion equations
Copyright policy )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.
- University of Seville Spain
- Ho Chi Minh City University of Science Viet Nam
- Vietnam National University, Ho Chi Minh City Viet Nam
regularity estimates, Smoothness and regularity of solutions to PDEs, regularization and error estimate, Fractional partial differential equations, A priori estimates in context of PDEs, Fractional nonclassical diffusion equation, Fractional derivatives and integrals, well-posedness, Ill-posed problems for PDEs, Ultraparabolic equations, pseudoparabolic equations, etc., ill-posedness
regularity estimates, Smoothness and regularity of solutions to PDEs, regularization and error estimate, Fractional partial differential equations, A priori estimates in context of PDEs, Fractional nonclassical diffusion equation, Fractional derivatives and integrals, well-posedness, Ill-posed problems for PDEs, Ultraparabolic equations, pseudoparabolic equations, etc., ill-posedness
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