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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0&gt;\alpha \le 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and show the ill-posedness in the sense of Hadamard in the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm is first established.