Space–time fractional Cauchy problem in spaces of generalized functions

The Cauchy problem for an equation with fractional derivatives in Bessel potential spaces

“…(16) Since F 0 * G 0 = f * G 1 satisfies (3), from (5) (7) follows from this estimate, (15), and (16). Thus, we have proven that a solution (6) to (2) belongs to C α,β ([0, T ]; H s,p (R n )).…”

“…Definition [1,15,16]. A pair of functions (G 0 (x, t), G 1 (x, t)) is called a Green's vector-function of (1) (and (2) as well) if for every function F 0 (x, t) bounded and Hölder continuous in the variables x ∈ R n for every t ∈ [0, T ] and every bounded continuous function u 0 (x) the function…”

“…of order β ∈ (0, 1) for these and more general equations and β ∈ (1, 2) have been studied by many authors [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16] (for α = 2, the bibliography can be found in [1]). Here the expression (−Δ) α/2 is defined with the use of the Fourier transform as follows:…”

“…Solvability of (1) for u 0 and F 0 from the space D of distributions is given in [15]. The character of singularities at t = 0 is established in dependence on the order of singularities of a distribution in the initial condition and power singularities of the right-hand side.…”

An application of integral equation method to fractional Stefan problem (in Ukrainian)

“…Питання роз'вязностi рiзних задач для рiвняння аномальної дифузiї вивчалося, зокрема, в [8]- [13]. Особливо вiзначимо роботи [14], [15], в яких використовується метод граничних iнтегралiв, за допомогою якого задачi Дiрiхле i Робiна зведено до iнтегральних рiвнянь типу Вольтера.…”