The fundamental solution of a diffusion-wave equation of fractional order

Asymptotic properties of solutions of the fractional diffusion-wave equation

“…Under these assumptions, the problem (1.1),(2.1) possesses a classical solution u(t, x) [20,15]. This means that u(t, x) belongs to C 2 in x for each t > 0; u(t, x) belongs to C 1 in (t, x), and for any x ∈ R n the fractional integral…”

“…is the unique bounded classical solution of the problem (4.1); see [15,20]. Without the above smoothness assumptions, we can investigate the function (4.2) interpreting it as a generalized solution of the problem (4.1).…”

“…the change in the order of integration was justified by an estimate of the Wright function given in Lemma 1 of [20]. Next, we use the identity…”

“…While the qualitative behavior of the kernel Y (α) is given by (2.5), to study u 3 we need an explicit representation [20]:…”

“…Its properties are intermediate between those of the classical heat and wave equations. This equation and its generalizations have been studied by many authors (see [14,15,17,20] for further references) who put their emphasis either on properties similar to those of parabolic equations, like the regularity properties, or those resembling hyperbolic equations, like the exponential decay of the fundamental solution outside the fractional light cone.…”