ASYMPTOTIC BEHAVIORS OF FUNDAMENTAL SOLUTION AND ITS DERIVATIVES TO FRACTIONAL DIFFUSION-WAVE EQUATIONS

Abstract

Let p (t, x) be the fundamental solution to the problem partial derivative(alpha)(t) u = -(-Delta)(beta)u, alpha is an element of (0, 2), beta is an element of (0, infinity). If alpha, beta is an element of (0, 1), then the kernel p (t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives D-x(n) (-Delta(x))(gamma) D-t(sigma) I(delta)(t)p(t, x), for all n is an element of Z(+), gamma is an element of [0, beta], sigma, delta is an element of [0, infinity), where D-x(n) is a partial derivative of order n with respect to x, (-Delta(x))(gamma) is a fractional Laplace operator and D-t(sigma) and I-t(delta) are Riemann-Liouville fractional derivative and integral respectively.