REPRESENTATION OF SOLUTIONS AND ASYMPTOTIC BEHAVIOR FOR NONLOCAL DIFFUSION EQUATIONS DESCRIBING TEMPERED LÉVY FLIGHTS

We investigate the Cauchy problem of the time-space nonlocal diffusion equation which describes tempered Lévy flight. The time derivative is defined in the Caputo sense and the spatial derivative is taken as a generalization of the fractional Laplacian. Representations and asymptotic behaviors for fundamental solutions of the nonlocal diffusion equation are considered. Then, we use the fundamental solution to obtain the representation formula of solutions of the Cauchy problem. Lastly, the quantitative decay rates for solutions are proved by employing the Fourier analysis technique.