Nonlocal abstract diffusion equations and applications

Here, the Cauchy problem for linear and nonlinear nonlocal diffusion equations are studied. The equation involve a convolution integral operators with a general kernel operator functions whose Fourier transform are operator functions defined in a Hilbert space H together with some growth conditions. By assuming enough smoothness on the initial data and the operator functions, the local and global existence, uniqueness and L^{p}-regularity properties of solutions are established. We can obtain a different classes of nonlocal diffusion equations by choosing the space H and linear operators which occur in a wide variety of physical systems