Local Well-posedness of Nonlinear Time-fractional Diffusion Equation

We study the local well-posedness of the following time-fractional nonlinear diffusion equation {(C)D(0,t)(alpha,lambda)u - Delta u = vertical bar u vertical bar(p-1)u, (x,t) is an element of R-n x (0,T], u(x, 0) = u(0)(x), x is an element of R-n, where 0 < alpha < 1, lambda >= 0, T < infinity, p > 1 , u(0) is an element of C-0(R-n) and D-c(0,t)alpha,lambda denotes Caputo tempered fractional derivative of order alpha. The local existence and uniqueness results are obtained from heat kernel and fixed point theorem. Then, we extend the solution to establish a maximal mild solution. Moreover, we provide estimate for continuous dependence on initial condition.