On a fractional reaction–diffusion equation

The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation

“…Indeed, applying D α to both sides of the equation in (27), one obtains the equation in (1), and evaluating the equation in (27) at t = 0, one gets u t (0, x) = 0. Problem (27) has been recently studied in [3,13]. In space dimension n = 1 it was first studied by Y. Fujita [22].…”

“…This problem, that represents the heat conductor model with memory [26,40], was originally studied by Y. Fujita [22] for n = 1 (see also [45]). Fujita's method produces an handle representation of solution via multiplier operators and it have been used to study (1) in several directions, see [2,3,13,28].…”

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

“…Numerical solutions of the alternative representation of such direct problem have been studied in [35,36,33,32]. In contrast, the problem of recovering the function u at previous time t ∈ [0, T ) as in (1) is called the backward problem. This kind of equation arises in practical situations in which the initial density of the diffusing substance is not available and we can only measure the density at positive time.…”

“…Motivated by the above, in this paper, we study problem (1) and aim to provide an approximate solution. In reality, it is impossible to get the exact final data ϕ and we only have the noisy physical measurement ϕ.…”

“…We first give some preliminaries which are needed for this paper in Section 2. In Section 3, we establish an integral formulation for the solution of problem (1). In Section 4, we prove that the present problem is not well-posed and then we construct an approximate regularized solution for the 2-dimensional problem using the Fourier truncated method.…”