Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive Medium.

Let 0 < α < 1, b, T be positive real numbers, Lαu = ut - (…), where … denotes the Riemann-Liouville fractional derivative. This paper consider the problem Lαu(x, t) = δ(x - b)f(u(x, t)) in (-∞,∞) x (0, T], subject to initial and boundaries condition {u(x, 0) = φ(x) in (-∞, ∞), with φ (x) → as |x| → to u(x, t) → 0 for 0 < t ≤ T, as |x| → ∞, where δ(x - b) is the Dirac delta function, f and φ are given functions. We assume that φ ≥ 0, f (0) ≥ 0, f'(u) > 0, f"(u) > 0 for u > 0. By using Green's function, the problem is converted into an integral equation. It is shown that there exists tb such that for 0 ≤ t ≤ tb, the integral equation has a unique nonnegative continuous solution u; if tb is finite, then u is unbounded in [0, tb). Then, u is proved to be the solution of the original problem. [ABSTRACT FROM AUTHOR]