Local existence and nonexistence for fractional in time reaction–diffusion equations and systems with rapidly growing nonlinear terms. (September 2022) - Suzuki, Masamitsu
Abstract: We study the fractional in time reaction–diffusion equation ∂ t α u = Δ u + f ( u ) in R N × ( 0, T ), u ( x, 0 ) = u 0 ( x ) in R N, where 0 < α < 1, N ≥ 1, T > 0 and u 0 ≥ 0 . The fractional derivative ∂ t α is meant in a generalized Caputo sense. We mainly consider the case where f has an exponential or a superexponential growth, and u 0 has a singularity. We obtain integrability conditions on u 0 which explicitly determine local in time existence/nonexistence of a nonnegative mild solution. Moreover, our analysis can be applied to time fractional systems.