2019/10/20 by Ngoc, Tran Bao, Kian, Yavar, Tuan, Nguyen Huy
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.09006
The initial inverse problem of finding solutions and their initial values (t = 0) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time (t = T). Our work focuses on the existence and regularity of mild solutions in two cases: \beginitemize \item[--] The first case: The nonlinearity is globally Lipschitz and uniformly bounded which plays important roles in PDE theories, and especially in numerical analysis. \item[--] The second case: The nonlinearity is locally critical which widely arises from the Navier-Stokes, Schrödinger, Burgers, Allen-Cahn, Ginzburg-Landau equations, etc. \enditemize Our solutions are local-in-time and are derived via fixed point arguments in suitable functional spaces. The key idea is to combine the theories of Mittag-Leffler functions and fractional Sobolev embeddings. To firm the effectiveness of our methods, we finally apply our main results to time fractional Navier-Stokes and Allen-Cahn equations.