Well-Posedness and Regularity Analyses for Nonlocal Nonautonomous System (CSIAM Transactions on Applied Mathematics, 2025)
The time-space nonlocal evolution equations are powerful implementationfor modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equationwhere $χ$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a boundedreaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced bya symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined byis the unique mild as well as weak solution. By further analysis, one can get that theabove stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the correspondingdynamical behaviors are displayed.
Keywords: Stochastic representation; time fractional nonautonomous equation; well-posedness; regularity