In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter $\epsilon$. The $L^p-L^q$ estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the $C_0-$semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices $p\in [1,+\mathrm{\infty })$ and $s\in (1,+\mathrm{\infty })$, the mild solution of the approximation problem converges to the solution of the associated limit problem in $L^p((0,T),L^s({\mathbf{R}}^n))$ as $\epsilon \to 0^{+}$. The resolvent operator family and a set of kernel $k(t)$ assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as $\alpha \to 2^{-}$.

This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $r$ and the nonlinear parameter $p$, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $r$ and $p$. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $r=2$ and $p=1$. Second, we consider the corresponding equation with a stationary structure.

In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.

In this paper we consider some inverse problems of determining the diffusion matrix between different structures for the time fractional diffusion equation featuring a Caputo derivative. We first study an inverse problem of determining the diffusion matrix in the period structure using data from the corresponding homogenized equation, then we investigate an inverse problem of determining the diffusion matrix in the homogenized equation using data from the corresponding period structure of the oscillating equation. Finally, we establish the stability and uniqueness for the first inverse problem, and the asymptotic stability for the second inverse problem.