The existence of a uniform attractor in a space of higher regularity is proved for the multi-valued process associated with the nonautonomous reaction-diffusion equation on an unbounded domain with delays for which the uniqueness of solutions need not hold. A new method for checking the asymptotical upper-semicompactness of the solutions is used.

First, we introduce the concept of pullback $\omega$ -limit compactness for multi-valued processes, as an extension of the similar concept in the autonomous and nonautonomous framework. Next, we present the necessary and sufficient conditions (pullback dissipativeness and pullback $\omega$ -limit compactness) for the existence of a nonempty local bounded kernel (kernel sections are all compact, invariant and pullback attracting) of an infinite dimensional multi-valued process. In addition, we prove a result ensuring the existence of a uniform attractor and the uniform forward attraction of the inflated kernel sections of a family of multi-valued processes under the general assumptions of point dissipativeness and uniform $\omega$ -limit compactness. Finally, we illustrate the abstract theory with a nonlinear reaction-diffusion model in an unbounded domain.

Compartment models with classical derivatives have diverse applications and attracted a lot of interest among scientists. To model the dynamical behavior of the particles that existed in the system for a long period of time with little chance to be removed, a power-law waiting time technique was introduced in the most recent work of Angstmann et al. [2]. The divergent first moment makes the power-law waiting time distribution less physical because of the finite lifespan of the particles. In this work, we take the tempered power-law function as the waiting time distribution, which has finite first moment while keeping the power-law properties. From the underlying physical stochastic process with the exponentially truncated power-law waiting time distribution, we build the tempered fractional compartment model. As an application, the tempered fractional SEIR epidemic model is proposed to simulate the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. (Colombia). Some analysis and numerical simulations are carried out around the equilibrium behavior.

We consider a class of nonlinear delay lattices $$ \ddot{u}_i(t)+(-1)^p\triangle^pu_i(t)+\lambda u_i(t)+\dot{u}_i(t)=h_i(u_i(t-\rho(t)))+f_i(t),~~~i \in \mathbb{Z}, $$ where $\lambda$ is a real positive constant, $p$ is any positive integer and $\triangle$ is the discrete one-dimensional Laplace operator. Under suitable conditions on $h$ and $f$ we prove the existence of pullback attractors for the multi-valued process associated with the system for which the uniqueness of solutions need not hold.

In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order $\alpha\in(0,1)$. We first prove the local and global existence, decay and regularity properties of mild solutions when the initial data belongs to $C([-h,0];D(A_r^\varepsilon))$. The fractional resolvent operator theory and some techniques of measure of noncompactness are successfully applied to obtain the results.