In this article, an initial-/boundary-value problem for a nonlocal diffusion system is treated. In the first part of the article, the unique solvability of the problem is established, and the asymptotic behavior of solutions is discussed by means of some key estimates. For the problem, it is also proved that the corresponding initial-value problem for a system of ordinary differential equations are of crucial importance. The second part of the article is devoted to the analysis of some nonlocal reaction-diffusion systems which are obtained as the special case where the coefficient matrices in the original system are diagonal. Lotka-Volterra prey-predator interaction and competitive interaction for two species are taken as fundamental examples of reaction kinetics; the stability and asymptotic behavior of solutions to these ecological models are inspected.

Let $\Omega\subseteq {\mathbb{R}^{N}}$ be a bounded open set, $N\geq 2$, and let $p>1$; we study the asymptotic behavior with respect to the time variable $t$ of the entropy solution of nonlinear parabolic problems whose model is $$ \begin{cases} u_{t}(x,t)-\Delta_{p} u(x,t)=\mu & \text{in}\ \Omega\times(0,T),\\ u(x,0)=u_{0}(x) & \text{in}\ \Omega, \end{cases} $$ where $T>0$ is any positive constant, $u_0 \in L^{1}(\Omega)$ a nonnegative function, and $\mu\in \mathcal{M}_{0}(Q)$ is a nonnegative measure with bounded variation over $Q=\Omega\times(0,T)$ which does not charge the sets of zero $p$-capacity; moreover, we consider $\mu$ that does not depend on time. In particular, we prove that solutions of such problems converge to stationary solutions.

We study the barotropic compressible Navier-Stokes equations in a bounded or an unbounded domain $\Omega $ of $ \mathbf{R}^3$. The initial density may vanish in an open subset of $\Omega$ or be positive but vanish at space infinity. We first prove the local existence of solutions $(\rho^{(j)}, u^{(j)})$ in $C([0,T_* ]; H^{2(k-j)+3} \times D_0^1 \cap D^{2(k-j)+3} (\Omega ) )$, $0 \le j \le k, k \ge 1$ under the assumptions that the data satisfy compatibility conditions and the initial density is sufficiently small. To control the non-negativity or decay at infinity of density, we need to establish a boundary-value problem of a $(k+1)$-coupled elliptic system which may not be, in general, solvable. The smallness condition of the initial density is necessary for the solvability of the elliptic system; this is not necessary when the initial density has positive lower bound. Secondly, we prove the global existence of smooth radially symmetric solutions of isentropic compressible Navier-Stokes equations by controlling every regularity with $|\rho|_{L^\infty}$.