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We present a domain decomposition method on a semilinear hyperbolic system with multiple relaxation times. In the region where the relaxation time is small, an asymptotic equilibrium equation can be used for computational efficiency. An interface condition based on the sign of the characteristic speed at the interface is provided to couple the two systems in a domain decomposition setting. A rigorous analysis, based on the Laplace Transform, on the $ L^2$ error estimate is presented for the linear case, which shows how the error of the domain decomposition method depends on the smaller relaxation time, and the boundary and interface layer effects. The given convergence rate is optimal. We present a numerical implementation of this domain decomposition method, and give some numerical results in order to study the performance of this method.

We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.

Macrophage activation and persistent inflammation contribute to the pathological process of spinal cord injury (SCI). It was reported that M2 macrophages were induced at 37 days after SCI but M2 markers were reduced or eliminated after 1 week. By contrast, M1 macrophage response is rapidly induced and then maintained at injured spinal cord. However, factors that modulate macrophage phenotype and function are poorly understood. We developed a model to distinguish bonemarrow derived macrophages (BMDMs) from residential microglia and explored how BMDMs change their phenotype and functions in response to the lesionrelated factors in injured spinal cord. Infiltrating BMDMs expressing higher Mac2 and lower CX3CR1 migrate to the epicenter of injury, while microglia expressing lower Mac2 but higher CX3CR1 distribute to the edges of lesion. Myelin debris at the lesion site switches BMDMs