We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neu-mann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coef-ficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κthe asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.

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In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

Given a compact Riemannian manifold $M$, we consider a warped product manifold $\bar M = I \times_h M$, where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalize the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $\sigma_k(\kappa)=\psi(V,\nu(V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.

The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier–Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semiimplicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semiimplicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.