We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

In this paper, a diffuse-interface immersed boundary method (IBM) is proposed to treat three different thermal boundary conditions (Dirichlet, Neumann, Robin) in thermal flow problems. The novel IBM is implemented combining with the lattice Boltzmann method (LBM). The present algorithm enforces the three types of thermal boundary conditions at the boundary points. Concretely speaking, the IBM for the Dirichlet boundary condition is implemented using an iterative method, and its main feature is to accurately satisfy the given temperature on the boundary. The Neumann and Robin boundary conditions are implemented in IBM by distributing the jump of the heat flux on the boundary to surrounding Eulerian points, and the jump is obtained by applying the jump interface conditions in the normal and tangential directions. A simple analysis of the computational accuracy of IBM is developed. The analysis indicates that the Taylor-Green vortices problem which was used in many previous studies is not an appropriate accuracy test example. The capacity of the present thermal immersed boundary method is validated using four numerical experiments: (1) Natural convection in a cavity with a circular cylinder in the center; (2) Flows over a heated cylinder; (3) Natural convection in a concentric horizontal cylindrical annulus; (4) Sedimentation of a single isothermal cold particle in a vertical channel. The numerical results show good agreements with the data in the previous literatures.

In this article, we present a numerical scheme based on a finite element method in orderto solve a time-dependent convection-diffusion equation problem and satisfy someconservation properties. In particular, our scheme is able to conserve the total energyfor a heat equation or the total mass of a solute in a fluid for a concentration equation,even if the approximation of the velocity field is not completely divergence-free. Weestablish a priori errror estimates for this scheme and we give some numerical exampleswhich show the efficiency of the method.

We study a nonlinear diffusion equationut = uxx + f(u)with Robin boundary condition at x = 0 and with a free boundary conditionat x = h(t), whereh(t) > 0 is a moving boundary representing theexpanding front in ecology models. For anyf ∈ C1 with f(0) = 0, weprove that every bounded positive solution of this problem converges to a stationary one.As applications, we use this convergence result to study diffusion equations withmonostable and combustion types of nonlinearities. We obtain dichotomy results and sharpthresholds for the asymptotic behavior of the solutions.

In this paper, a lower bound is established for the local energy of partial sum of eigenfunctions for Laplace-Beltrami operators (in Riemannian manifolds with low regularity data) with general boundary condition. This result is a consequence of a new pointwise and weighted estimate for Laplace-Beltrami operators, a construction of some nonnegative function with arbitrary given critical point location in the manifold, and also two interpolation results for solutions of elliptic equations with lateral Robin boundary conditions.