We generalize previous uniqueness results on a semilinear elliptic equation with zero Dirichlet boundary condition and superlinear, subcritical nonlinearity. Our proof is based on a bifurcation approach and a Pohozaev type integral identity, which greatly simplifies the previous arguments.

This paper is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including one not previously considered. These problems share several common difficulties including the following: (a) The steady state solution which appears in the coefficients of the relevant differential equations ...

To avoid the order reduction when third order implicit-explicit Runge-Kutta time discretization is used together with the local discontinuous Galerkin (LDG) spatial discretization, for solving convection-diffusion problems with time-dependent Dirichlet boundary conditions, we propose a strategy of boundary treatment at each intermediate stage in this paper. The proposed strategy can achieve opt...

A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂t + I −Δ, but also f...

In this paper, we deal with controllability properties of linear and nonlinear Korteweg–de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact contro...

We consider bosonic noncritical strings as QCD strings and we present a basic strategy to construct them in the context of Liouville theory. We show that Dirichlet boundary conditions play important roles in generalized Liouville theory. Specifically, they are used to stabilize the classical configuration of strings and also utilized to treat tachyon condensation in our model. We point out that...

We study the boundary controllability of a nonlinear Korteweg–de Vries equation with the Dirichlet boundary condition on an interval with a critical length for which it has been shown by Rosier that the linearized control system around the origin is not controllable. We prove that the nonlinear term gives the local controllability around the origin.

In this paper we investigate numerically positive solutions of the equation −Δu = λu+u with Dirichlet boundary condition in a boundary domain Ω for λ > 0 and 0 < q < 1 < p < 2∗, we will compute and visualize the range of λ, this problem achieves a numerical solution. Keywords—positive solutions; concave-convex; sub-supersolution method; pseudo arclength method.

In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in H1/2(∂Ω). The tools used are those of the unfolding method in periodic homogenization.

This paper will give the numerical difference scheme with Dirichlet boundary condition, and prove stability and convergence of the difference scheme, final numerical experiment results also confirm effectiveness of the algorithm. KeywordsFractional derivative; Numerical difference scheme; The gradient regularization method.