We show that there is no blowup solutions, for positive viscosity constant , to the equation f xxt f xxxx +ff xxx f x f xx = 0; x 2 (0; 1); t > 0 with (i) periodic boundary condition, or (ii) Dirichlet boundary condition f = f x = 0 or (iii) Neumann boundary condition f = f xx = 0 on the boundary x = 0; 1. Furthermore we show that every solution decays to the trivial steady state as t goes to i...

We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on a finite interval (0, 2π). The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove th...

We consider bosonic noncritical strings as QCD string. We present an basic strategy to construct them, in the context of Liouville theory. We show that Dirichlet boundary condition fills important roles in generalized Liouville theory. Specifically, it is used to stabilize the classical configuration of strings, and also utilized to treat the tachyon condensation, in our model. We point out tha...

We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton-Jacobi Equation ut−∆u+|Du| m = f in Ω×(0,+∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN . We mainly focus on the superquadratic case (m > 2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial...

An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-square value of the solution at the boundaries. Application of the MSD boundary condition to the nonlinear Schrödinger equation is shown, a...

We discuss various issues concerning the behaviors near the boundary (σ = 0, π) and the midpoint (σ = π/2) of the open string coordinate X(σ) and its conjugate momentum P (σ) = −iδ/δX(σ) acting on the matter projectors of vacuum string field theory. Our original interest is in the dynamical change of the boundary conditions of the open string coordinate from the Neumann one in the translational...

Motivated by numerous models of representing trust and distrust within a network ranking system, we examine a quantitative vertex ranking with consideration of the influence of a subset of nodes. We propose and analyze a general ranking metric, called Dirichlet PageRank, which gives a ranking of vertices in a subset S of nodes subject to some specified conditions on the vertex boundary of S. In...

The spectral properties of curved quantum wires with Dirichlet boundary condition were widely investigated (e.g. [1], [2], [3]). It was shown that any small curvature of the tube in dimensions 2 and 3 produces at least one positive eigenvalue below the essential spectrum threshold. The problem of the existence of such eigenvalues in the straight quantum waveguides with a combination of Dirichle...

We design an artificial boundary condition for the steady incompressible Navier-Stokes equations in streamfhction-vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived fiom the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier-Stokes equations is presented. The artificial boundary conditio...

We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this set the Dirichlet boundary condition is imposed while on the rest part of the boundary we impose the Neumann condition. For the case of homogenized Neumann or...