Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn-Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.     

Initial boundary value problem for generalized logarithmic improved Boussinesq equation

In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.     

A generalized Minkowski problem with Dirichlet boundary condition

We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.

     

A fourth-order boundary value problem for a Sturm-Liouville type equation

An existence result of multiple solutions for a fourth-order Sturm-Liouville boundary value problem with variable parameters is established. As a consequence, three solutions for a boundary value problem with a fourth-order equation in a complete form are obtained. Our approach is based on variational methods.     

Some blow-up results for a generalized Ginzburg-Landau equation

We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rⁿ, n = 1{\Bbb R}^n, n = 1 or 2.     

A boundary value problem for a differential equation of second order

We find the spectrum and prove a theorem on the expansion of an arbitrary function satisfying certain smoothness conditions in terms of the root functions of a boundary value problem of the type ?y″+q(x)+a/x²y=λy, y(0)=0, M(λ) y(a)+N(λ) y(b)=0, where 0   

A mixed boundary value problem for Chaplygin's hodograph equation

In this paper we will prove existence, uniqueness and regularity of a classical solution to a mixed boundary value problem for Chaplygin's hodograph equation, which is degenerate elliptic on a part of the boundary. This problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane. The proof depends on Perron's method and some techniques from linear elliptic equations.     

Uniqueness for the homogeneous Dirichlet Willmore boundary value problem

We give a sufficient condition for curves on a plane or on a sphere such that if these give the boundary of a Willmore surface touching tangentially along the boundary the plane or the sphere respectively, the surface is necessarily a piece of the plane or a piece of the sphere. The condition we require is that the curves bound a strictly star-shaped domain with respect to the Euclidean geometry in the plane and with respect to the spherical geometry in the sphere, respectively.     

On boundary value problems for a discrete generalized Emden-Fowler equation

In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory.     

On a boundary value problem for a higher-order elliptic equation

For an elliptic 2lth-order equation with constant (and only leading) real coefficients, we consider the boundary value problem in which the (k _j ? 1)st normal derivatives, j = 1,..., l, are specified, where 1 ≤ k ₁ < ... < k _l . If k _j = j, then it becomes the Dirichlet problem; and if k _j = j + 1, then it becomes the Neumann problem. We obtain a sufficient condition for this problem to be Fredholm and present a formula for the index of the problem.     

Dirichlet problem for the generalized Laplace equation with the Caputo derivative

For a partial differential equation with the Caputo fractional derivative with respect to one of two independent variables, we solve the Dirichlet problem in a rectangular domain. The considered equation becomes the Laplace equation if the order of the fractional derivative is equal to 2. By using a method based on the completeness of the system of eigenfunctions of the Sturm-Liouville problem, we prove the uniqueness of the solution.     

BOUNDARY VALUE PROBLEM FOR A GENERALIZED LIENARD EQUATION

By coincidence degree, the existence of solution to the boundary value problem of a generalized Liénard equation