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.     

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.     

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.     

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   

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.     

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 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.     

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

Initial boundary value problem for generalized Zakharov equations

This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in (2+1) dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.     

A singular characteristic boundary value problem for the Euler-Poisson-Darboux equation

Summary The paper considers a singular characteristic boundary value problem for the well known EPD equation. By using a modified Riemann method, a formula is obtained which is shown to provide a weak solution of the problem. The weak solution is then shown to be a solution in the classical sense by investigating its differentiability properties. The research of the second author has been supported by NSF research grant GP-11543. Entrata in Redazione il 10 novembre 1971.     

A three-point normal boundary value problem for an operator-differential equation

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 100–1005, September–October, 1994.     

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.