A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

Global existence for rough solutions of a fourth-order nonlinear wave equation

In this paper, we prove that the cubic fourth-order wave equation is globally well-posed in Hs(Rn) for by following the Bourgain's Fourier truncation idea in Bourgain (1998) [2]. To avoid some troubles, we technically make use of the Strichartz estimate for low frequency part and high frequency part, respectively. As far as we know, this is the first result on the low regularity behavior of the fourth-order wave equation.     

Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions

An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

     

Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds

In this paper, we study the local gradient estimate for the positive solution to the following equation:

     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

Explicit decay estimates for solutions to nonlinear parabolic systems

The aim of this paper is to investigate a class of nonlinear parabolic systems with initial and boundary values of Dirichlet type, when the nonlinearities depend on the gradient of the solution. Sufficient conditions on data are established in order to preclude blow up and to deduce that the solution decays exponentially in time. Moreover, an upper bound of its gradient is derived.     

Explicit decay estimates for solutions to nonlinear parabolic systems

The aim of this paper is to investigate a class of nonlinear parabolic systems with initial and boundary values of Dirichlet type, when the nonlinearities depend on the gradient of the solution. Sufficient conditions on data are established in order to preclude blow up and to deduce that the solution decays exponentially in time. Moreover, an upper bound of its gradient is derived.     

Uniqueness of positive radial solutions for Dirichlet problems on annular domains

This paper deals with the uniqueness of positive radial solutions to Dirichlet problems on annular domains in Rn, n?3. As an application we can obtain the results to equation Δu+up−α₁u−α₀=0, where p>1, α₁?0, α₀?0 and α₁+α₀>0.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Integral estimates of the solutions to the helmholtz equation in unbounded domains

The following boundary value problem is studied:

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval

By employing the monotone iterative technique, we not only establish the existence of the unique solution for a fractional integral boundary value problem on semi-infinite intervals, but also develop an explicit iterative sequence for approximating the solution and give an error estimate for the approximation, which is an important improvement of existing results.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

Sign-changing solutions of nonlinear elliptic equations

In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be briefly revisited.      

Space-time estimates for the wave equations with singular potentials

We study the regularity of the solutions for the wave equations with potentials that are time-dependent and singular. The size of the potentials is exactly a function of the spatial dimension rather than being small enough in the known results. Based on a weighted L² estimate for the solutions, we prove the local regularity and the Strichartz estimates. The solvability of the equation is also studied.     

EXISTENCE OF SOLUTIONS OF NONLINEAR TWO AND THREE-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, the two and three-point boundsry problems (with nonlinear boundary conditions)for the genaral noniinear equations of fourth order are discussed.We have set some grups of the assurnpion coditions and proved the existence of solutins for corresponding boundary value problems under these conditons.