Existence and uniqueness of solutions for the Lane, Emden and Fowler type problem

This paper deals with the existence and uniqueness of a positive solution to the semilinear elliptic equation of the Lane, Emden and Fowler type.     

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.     

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.     

Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a fourth-order nonlinear parabolic equation.     

Uniqueness of positive solutions for a boundary blow-up problem

In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem

where Ω is a C² bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/t^p is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x).     

Existence and uniqueness of solutions for a second order differential equation with integral boundary conditions

In this paper, we consider the following second order differential equation

     

Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with finitely many deviating arguments

In this paper, we consider a kind of Rayleigh equation with finitely many deviating arguments of the form

     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation

In this paper, we are concerned with a singular parabolic equation in a smooth bounded domain Ω⊂RN subject to zero Dirichlet boundary condition and initial condition φ?0. Under the assumptions on μ, φ and f(x,t), some existence and uniqueness results are obtained by applying parabolic regularization method and the sub-supersolutions method. We also discuss the asymptotic behaviors of solutions in the sense of and L^∞(0,T;L²(Ω)) norms as μ→0 or μ→∞. As a byproduct we obtain the existence of solutions for some problems which blow up on the boundary.     

Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systems

with null Dirichlet boundary conditions in a smooth bounded domain ΩR^N, where p,q>1, , and a,b>0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=B_R={xR^N:|x|<R}(R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.     

Existence and uniqueness of local solutions to a class of quasilinear degenerate parabolic systems

In this paper, the authors study an initial and boundary value problem to a system of evolution p-Laplacian equations coupled with general nonlinear terms. The authors use the method of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and hence obtain the existence of solutions to a regularized system of equations. Then the existence of solutions to the system of evolution p-Laplacian equations is obtained with the use of a standard limiting process. The uniqueness of the solution is also proven.     

Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness

We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u^−pv^−q,−Δv=u^−rv^−s,u>0,v>0,x∈Ω,u|_∂Ω=v|_∂Ω=0, where Ω is a bounded domain with smooth boundary in R^N, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318].     

Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system

An elliptic system is considered in a smooth bounded domain, subject to Dirichlet boundary conditions of three different types. Based on the construction of certain upper and sub-solutions, we obtain some conditions on the parameters ai,bi,ci (i=1,2) and the exponents m,n,p,q to ensure the existence of positive solutions. Furthermore, uniqueness and boundary behavior of positive solutions is also discussed.     

Existence,uniqueness and positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].     

Existence and uniqueness for a stochastic age-structured population system with diffusion

In this paper, a class of stochastic age-dependent population dynamic system with diffusion is introduced. Existence and uniqueness of strong solution for a stochastic age-dependent population dynamic system in Hilbert space are established. The analysis use Barkholder–Davis–Gundy’s inequality, Itô’s formula and some special inequalities for our purposes.     

Existence and asymptotic behavior of entire blow-up solutions for quasilinear elliptic equation

In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ _p u ≡ div(|?u| ^p?2?u) = ρ(x)f(u), x ∈ R ^N . No monotonicity condition is assumed upon f(u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques.     

Existence and multiplicity of symmetric positive solutions for three-point boundary value problem

In this paper, we are concerned with the existence and multiplicity of symmetric positive solutions for the following second-order three-point boundary value problem