GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON R^(3)×T WITH CUBIC NONLINEARITIES

In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.

BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS

In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

Multi-solution of Semilinear Elliptic Equation on R^N

In this paper, we give a Z 2 index theory of generalized critical point. By this theory we get a sufficient condition on the existing result of a class functional, as an example we give a new result that the equation-Δu+u=｜u｜ p-1 u, x∈R N,(1)has infinite solutions.

IDENTIFICATION OF PARAMETERS IN SEMILINEAR PARABOLIC EQUATIONS

An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.

HOMOGENIZATION OF SOME LINEAR AND SEMILINEAR SCHRDINGER EQUATIONS WITHREAL POTENTIAL

This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear Schrodinger equations and gives correctors for the homogenization of linear and semilinear Schrodinger equations.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.

PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.

AN EXISTENCE THEOREM OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS ON R^N

An existent theorem is obtained for nonzero W-1,W-2(R-N) solutions of the following equations on R-N -Delta u + b(x)u = f(x,u), x is an element of R-N, where b is periodic for some variables and coercive for the others, f is superlinear.

SEPARATION PROPERTY OF POSITIVE RADIAL SOLUTIONS FOR A GENERAL SEMILINEAR ELLIPTIC EQUATION

The asymptotic behavior at infinity and an estimate of positive radial solutions of the equation △u ＋ sum from i=1 to k cirli upi = 0, x ∈ Rn,（0.1）are obtained and the structure of separation property of positive radial solutions of Eq. （0.1） with different initial data α is discussed.

SINGULAR POSITIVE RADIAL SOLUTIONS FOR A GENERAL SEMILINEAR ELLIPTIC EQUATION

The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.

STABILITY RESULTS OF RANDOM IMPULSIVE SEMILINEAR DIFFERENTIAL EQUATIONS

In this paper, we study the existence, uniqueness, continuous dependence, Ulam stabilities and exponential stability of random impulsive semilineax differential equations under sufficient condition. The results are obtained by using the contraction mapping principle. Finally an example is given to illustrate the applications of the abstract results.

THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.

Nonexistence of Global Solutions for a Semilinear Double-Wave Equation with Nonlinearity of Derivative Type

In this paper,we study the blow-up of solutions to a semilinear double-wave equation with nonlinearity of derivative type.By using the iteration method and the differential inequality techniques,we can get the estimates of the lifespan and the blow-up of solutions in the subcritical case under some assumptions.

Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions

We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy （which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball） and on Strichartz＇s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.

POSITIVE EQUILIBRIUM SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS

The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions： ＂subsolution ≮ supersolution＇, the existence and stability/instability of equilibrium solutions are obtained.

Existence of Single-Peaked Solution of a Semilinear Elliptic Problem

We discussed the Dirichlet problem of semilinear elliptic equation (P β,ε):β 2Δu=u p +εu,u>0, in Ω;u=0, on ?Ω, where Ω?R N (N≥4) is smooth and bounded domain, ,β,ε>0. We have proved that there exist positiveε 0 andε 1, such that when 0?ε?ε 0,β>√ε 1, (P β,ε) has a single-peaked solutionu β,ε; furthermore, |?u β0|2?0 in the sense of measure as ε→0 and β→0.

Kamenev-type Oscillation Criteria for Semilinear Elliptic Differential Equations

Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.

A Class of Shock Solutions for the Semilinear Singularly Perturbed Boundary Value Problem

The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.