Exact Controllability to Trajectories for a Semilinear Heat Equation with a Superlinear Nonlinearity

In this paper, we consider exact controllability to trajectories for a semilinear heat equation with nonlinearity that has superlinear growth at infinity with Dirichlet boundary conditions in a bounded domain of ? ^N . The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem and generalize the previous results.     

Insensitizing controls for a nonlinear parabolic equation with a nonlinear term involving the state and the gradient in unbounded domains

In this paper, we consider the existence of insensitizing control for a semilinear heat equation involving gradient terms in unbounded domain Ω. In this case, we prove the existence of controls insensitizing the L²-norm of the observation of the solution in an open subset of the domain. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

临界非齐次双调和方程的多解存在性

该文讨论了下列边值问题Δ2 u =λu |u|p- 1u μf (x) ,x∈Ω ,μ >0 ;u| Ω =0 , u n Ω =0 .的多解存在性和非存在性 .其中 :Ω RN是有界光滑区域 ,N≥ 5,λ∈ R1,P =N 4N - 4,f(x)是Ω中的非负不恒为零的连续函数 ,Δ2 =ΔΔ表示 N维双调和算子 .     

Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u₀) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u₀) < 0.     

Nonlinear degenerate parabolic equations with Neumann boundary condition

We study Neumann problem for a class of nonlinear degenerate parabolic PDE. A typical nonlinearity we have in mind is, for instance, β(u)=−1/u(u>0). We establish a necessary and sufficient condition on given data for existence of solution.     

Existence of Time Periodic Solutions to Boundary Value Problem of One-dimensional Semilinear Viscoelastic Dynamic Equation with Memory

In this paper, we prove the existence of time-periodic solutions to the boundary value problem of semilinear one-dimensional dynamical equation for viscodastic materials.     

Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity

We consider the following semilinear elliptic equation with singular nonlinearity:

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 ...     

半线性椭圆方程的多解存在性

本文考虑具有凹凸非线性和临界Sobolev指数的半线性椭圆方程的正解和多重解（可能非正）的存在性。     

Exact Controllability of the Superlinear Heat Equation

The exact internal and boundary controllability of parabolic equations with superlinear nonlinearity is studied. Accepted 3 January 2000     

Existence of Solutions for a Wave Equation with Non-monotone Nonlinearity and a Small Parameter

We provide sufficient conditions for the existence of solutions to a semilinear wave equation with non-monotone nonlinearity involving a small parameter. Our results are based on the analysis of a an operator that characterizes the projection onto the kernel of the wave operator subject to periodic-Dirichlet boundary conditions. Such a kernel is infinite dimensional which makes standard compactness arguments inapplicable.     

一类带参数的半线性椭圆方程正径向解的存在性研究

主要讨论一类带参数的半线性椭圆型方程在一定的边值条件下的解的存在性问题.主要利用上下解方法和不动点定理解决方程解的存在性问题,重点讨论了解存在的充分条件,并以定理的形式给出具体的证明.     

Quenching Time for a Semilinear Heat Equation with a Nonlinear Neumann Boundary Condition

In this paper we consider the finite time quenching behavior of solutions to a semilinear heat equation with a nonlinear Neumann boundary condition. Firstly, we establish conditions on nonlinear source and boundary to guarantee that the solution doesn＇t quench for all time. Secondly, we give sufficient conditions on data such that the solution quenches in finite time, and derive an upper bound of quenching time. Thirdly, undermore restrictive conditions, we obtain a lower bound of quenching time. Finally, we give the exact bounds of quenching time of a special example.     

关于半线性热方程的死角性质

本文考虑下述问题：这里，．本文证明了在发生死角时只有有限个死角点．     

半线性热方程精确内部能控性

主要采用了构造辅助系统u′ △u=0的方法，同时也利用了强单调算子 的满射性。在条件（H)的修改下，得到在有限时间内的精确可控性，研究半线性热方程系统ey/et=△y-f(x,t,y,倒△y) h(x,t)的精确内部能控性。