Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity

We study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, we obtain the existence of global solution and blow-up at +∞ under some suitable conditions. On the other hand, the results for decay estimates of the global solutions are also given. Our result in this paper means that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of semilinear heat equations.     

关于带混合边界条件的半线性反应扩散系统解的整体存在性问题

本文利用Ｇｒｏｎｗａｌ不等式及对偶技巧，研究了一类带混合边界条件的半线性反应扩散系统解的整体存在性问题，给出一些控制条件，突破了以往加在“Ｍｏｒ ｇａｎ”和上的线性控制，得到解的整体存在性，很大程度上改进和推广了以往结果，使该问题的研究前进了一大步     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.     

Global existence for a nonlinear theory of bubbly liquids

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall-like inequality are then derived to prove global existence.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Large time behavior of solutions to semilinear systems of wave equations

This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that ``modified' free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non–existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. The first author was partially supported by Grant-in-Aid for Science Research (14740114), JSPS.     

Existence of a nontrivial solution for a class of hemivariational inequality problems at double resonance

In this paper, we discuss a class of semilinear elliptic hemivariational inequality problems. By using the nonsmooth minimax principle for locally Lipschitz functions, we establish the existence of a nontrivial solution for the semilinear elliptic hemivariational inequality problem, where incomplete double resonance occurs at infinity between two distinct consecutive eigenvalues.     

THE EXISTENCE OF THE SOLUTIONS FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS

The author obtains the existence of the local and global solution for a class of semilinear evolution equations in a Hilbert spcae, and uses the results to prove the existence of the solution for semilinear parabolic partial differential equations in R~n.     

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of -regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of -regular solutions is given. We also formulate sufficient conditions to construct a piecewise -regular solutions (continuation beyond maximal time of existence for -regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for in is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2mth order semilinear parabolic initial boundary value problem in a Hilbert space .     

关于猝灭问题的一些结果及其应用

本文研究一类含奇异项的半线性抛物方程的初边值问题,给出了该问题的古典解整体存在或发生猝灭现象的判据以及猝灭解的生命跨度估计。同时,还用上述结果研究了一类含超Sobolev临界指标的半线性椭圆方程的Dirichlet边值问题,获得了正解的存在性结果。     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Existence of solution for a singular critical elliptic equation

In this paper, a singular semilinear elliptic problem involving the critical Sobolev exponent is studied by variational method, the existence of a solution is proved under certain conditions. The Hardy inequality is used and plays an important role in the discussion.     

Remarks on infinite energy solutions of nonlinear wave equations

We first consider the existence of a solution of the critical semilinear wave equation in Besov space which extends the results in [P. Germain, Global infinite energy solutions of critical semilinear wave equation, Revista Matematica Iberoamericana 24 (2) (2008) 463-497] to general dimensions. Next we derive the existence and uniqueness of global solutions for a semilinear wave equation in Marcinkiewicz space.     

Existence of solution for the n-dimension second order semilinear hyperbolic equations

In the paper we establish the local and global existence of solution for the n-dimensional second order semilinear hyperbolic equation with a strongly singular coefficient which appears in the boundary-value problems of fluid dynamics. Based on the analysis about the loss of regularity on the line t=0 for the solution of the corresponding linear equation and the decay at infinity which caused by the singular coefficient, we obtain the existence of a small solution for the semilinear equation by use of fixed point theorem.     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

The Carleman Inequality and Its Application to Periodic Optimal Control Governed by Semilinear Parabolic Differential Equations

This paper deals with optimal control problems for semilinear parabolic differential equations, which may be governed by nonmonotone operators and have no global solution, with periodic inputs. The Pontryagin maximum principle is obtained and the Carleman inequality for the backward linearized adjoint system associated with the state system is established.     

On the dynamics of a strongly singular parabolic equation

In this paper we study the semiflow defined by a semilinear parabolic equation, in which both the diffusion and the reaction term present strong order of singularity at the origin. We justify the existence of a global branch of nontrivial equilibrium solutions for subcritical nonlinearities. The approach, based on the critical Caffarelli–Kohn–Nirenberg inequality, follows the arguments of  and .     

外区域上半线性波动方程解的整体存在性

该文研究带耗散项的线性和半线性波动方程外问题. 首先利用一个Sobolev型不等式得到了线性耗散波动方程在外区域上的整体能量衰减估计, 此结果用来证明非线性项为|u|^p (2+) 的半线性波动方程解的整体存在性. 为此, 该文主要研究N维(3≤ N≤7)外区域上球对称解的情形.