Semigroups of locally Lipschitz operators associated with semilinear evolution equations

In this paper we introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with respect to their initial data. The result obtained is applied to the initial-boundary value problem for the complex Ginzburg–Landau equation.     

Anti-periodic solutions for semilinear evolution equations

In this paper, we study the existence problem of anti-periodic solutions for the following first order evolution equation:

     

Positive solutions of a semilinear elliptic equation with singular nonlinearity

This paper is devoted to the study of positive solutions of the semilinear elliptic equation Δu+K(|x|)u−p=0, x∈Rn with n?3 and p>0. Asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has infinitely many positive solutions with fast growth.     

Existence of anti-periodic mild solutions for semilinear evolution equations

In this paper we study the existence of anti-periodic mild solutions for a class of semilinear evolution equations in Banach spaces and extend some related results in this direction. An example is given to illustrate our results.     

Fréchet differentiability of solution mappings for semilinear second order evolution equations

In this paper we establish the strong Fréchet differentiability of maps from the set of initial values and forcing terms into the set of solutions for semilinear second order evolution equations. Also, under the weaker conditions of nonlinear terms we establish the strong Gâteaux differentiability of the solution maps. An application of results to semilinear strongly damped wave equations is given.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Determining nodes for semilinear parabolic equations

We are concerned with the determination of the asymptotic behavior of strong solutions to the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set. More precisely, if the asymptotic behavior of the strong solution is known on a suitable finite set which is called determining nodes, then the asymptotic behavior of the strong solution itself is entirely determined. We prove the above property by the energy method.     

Regional controllability of semilinear degenerate parabolic equations in bounded domains

In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of ‘regional null controllability’, showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.     

Three positive solutions of nonhomogeneous semilinear elliptic equations

In this paper, we use new analyses to assert that there are three positive solutions of Eq. (1.1) in infinite cylinder domain with hole .     

Bounded positive solutions of semilinear elliptic equations

In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of semilinear elliptic equations in exterior domain of Rn, n?3.     

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R ³×R ⁺ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp. Received April 1999 – Accepted September 1999     

On noncoercive semilinear equations

In this article some noncoercive semilinear equations are investigated. Here, a subset of a right side for which the corresponding boundary value problems are uniquely solvable is described.     

On blow-up at space infinity for semilinear heat equations

A nonnegative blowing up solution of the semilinear heat equation ut=Δu+up with p>1 is considered when initial data u₀ satisfies

     

Almost‐periodic solutions to quasilinear evolution equations with a nonlocal nonlinearity

We prove an existence and uniqueness result for almost‐periodic solutions to the quasilinear evolution equations (1) and (5). Copyright © 2006 John Wiley & Sons, Ltd.     

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.