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Existence and uniqueness of mild and strong solutions of nonlinear Volterra integrodifferential equations in Banach spaces

Tidke H. L., Dhakne M. B.

Demonstratio Mathematica

2010

Vol. 43, nr 3

643-652

In this paper we prove the existence and uniqueness of mild and strong solutions of a nonlinear Volterra integrodifferential equation with nonlocal condition. Our analysis is based on semigroup theory and Banach fixed point theorem and inequalities are established by Gronwall and B. G. Pachpatte.

Some results for the reflection problems in Hilbert spaces

Barbu V., Da Prato G.

Control and Cybernetics

2008

Vol. 37, no 4

797-810

This work is concerned with existence and uniqueness of a solution of a stochastic variational inequality on closed convex bounded subsets with nonempty interior and smooth boundary of a Hilbert space H (the reflection problem).

Stability of higher-level energy norms of strong solutions to a wave equation with localized nonlinear damping and a nonlinear source term

Lasiecka I., Toundykov D.

Control and Cybernetics

2007

Vol. 36, no 3

681-710

We derive global in time a priori bounds on higher-level energy norms of strong solutions to a semilinear wave equation: in particular, we prove that despite the influence of a nonlinear source, the evolution of a smooth initial state is globally bounded in the strong topology ∼ H2 x H1. And the bound is uniform with respect to the corresponding norm of the initial data. It is known that an m-accretive semigroup generator moriotoni-cally propagates smoothness of the initial condition; however, this result does not hold in general for Lipschitz perturbations of monotone systems where higher order Sobolev norms of the solution may blowup asymptotically as t → ∞. Due to nonlinearity of the system, the only a priori global-in-time bound that follows from classical methods is that on finite energy: ∼ H1 x L2. We show that under some correlation between growth rates of the damping and the source, the norms of topological order above the finite energy level remain globally bounded. Moreover, we also establish this result when damping exhibits sublinear or superlinear growth at the origin, or at infinity, which has immediate applications to asymptotic estimates on the decay rates of the finite energy. The approach presented in the paper is not specific to the wave equation, and can be extended to other hyperbolic systems: e.g. plate, Maxwell, and Schrodinger equations.