Study on the Controllability in a Steady State for the Quasilinear Wave Equations with Weak Decay

摘要

A new method of investigating the so-called quasilinear strongly-damped wave equations is proposed in this paper, concerned with controllability in a steady state for the Quasilinear Wave Equations with weak decay. A so-called energy perturbation method to establish weak controllability of solutions in terms of energy norm for a class of nonlinear functions is presented. This method establishes the existence and uniqueness of energy solutions and the existence of finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity. Controllability in a steady state with the help of differential inequalities by estimating the relationship between energy inequalities and attenuating property of weak solutions is demonstrated. A small positive number is determined and derived differential inequalities by using a perturbation of energy.