Stability of solutions to some evolution problem

TL;DR

This paper investigates the long-term behavior of solutions to a class of abstract differential equations with dissipative operators, nonlinear terms, and external forcing, providing conditions for existence, boundedness, and decay of solutions.

Contribution

It establishes new sufficient conditions ensuring global existence, boundedness, and decay of solutions for evolution problems with nonlinear and time-dependent operators.

Findings

Solutions exist for all time under specified conditions.

Solutions are uniformly bounded and tend to zero as time approaches infinity.

A key nonlinear inequality is developed to analyze solution behavior.

Abstract

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -\gamma(t)(u,u)$, $\gamma(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\le c_0\|u\|^p$, $p>1$, $c_0,p$ are constants, $\|b(t)\|\le \beta(t),$ $\beta(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $\|u(t)\|$ is given. This bound implies the relation $\lim_{t\to \infty}\|u(t)\|=0$ under suitable conditions on $\gamma(t)$ and $\beta(t)$. The basic technical tool in this work is the following nonlinear inequality: $$ \dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0. $$