Large-time behavior of solutions to evolution problems

TL;DR

This paper investigates the long-term behavior of solutions to a class of abstract differential equations in Hilbert spaces, providing conditions for existence, boundedness, and decay of solutions over time.

Contribution

It offers new sufficient conditions ensuring global existence, uniform boundedness, and decay of solutions to evolution problems with time-dependent operators and nonlinearities.

Findings

Solutions exist for all time under specified conditions.

Solutions are uniformly bounded on the positive real axis.

Solutions tend to zero as time approaches infinity.

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 $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $H$ is a Hilbert space, $t\in \R_+:=[0,\infty)$, $A(t)$ 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 positive 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 $\beta(t)$.