Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity (Q2730790)

The authors estimate the rate of decay of the difference between a solution to the evolution equation \(au_{tt}+bu_{t}-\triangle u=f(x,u), x\in \Omega\subset \mathbb{R}^n, t>0\) satisfying homogeneous Dirichlet condition on the boundary of \(\Omega\) and its limiting equilibrium point (for \(t\rightarrow \infty \)). Moreover, the optimality of the obtained estimates is discussed. The equation involves three types of problems: parabolic for \(a=0,b=1,\) elliptic for \(a=-1\) and hyperbolic for \(a=1,b=0.\)