We consider a class of nonlinear higher-order evolution inequalities posed in $(0,\infty)\times B_1\backslash\{0\}$, subject to inhomogeneous Dirichlet-type boundary conditions, where B1 is the unit ball in $\mathbb{R}^N$. The considered class involves differential operators of the form

\begin{equation*}\mathcal{L}_{\mu_1,\mu_2}=-\Delta +\frac{\mu_1}{|x|^2}x\cdot \nabla +\frac{\mu_2}{|x|^2},\qquad x\in \mathbb{R}^N\backslash\{0\},\end{equation*}

$\mu_1\in \mathbb{R}$

$\mu_2\geq -\left(\frac{\mu_1-N+2}{2}\right)^2$

Let $\tau _{D}(Z) $ be the first exit time ofiterated Brownian motion from a domain $D \subset \mathbb{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau _{D}(Z)>t]$ be itsdistribution. In this paper we establish the exact asymptotics of $P_{z}[\tau _{D}(Z)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116(2006) 905–916], for $z\in D$ 
 $ \displaystyle \lim_{t\to\infty}t^{-1/2}\exp\left(\frac{3}{2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}\right)P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$ 
where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}\left(\psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$ . Here λD is thefirst eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ inD, and ψ is the eigenfunction corresponding toλD . We also study lifetime asymptotics of Brownian-time Brownian motion, $Z^{1}_{t} = z+X(|Y(t)|)$ , where X t and Y t are independentone-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.