We consider the one-dimensional equations for the double-wall carbon nanotubes modeled by coupled Timoshenko elastic beam system with nonlinear arbitrary localized damping. We prove that the damping placed on an arbitrary small support, unquantized at the origin, leads to uniform decay rates (asymptotic in time) for the energy function. This study generalizes and improves previous literature outcomes.

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \left\{\begin{array}{ll}\partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) & \hbox{ in }~ \mathbb{H}^{n}\times (0, T), \\\quad u =u_{0} & \hbox{ in }~ \mathbb{H}^{n}\times \{0\}.\end{array}\right.\tag{0.1}\end{align}We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, (0.1) exhibitsFujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e.,there exists a critical exponent $\mu^*$ such that if $\mu > \mu^*$, then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$, there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that(0.1)with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

The asymptotic behavior of solutions of twodimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations inunbounded domains is discussed in this work (for example, Poincaré domains).We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorptionexponent $r\in[1,3]$) perturbed by an additive noise on Poincaré domains $\mathcal O$. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1(\mathcal O) $ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these results to generalunbounded domains is also provided. Finally, for 2D SCBF equations forced byadditive one-dimensional Wiener noise, we prove the upper semicontinuity ofthe random attractors, when the domain changes from bounded to unbounded (Poincaré) domain.

Erratum for the article [1]: G. Vinti and L. Zampogni, A unifying approach to convergence of linear sampling type operators in Orlicz spaces, Advances in Differential Equations, 16 (2011), No. 5-6, 573-600.