This work is concerned with the exact controllability/obser-vability of abstract cascade hyperbolic systems by a reduced number of controls/observations. We prove that the observation of the last component of the vector state allows one to recover the initial energies of all of its components in suitable functional spaces under a necessary and sufficient condition on the coupling operators for cascade bidiagonal systems. The approach is based on a multi-level energy method which involves $n$-levels of weakened energies. We establish this result for the case of bounded as well as unbounded dual-control operators and under the hypotheses of partial coercivity of the $n-1$ coupling operators on the sub-diagonal of the system. We further extend our observability result to mixed bidiagonal and non-bidiagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying the HUM method, we derive the corresponding exact controllability results for $n$-coupled bidiagonal cascade and $n+p$-coupled mixed cascade systems. Using the transmutation method for the wave operator, we prove that the corresponding heat (respectively Schr\"odinger) multi-dimensional cascade systems are null-controllable for control regions and coupling regions which are disjoint from each other and for any positive time for $n \le 5$ for dimensions larger than $2$, and for any $n \ge 2$ in the one-dimensional case. The controls can be localized on a subdomain or on the boundary, and in the one-dimensional case the coupling coefficients can be supported in any non-empty subset of the domain.

We consider a dynamical nonlinear model for shallow shells of the Marguerre--Vlasov's type in the presence of thermal effects. Results on existence and uniqueness of global weak solutions are already available. We consider the above model depending on a parameter $\varepsilon>0$ and study its weak limit as $\varepsilon\rightarrow 0^+$. The limit model turns out to be a nonlinear Timoshenko's equation with thermal effects on the manifold (the shell). We also analyze the asymptotic behavior of the total energy of the nonlinear model of Marguerre--Vlasov's type with thermal effects.

In this paper, we show the global existence of the classical solutions to the Boussinesq equations with fractional diffusion by using energy methods, the Fourier localization technique, and Bony's paraproduct decomposition.

We prove existence and stability results for a two-parameter family of solitary-wave solutions to a system in which an equation of nonlinear Schr\"odinger type is coupled to an equation of Korteweg--de Vries type. Such systems model interactions between short and long dispersive waves. The results extend earlier results of Angulo, Albert and Angulo, and Chen. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we establish the sub-additivity of the problem with respect to both constraint variables jointly.

In this paper we are concerned with an initial--boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of long waves generated by a wave maker in a channel. It is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space $L^2(e^{2bx}\,dx)$ for $b>0$. The stabilization problem is studied using a Lyapunov approach, while the well-posedness result is obtained combining fixed-point arguments and energy-type estimates.

We prove global existence and stability for the Keller--Segel equations with small initial values in the critical Morrey space.