The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms $u_{*}^{\text{'}}\left(\right|u>u\left(x\right)\left|\right)$ and $|u>u(x\left)\right|$, which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem.

We establish the global existence and uniqueness of smooth solutions to a nonlinear Alfvén wave equation arising in a finite-beta plasma. In addition, the spatial asymptotic behavior of the solution is discussed.

This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in ${ℝ}^n$. Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound $$\parallel u\left(t\right)\parallel \ge {(t+1)}^{-\frac{n+4}{2}}.$$