In this paper, convergence rates of solutions towards stationary solutions for the outflow problem of planar magnetohydrodynamics (MHD) are investigated. Inspired by the relationship between MHD and Navier-Stokes, we prove that the global solutions of the planar MHD converge to the corresponding stationary solutions of Navier-Stokes equations. We obtain the corresponding convergence rates based on the weighted energy method when the initial perturbation belongs to some weighted Sobolev space.

We explicitly give all stationary solutions to the focusing cubic NLS on the line, in thepresence of a defect of the type Dirac’s delta or delta prime. The models proveinteresting for two features: first, they are exactly solvable and all quantities can beexpressed in terms of elementary functions. Second, the associated dynamics is far frombeing trivial. In particular, the NLS with a delta prime potential shows two symmetrybreaking bifurcations: the first concerns the ground state and was already known. Thesecond emerges on the first excited state, and up to now had not been revealed. Wehighlight such bifurcations by computing the nonlinear and the no-defect limits of thestationary solutions.

In this paper, we consider the bi-stable equation proposed by Rosenau to replace the Allen–Cahn equation in the case of large gradients. We discuss the bifurcation problem for stationary solutions of this equation on an interval as the diffusion coefficient and the length of the interval are varied, concentrating on classical solutions.

The non-local equation

$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$

is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.

AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99