Springer Science and Business Media LLC

The existence of a rich structure of homoclinic and heteroclinic solutions is established for a family of Hamiltonian systems that serve as a simpler model for the multiple pendulum system. The proof is based on recently developed arguments from the calculus of variations that have proved useful in finding actual solutions of an equation near approximate solution.

In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure $$\mathcal P$$ defined on the interval $$[-1,1]$$ . Under Dirichlet boundary conditions we establish the exclusion of boundary blow-up points, we show that the concentration mass does not have residual $$L^1$$ -terms (“residual vanishing”) and we determine the location of blow-up points in terms of Kirchhoff’s Hamiltonian. We allow $$\mathcal P$$ to be a general Borel measure, which could be “degenerate” in the sense that $$\mathcal P(\{\alpha _-^*\})=0=\mathcal P(\{\alpha _+^*\})$$ , where $$\alpha _-^*=\min \mathrm {supp}\mathcal P$$ and $$\alpha _+^*=\max \mathrm {supp}\mathcal P$$ . Our main results are new for the standard sinh-Poisson equation as well.

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$\begin{aligned} \partial _t u=[|D u|^q+a(x,t)|D u|^s]\left( \Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle \right) , \end{aligned}$$ where $$1