A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart

We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.Comment: In this third version, we clarified that the approach of Fila and Yanagida [Tohoku Math. J. (2011)] also works in the subcritical regim

On periodic orbits in a slow-fast system with normally elliptic slow manifold

In this note we consider the bifurcation of a singular homoclinic orbit to periodic ones in a 4-dimensional slow-fast system of ordinary differential equations, having a 2-dimensional normally elliptic slow manifold, originally studied by Feckan and Rothos. Assuming an extra degree of differentiability on the system, we can refine their perturbation scheme, in particular the choice of approximate solution, and obtain improved estimates

A weighted Hardy-Sobolev-Maz’ya inequality

We provide a weighted extension of a Hardy-Sobolev-Maz’ya inequality that is due to Filippas, Maz’ya and Tertikas

Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation

We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a stronger result in all dimensions

The heteroclinic connection problem for general double-well potentials

By variational methods, we provide a simple proof of existence of a heteroclinic orbit to a second order Hamiltonian ODE that connects the two global minima of a double-well potential. Moreover, we consider several inhomogeneous extensions