We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-type scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-type scheme, that under certain conditions – not too far from optimal - convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale (at best) with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.

We study travelling wave solutions to nonlinear Schrödinger and Klein-Gordon equations on complete Riemannian manifolds, which have a bounded Killing field $X$. For a natural class of power-type nonlinearities, we use standard variational techniques to demonstrate the existence of travelling waves on complete weakly homogeneous manifolds. If the manifolds in question are weakly isotropic, we prove that they have genuine subsonic travelling waves, at least for a non-empty set of parameters. Finally we establish that a slight perturbation of the Killing field $X$ will result in a controlled perturbation of the travelling wave solutions (in appropriate $L^p$-norms).

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. We consider also elliptic equations on $M$ with similar conditions at infinity.

In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & \text{in}\ \Omega , \\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega, \end{array} \right. \end{equation*} where $s\in(0,1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if $f(x,u)$ satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x,u)$ is odd in $u$.

The aim of this paper is to prove the existence of almost global weak solutions for the unsteady nonlinear elastodynamics system in dimension $d=2$ or $3$, for a range of strain energy density functions satisfying some given assumptions. These assumptions are satisfied by the main strain energies generally considered. The domain is assumed to be bounded, and mixed boundary conditions are considered. Our approach is based on a nonlinear parabolic regularization technique, involving the $p$-Laplace operator. First we prove the existence of a local-in-time solution for the regularized system, by a fixed point technique. Next, using an energy estimate, we show that if the data are small enough, bounded by $\varepsilon >0$, then the maximal time of existence does not depend on the parabolic regularization parameter, and the behavior of the lifespan $T$ is $\gtrsim \log (1/\varepsilon)$, defining what we call here almost global existence. The solution is thus obtained by passing this parameter to zero. The key point of our proof is due to recent nonlinear Korn's inequalities proven by Ciarlet and Mardare in $W^{1,p}$ spaces, for $p>2$.