We study integral functionals defined on scalar Sobolev spaces of the form$ E[f]:u\mapsto \displaystyle \int _\Omega f(x, u(x), \nabla u(x)) d x, $with an emphasis on the non-convex case and the difficulties involved in preventing the Lavrentiev phenomenon. We determine a formulation of the lower semicontinuous envelope of $ E[f] $ with respect to various topologies and with fixed Lipschitz Dirichlet boundary conditions.

In a separable real Hilbert space, we study the problem of minimizing a convex function with Lipschitz continuous gradient in the presence of noisy evaluations. To this end, we associate a stochastic Heavy Ball system, incorporating a friction coefficient, with the optimization problem. We establish existence and uniqueness of trajectory solutions for this system. Under a square integrability condition for the diffusion term, we prove almost sure convergence of the trajectory process to an optimal solution, as well as almost sure convergence of its time derivative to zero. Moreover, we derive almost sure and expected convergence rates for the function values along the trajectory towards the infimal value. Finally, we show that the stochastic Heavy Ball system is equivalent to a Su-Boyd-Candès-type system for a suitable choice of the parameter function, and we provide corresponding convergence rate results for the latter.In the second part of this paper, we extend our analysis beyond the optimization framework and investigate a monotone equation induced by a monotone and Lipschitz continuous operator, whose evaluations are assumed to be corrupted by noise. As before, we consider a stochastic Heavy Ball system with a friction coefficient and a correction term, now augmented by an additional component that accounts for the time derivative of the operator. We establish analogous convergence results for both the trajectory process and its time derivative, and derive almost sure as well as expected convergence rates for the decay of the residual and the gap function along the trajectory. As a final result, we show that a particular instance of the stochastic Heavy Ball system for monotone equations is equivalent to a stochastic second-order dynamical system with a vanishing damping term. Remarkably, this system exhibits fast convergence rates for both the residual and gap functions.

This paper investigates traveling wave solutions for the spatial diffusion predator-prey model with Allee effects. From the perspective of singular perturbations, we demonstrate the existence of the traveling fronts connecting periodic orbits and the positive equilibrium. On the one hand, we use geometric singular perturbation theory and exchange lemma to study the existence of periodic wave trains, including the degenerate case in which the periodic orbits pass through the small neighbourhoods of two folds on the critical manifold. We utilize the Fenichel normal form and central manifold theory to analyze the flows near the folds where normal hyperbolicity fails. Additionally, we derive the period functions of these periodic orbits. On the other hand, by invariant manifold theory, we prove the transversal intersection between the stable manifolds of positive equilibrium and the unstable manifolds of the periodic orbit. In particular, we identify two types of traveling fronts: pulled and pushed fronts connecting periodic orbits to the positive equilibrium with the steepest decay. Finally, numerical simulations illustrate these theoretical results.

This paper investigates the vortex confinement property of the two-point vortex system in a planar domain. We compute the time over which initial point vortices around a stable stationary point remain within a slightly larger ball. In particular, we show that this concentration persists indefinitely regardless of the vorticity strengths. In the borderline of the stability condition, we show that this time becomes a power law if, in addition, one relaxes the size of the stability ball.

Given a one-parameter family of flows over a parameter interval $ \Lambda $, assuming there is a continuation of Morse decompositions over $ \Lambda $, Reineck (1988) defined a singular transition matrix to show the existence of a connecting orbit between some Morse sets at some parameter points in $ \Lambda $. This paper aims to extend the definition of a singular transition matrix in cases where there is no continuation of Morse decompositions over the parameter interval. This extension will help study the bifurcation associated with the change of Morse decomposition from a topological dynamics viewpoint.

In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with compact dynamics, which we defined as a semiflow $ F $ with a compact global trapping region. In this follow-up, we restrict to the important case where the phase space $ X $ is locally compact and we move the focus on the concept of global attractor, a maximal compact set that attracts every compact subset of $ X $. A semiflow $ F $ can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that $ F $ has compact dynamics if it has a global attractor. We show that most of the qualitative properties of $ F $ are inherited by the restriction of $ F $ to its global attractor and that, in case of Conley's chains stream of $ F $, the qualitative behavior of $ F $ and of its restriction to its global attractor coincide. Moreover, if $ F $ is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow $ F $ with compact dynamics over a locally compact space, the graphs of the prolongational relation of $ F $ and of every stream of $ F $ are connected if the global attractor is connected.

This work studies the dynamics of solutions to the sine-Gordon equation posed on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex, and endowed with boundary conditions at the vertex of $ \delta $-type. The latter generalize conditions of Neumann-Kirchhoff type. The purpose of this analysis is to establish the existence and instability of stationary solutions which we have called single-lobe kink state profiles, which consist of a symmetric stationary solution of subluminal-type profile in the finite ring of the tadpole, coupled with a decaying kink-profile at the infinite edge of the graph. It is proved that such stationary profile solutions are linearly (and nonlinearly) unstable under specific restrictions and by the flow of the sine-Gordon model on the graph. The extension theory of symmetric operators, Sturm-Liouville oscillation results and a splitting eigenvalue method are fundamental ingredients in the stability analysis. The local well-posedness of the sine-Gordon model in an appropriate energy space is also established. The theory developed in this investigation constitutes the first step to understand the dynamics of the sine-Gordon equation on a tadpole graph.

We prove Hölder continuity of Lyapunov exponents for general linear cocycles varying measures on the base in Wasserstein distance under the assumption of uniform large deviations type (LDT) estimates. This is a measure version of the abstract continuity theorem (ACT) established by Duarte-Klein [23]. The main obstacle here lies in the fact that the magnitude of the exceptional sets in LDT estimates is constantly changing when the base measures deviate. We overcome this via a combination of a Urysohn-type lemma and properties of Wasserstein distance in every iteration step. Our measure version of ACT, combined with the original work of Duarte-Klein [23], provides a complete scheme for proving joint Hölder continuity of Lyapunov exponents with respect to both measure and fiber which resolves all parameter dependence. This continuity theorem is general and applicable to a wide range of mathematical models, including product of random matrices and cocycles essentially generated by shifts. In particular, it applies to associated Schrödinger operators which are central objects in the study of mathematical physics.

In this paper, we study the derivative nonlinear wave equation$ u_{tt}-\Delta u+mu+\Lambda^{\alpha}f(\Lambda^{\alpha}u) = 0,\quad x\in\mathbb{T}^{d}: = (\mathbb{R}/2\pi\mathbb{Z})^{d} $with small-amplitude initial data of size $ \varepsilon $ in Sobolev spaces, where $ \frac{1}{2} < \alpha\leq\frac{3}{4} $, the operator $ \Lambda = (-\Delta+m)^{\frac{1}{2}} $ with $ m > 0 $, and $ f $ is an analytic function of order at least two at the origin. After performing any finite steps of Birkhoff normal form process and one step of vector field transformation, we establish the long time behavior of solutions. Due to the unboundedness of the nonlinear vector field, we provide different coefficient estimates and energy estimates tailored to the low and high frequencies. We emphasize that after the Birkhoff normal form process, certain terms are still not controlled via the standard energy estimates. We therefore construct a vector field transformation to eliminate these terms, while ensuring that all newly generated terms are bounded.

In this paper, we consider radially symmetric solutions of the degenerate cross-diffusion system with flux limitation diffusion and logistic source,$ \begin{equation*} \begin{cases} u_t = \nabla\cdot\Big( \frac { u \nabla u }{\sqrt{u^2+ |\nabla u|^2}} \Big) - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w) + f(u), & \\ 0 = \Delta v +\alpha u - m_{1}(t), & \\ 0 = \Delta w +\gamma u - m_{2}(t), & \end{cases} \end{equation*} $in $ \Omega \times (0,\infty) $, with $ \Omega $ a ball in $ \mathbb{R}^N $, $ N\geq 1 $, and subjected to no-flux boundary conditions. The logistic dampening satisfies $ f(u) = \lambda u - \mu u^{k} $ with $ \lambda,\, \mu $ positive constants and $ k \geq 1 $. If $ N\geq 3 $ and $ \chi\alpha-\xi\gamma>0 $, under certain smallness conditions on logistic degradation, we demonstrate that the solution $ u(x,t) $ exhibits blow-up behavior in $ L^{\infty} $-norm at a finite time. Moreover, for some $ p>N $, we prove that the solution also blows up in $ L^p $-norm. On the other hand, if $ \chi\alpha-\xi\gamma<0 $, or if $ \chi\alpha-\xi\gamma>0 $ and $ k $ is large, we prove that the solution is global in time.