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We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L $\infty$ norm of these solutions.

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain $\Omega_\epsilon:=\Omega\setminus \displaystyle \overline{B(\xi,\epsilon)}$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(\xi,\epsilon)$ is a ball centered at $\xi\in \Omega$ with radius $\epsilon>0$. Precisely, given a small parameter $\rho>0$ and any integer $m\ge 2$, there exist a radius $\epsilon=\epsilon(\rho)>0$ small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution $u_\rho$ with an asymptotic profile as a sign-changing tower of $m$ singular Liouville bubbles centered at the same $\xi$ and with $\epsilon(\rho)\to 0^+$ as $\rho$ approaches to zero., Comment: arXiv admin note: text overlap with arXiv:1904.00127, arXiv:1909.00905

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi \cite{Miyachi} and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space., 9 pages; In this new version, we extend the fractional chain rule to the Hardy space

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics., 41 pages

We deal with some quasilinear elliptic problems posed in a bounded smooth convex domain Ω⊂RN (N≥3), namely {−Δu=λu+μ(x)|∇u|q+f(x),x∈Ω,u=0,x∈∂Ω, where the data satisfy μ∈L∞(Ω),μ⪈0;f∈Lp(Ω),p>N,f⪈0 and 1, Grant PID2021-122122NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Junta de Andalucía, Consejería de Transformación Económica, Industria, Conocimiento y Universidades-Unión Europea grant P18-FR-667. Second and third author supported by Junta de Andalucía FQM-116. First author supported by Junta de Andalucía FQM-194 and CDTIME. Third author supported by Junta de Andalucía, Consejería de Transformación Económica, Industria, Conocimiento y Universidades grant UAL2020-FQMB2046. The authors would like to thank the reviewers of the manuscript as their comments and suggestions have contributed deeply to the improvement not only in clarity but also in the quality of the results obtained

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn--Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$., 41 pages

We study the asymptotic emergent dynamics of two models that can be thought of as extensions of the well known Schr\"odinger-Lohe model for quantum synchronization. More precisely, the interaction strength between different oscillators is determined by intrinsic parameters, following Cucker-Smale communication protocol. Unlike the original Schr\"odinger-Lohe system, where the interaction strength was assumed to be uniform, in the cases under our consideration the total mass of each quantum oscillator is allowed to vary in time. A striking consequence of this property is that these extended models yield configurations exhibiting phase, but not space, synchronization. The results are mainly based on the analysis of the ODE systems arising from the correlations, control over the well known Cucker-Smale dynamics, and the dynamics satisfied by the quantum order parameter., Comment: 18 pages, minor changes and submitted

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse-Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincare-Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. A number of nontrivial examples admitting a unique limit measure are provided, which include monostable, multistable systems and those possessing infinite equivalent classes., Comment: 37 pages, 12 figures