We consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the “interior” wave equation on the “boundary” wave equation and show that it generates a structural damping.
Based on the recent surprising work on the symmetry breaking phenomenon of the Allen–Cahn equation [11, 12], we consider the one-dimensional parabolic sine‑Gordon equation with periodic boundary conditions. Particularly, we derive a strong dependence of the non-trivial steady states on the diffusion coefficient $\kappa$ and provide some description on them for $0 \lt \kappa \lt 1$. To further investigate the property of energy associated to the steady states, we give a complete classification and prove the monotonicity of the ground state energy with respect to the diffusion constant $\kappa$. Finally, we identify the exact decay rate of the solution to the parabolic equation together with the explicit leading term for $\kappa \geq 1$.
We consider the nonlinear Schrödinger equation with $L^2$-supercritical and $H^1$-subcritical power type nonlinearity. Duyckaerts and Roudenko [8] and Campos, Farah, and Roudenko [3] studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.
We study an initial-boundary-value problem for three-dimensional nonhomogeneous magneto-micropolar fluid equations without angular viscosity. Using linearization and Banach’s fixed point theorem, we prove the local existence and uniqueness of strong solutions. Moreover, a regularity criterion is also obtained.
Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions in a suitable Banach space. The second one considers well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the potential and $\pi\textrm{–Camassa–Holm}$; generalised Boussinesq equations; and the modified Euler–Poisson system.
We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $\mathbb{R}^d \times \mathbb{T} , d \geq 3$. This result can be regarded as the analogue of [63, 65] in our setting and the waveguide analogue investigated in [28]. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.
A new configurational probability diffusion—CPD—equation that accounts for temperature dependent molecular dynamics for incompressible polymer fluids is introduced. We prove the existence of positive solutions for the corresponding variational formulation using Schauder’s fixed point theory. The polymer fluid macromolecules are modeled as Finitely Extensible Nonlinear Elastic dumbbells, also known as FENE chains.
In this paper, we study the Cauchy problem of the incompressible Bénard system with density-dependent viscosity on the whole three-dimensional space. We first construct a key priori exponential estimates by the energy method, and then we prove that there is a unique global strong solution for the 3D Cauchy problem under the assumption that initial energy is suitably small. In particular, it is not required to be smallness condition for the initial density which contains vacuum and even has compact support. Finally, we obtain the exponential decay rates for the gradients of velocity, temperature field and pressure.
In ellipsoidal coordinates, we study the motion of the wind in the steady atmospheric Ekman layer for the height-dependent eddy viscosities in the form of some quadratic, fourth and rational power functions. We construct the explicit solutions for these forms of the eddy viscosities by using suitable boundary conditions. Furthermore, we write down a formula of the angle between the wind vector and the geostrophic wind vector at any height.
We construct a new series of multi-component integrable PDE systems that contains as particular examples (with appropriately chosen parameters) and generalises many famous integrable systems including KdV, coupled KdV [1], Harry Dym, coupled Harry Dym [2], Camassa–Holm, multicomponent Camassa–Holm [14], Dullin–Gottwald–Holm and Kaup–Boussinesq systems. The series also contains integrable systems with no low-component analogues.