Initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves (that may serve as mathematical models wildfires) were considered. The existence and uniqueness of global strong solutions well their exponential decay have been established.

The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation ...

We present a heuristic model for the energy spectrum of the one-dimensional phase turbulence in the steady state of the Kuramoto-Sivashinsky equation. Our model contains an energy transfer mechanism from low- to high-wave-vector modes. The energy transfer is written as the sum of local and nonlocal interactions. Our analytical results show good agreement with numerical simulations, particularly...

Abstract: We study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on a finite domain. Firstly it shows that under the given boundary feedback conditions it admits a unique solution and the solution is stable. Secondly it proves that if the external excitation f is a time periodic function, then the system under the boundary conditions admits ...

In this article, we study the boundary controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval. The control acts on the first spatial derivative at the left endpoint. First, we prove that this control system is null controllable. It is done using a spectral analysis and the method of moments. Then, we introduce a boundary feedback law stabilizing to zero the solution o...

A nonlinear equation describing curved stationary flames with arbitrary gas expansion θ = ρfuel/ρburnt, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak nonlinearity. It is proved that in the scope of the asymptotic expansion for θ → 1, the new equation gives the true solution to the problem of stationary flame propagation with the accuracy ...