Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded domains

Korteweg-de Vries Equation in Bounded Domains

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence...

Partial Differential Equations on Bounded Domains

Linear Degenerate Parabolic Equations in Bounded Domains: Controllability and Observability

In this paper we study controllability properties of linear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of 'regional null controllability', showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate. keywords: l...

Classical , Nonlocal , and Fractional Diffusion Equations on Bounded Domains

The purpose of this paper is to compare the solutions of one-dimensional boundary value problems corresponding to classical, fractional and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion, when Fick’s first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick’s first law in terms of a nonlocal fl...

Nonlocal Elliptic Equations in Bounded Domains: a Survey

In this paper we survey some results on the Dirichlet problem { Lu = f in Ω u = g in R\Ω for nonlocal operators of the form