We propose a systemof partial differential equations with a single constant delay τ > 0 describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of R. For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of o...

The system is called a Benjamin-Ono-Boussinesq system because it can be reduced to a pair of equations whose linearization uncouples to a pair of linear Benjamin-Ono equations. Equations of type (1.1) are a class of essential model equations appearing in physics and fluid mechanics. To describe two-dimensional irrotational flows of an inviscid liquid in a uniform rectangular channel, Boussinesq...

We study the initial value problem for the quasi-geostrophic type equations ∂θ ∂t + u · ∇θ + (−∆)θ = 0, on R × (0,∞), θ(x, 0) = θ0(x), x ∈ R n , where λ(0 ≤ λ ≤ 1) is a fixed parameter and u = (uj) is divergence free and determined from θ through the Riesz transform uj = ±Rπ(j)θ, with π(j) a permutation of 1, 2, · · · , n. The initial data θ0 is taken in the Sobolev space L̇r,p with negative ind...

the main purpose of this article is to increase the efficiency of the least squares method in numerical solution of ill-posed functional and physical equations. determining the least squares of a given function in an arbitrary set is often an ill-posed problem. in this article, by defining artificial constraint and using lagrange multipliers method, the attempt is to turn -dimensional least squ...

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the...

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary...

We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...