In this note we consider a coupled parabolic-elliptic system. We show existence, uniqueness, and regularity of the system by using a contraction argument. The presented results have applications to the linear stability analysis of gravity-driven flow. Mathematics Subject Classification (2000). 35M10,35B05,35C15,76E20.

We establish the existence of a nonnegative fully nontrivial solution to non-variational weakly coupled competitive elliptic system. show that this kind solutions belong topological manifold Nehari-type, and apply degree-theoretical argument on derive existence.

Exact chirped elliptic wave solutions are obtained within the framework of coupled cubic nonlinear Helmholtz equations in presence non-Kerr nonlinearity like self steepening and frequency shift. It is shown that, for a particular combination shift parameters, associated nontrivial phase gives rise to chirp reversal across solitary profile. But different terms leads chirping but no reversal. The...

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmem...

This paper is concerned with the existence of multiple positive‎ ‎solutions for a quasilinear elliptic system involving concave-convex‎ ‎nonlinearities‎ ‎and sign-changing weight functions‎. ‎With the help of the Nehari manifold and Palais-Smale condition‎, ‎we prove that the system has at least two nontrivial positive‎ ‎solutions‎, ‎when the pair of parameters $(lambda,mu)$ belongs to a c...

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a transformation to a quasimonotone system. For the transformed system we may and will use maximum (sweeping) principle arguments to derive point...

In this paper a new Weierstrass semi-rational expansion method is developed via the Weierstrass elliptic function ℘(ξ; g2, g3). With the aid of Maple, we choose the coupled water wave equation and the generalized Hirota-Satsuma coupled KdV equation to illustrate the method. As a consequence, it is shown that the method is powerful to obtain many types of new doubly periodic solutions in terms o...

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method o...

We study well-posedness for elliptic problems under the form b(u)− div a(x, u,∇u) = f, where a satisfies the classical Leray-Lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u. A prototype case is the equation u− div ( | ∇u| ∇u ) = f . We have to assume that infx∈Ω, z∈R p(x, z) is greater than the space dimensionN . Then, under mild r...