Time Existence Of Solutions Research Articles

Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems {ie133-01} where {ie133-2},u is a complex valued function and φ is a real valued function. When (c 1,c 2) = (-1, 2) the system (*) is called DSI equation in the inverse scattering literature. Our purpose in this paper is to prove the local existence of a unique solution to (*) in the Sobolev spaceH 2(R 2) without the smallness condition on the data which were assumed in previous works [7], [17], [19], [26], Our result is a positive answer to Question 7 in [24].

Local existence in time of solutions to higher-order nonlinear dispersive equations

We study a system of nonlinear dispersive equations of the form $$ \partial_t u_k + \partial_x^{2j+1}u_k + F_k(u_1,\ldots ,u_n,\ldots , \partial_x^{2j}u_1, \ldots , \partial_x^{2j} u_n)=0, (t,x) \in \mathbf{R}\times h{\mathbf{R}}, $$ where $k=1,\ldots ,n$, $j\in \mathbf{N}$ and $F_k(\cdot)$ is a polynomial having no constant or linear terms. Local existence of solutions to the associated initial value problem is shown without a smallness condition on the data.

Local in time existence of solutions to the some linear constitutive model in hypoelasticity

Local in time existence of solutions to the some linear constitutive model in hypoelasticity