scholarly journals Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev space

1997 ◽  
Vol 40 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Nakao Hayashi ◽  
Hitoshi Hirata
Keyword(s):  
Sobolev Space ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Initial Value ◽  
Hyperbolic Case ◽  
Local Existence And Uniqueness ◽  
Usual Sobolev Space

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

2017 ◽  
Vol 147 (6) ◽  
pp. 1243-1277 ◽  
Author(s):  
Eiji Onodera
Keyword(s):  
Fourth Order ◽  
Energy Method ◽  
Initial Value Problem ◽  
Local Existence ◽  
Three Dimensional ◽  
Closed Curve ◽  
Initial Value ◽  
Curve Flow ◽  
Local Existence And Uniqueness ◽  
Loss Of Derivatives

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

scholarly journals Exponential decay of solutions with \(L^{p}\)-norm for a class to semilinear wave equation with damping and source terms

2020 ◽  
Vol 4 (2) ◽  
pp. 123-131
Author(s):  
Amar Ouaoua ◽  
◽  
Messaoud Maouni ◽  
Aya Khaldi ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Lyapunov Function ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Local Solution ◽  
Initial Value ◽  
Local Existence And Uniqueness ◽  
Decay Of Solutions ◽  
Uniqueness Of The Solution ◽  
Damping And Source Terms

In this paper, we consider an initial value problem related to a class of hyperbolic equation in a bounded domain is studied. We prove local existence and uniqueness of the solution by using the Faedo-Galerkin method and that the local solution is global in time. We also prove that the solutions with some conditions exponentially decay. The key tool in the proof is an idea of Haraux and Zuazua with is based on the construction of a suitable Lyapunov function.

MATHEMATICAL ANALYSIS OF A REACTIVE-DIFFUSIVE MODEL OF THE DISPERSAL OF A CHEMICAL TRACER WITH NONLINEAR CONVECTION

1995 ◽  
Vol 05 (01) ◽  
pp. 29-46 ◽  
Author(s):  
STEVE COHN ◽  
J. DAVID LOGAN
Keyword(s):  
Initial Value Problem ◽  
Local Existence ◽  
Heteroclinic Orbit ◽  
Traveling Wave Solutions ◽  
Sobolev Embedding ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Contraction Mapping Theorem ◽  
Weakly Nonlinear ◽  
Local Existence And Uniqueness

We formulate and analyze a nonlinear reaction-convection-diffusion system that models the dispersal of solutes, or chemical tracers, through a one-dimensional porous medium. A similar set of model equations also arises in a weakly nonlinear limit of the combustion equations. In particular, we address two fundamental questions with respect to the model system: first, the existence of wavefront type traveling wave solutions, and second, the local existence and uniqueness of solutions to the pure initial value problem. The solution to the wavefront problem is obtained by showing the existence of a heteroclinic orbit in a two-dimensional phase space. The existence argument for the initial value problem is based on the contraction mapping theorem and Sobolev embedding. In the final section we prove non-negativity of the solution.

Local wellposedness in Sobolev space for the inhomogeneous non-resistive MHD equations on general domain

2017 ◽  
Vol 19 (06) ◽  
pp. 1650055
Author(s):  
Weiren Zhao
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Mhd Equations ◽  
General Domain ◽  
Local Existence And Uniqueness ◽  
Uniqueness Of The Solution ◽  
Local Wellposedness

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [Formula: see text] bounded domain [Formula: see text] or [Formula: see text], if the initial data [Formula: see text] with [Formula: see text] satisfies [Formula: see text].

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