Continuous Dependence for the Pseudoparabolic Equation

M Yaman,Ş Gür

doi:10.1155/2010/872572

Abstract

We determine the continuous dependence of solution on the parameters in a Dirichlet-type initial-boundary value problem for the pseudoparabolic partial differential equation.

Highlights

A mixed-boundary value problem for the one-dimensional case of 1.1 appears in the study of nonsteady flow of second-order fluids 1 where u represents the velocity of the fluid
Equation 1.1 can be assumed as a model for the heat conduction involving a thermodynamic temperature θ u − αΔu and a conductive temperature u; see 2
Equations of the form 1.1 have been called pseudoparabolic by Showalter and Ting 3, because well posed initial-boundary value problems for parabolic equations are wellposed for 1.1

Summary

Introduction

We consider the following initial-boundary value problem: ut − αΔut − βΔu f u , x ∈ Ω, t > 0, 1.1 u x, 0 u0 x , x ∈ Ω, 1.2 u x, t 0, x ∈ ∂Ω, t > 0, 1.3 where α and β are positive constants, Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary ∂Ω, and f u is a given nonlinear function which satisfies. Equations of the form 1.1 have been called pseudoparabolic by Showalter and Ting 3 , because well posed initial-boundary value problems for parabolic equations are wellposed for 1.1. In certain cases, the solution of a parabolic initial-boundary value problem can be obtained as a limit of solutions to the corresponding problem for 1.1 when α goes to zero; see 4. In 5 , Karch proved well-posedness for a Cauchy problem for the pseudoparabolic 1.1

A Priori Estimates

Continuous Dependence on the Coefficient α

Continuous Dependence on the Coefficient β