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Sobolev-Galpern type equations

The following partial differential equation is called of Sobolev-Galpern type:
\begin{eqnarray*}u_t-\eta \Delta u_t-\nu \Delta u =f(x,u,\nabla u)\end{eqnarray*}
where $u=u(x,t), x\in G\subset \R^n, t\ge0, \eta,\nu$ non-negative constants. Several physical phenomena lead to boundary value problems or Cauchy problems of this pde. We will list here some of them:
 

1. $ n=1$; Non-steady flow of second-order fluids. Here $u$ represents the velocity of the fluid. Mixed boundary conditions are posed.
 

2.     Theory of seepage of homogenous fluids through a fissured rock. Here $\eta $ is a charcteristic of the fissured rock (increasing $\eta $ corresponds to a decreasing degree of fissuring.
 

3.     A model for heat conduction with a thermodynamic temperature $\Theta=u-\eta\Delta u$ and a conductive temperature $u$. (cf. Zeitschrift für Angewandte Math. Phys. 19 (1968) 614-627 (Chen P.J./Gurtin M.E.))
 

4.     Benjamin-Bona-Mahony-Burgers equation.
 

This type of equations is called pseudoparabolic which was introduced by R.E. Showalter and T.W. Ting in 1970, because well-posed initial-boundary-value problems for parabolic equations remain well-posed also for Sobolev-Galpern equations. For $\eta\to 0$ we get a class of parabolic equations.
 

A late Reference: G. Karch: Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations, MMAS, 20, 3, 271-289 (1997). 


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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg