I am interested in the interpretation of PDE that are not strictly elliptic, parabolic,or hyperbolic: ones that change type in sections of the domain, e.g. the generalised Tricomi equation
A(x)u,xx+u,yy=0
will be elliptic in regions where A(x)≥0 and hyperbolic when A(x)<0.
I am aware there is a great deal of literature surrounding the topic (e.g. transonic flows), but mostly regarding well-posedness and such, not so much about the actual physics of what is going on.
For a start, I usually think of hyperbolic equations as having wave-like components in their solutions -- this usually means there is some kind of 'time' involved. What does it mean for a PDE in spatial variables to be hyperbolic? It seems weird to me to think of 'travelling waves' in the (x,y) plane. What about an elliptic PDE involving both spatial and temporal variables?
Can someone explain what is physically happening here? In the non-linear case [A(x)→A(x,u)] does this kind of elliptic-hyperbolic transition imply something about shocks?
As a simple example, if we have A(x)=x and the domain is something like D={x,y:−a≤x≤a,−b≤y≤b}, what are the 'wave-like' portions doing in the x<0 region? What physics are they conveying?