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Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types. (English) Zbl 1078.35078

For the Tricomi type equation \[ K(y) \Delta_x u + u_{yy} + f(u) = 0, \tag \(*\) \] where \((x,y)\in \mathbb R^N\times \mathbb R,\) \(N\geq 1,\) \(yK(y)>0,\) \(K(0)=0,\) the basic changes of basis of independent variables such as translation, rotation, dilation and inversion is considered. For these cases the conservation laws are founded. The uniqueness theorem for the Tricomi problem for equation \((*)\) in a star-like domain \(\Omega\) with \(N=2,\) \(f(u) = C u | u| ^{p},\) is proved with the help of symmetry groups.

MSC:

35M10 PDEs of mixed type
58J70 Invariance and symmetry properties for PDEs on manifolds
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