Two generalized Tricomi equations

Abstract

AbstractIn this note we give existence results for the generalized Tricomi equations $${\mathcal {R}}u'' + {\mathcal {B}}u = f$$ R u ′ ′ + B u = f and $$({\mathcal {R}}u')' + {\mathcal {B}}u = f$$ ( R u ′ ) ′ + B u = f with suitable boundary data where $${\mathcal {R}}$$ R may be an operator (or a function) depending also on time assuming positive, null and negative sign, while $${\mathcal {B}}$$ B is an elliptic operator. To do that we also extend a result for equations like $$({\mathcal {R}}u')' + {\mathcal {A}}u' + {\mathcal {B}}u = f$$ ( R u ′ ) ′ + A u ′ + B u = f to equations like $${\mathcal {R}}u'' + {\mathcal {A}}u' + {\mathcal {B}}u = f$$ R u ′ ′ + A u ′ + B u = f and use these to derive the existence for the generalised Tricomi type equations mentioned above.