PDEs of this type are parabolic . PDEs of this type are called hyperbolic . The analyses of these three types of equations are quite distinct in character. These behaviours generalize to nonlinear PDEs as well. A general linear PDE may be viewed as seeking the kernel of a linear map defined between appropriate function spaces.
Modern approaches seek methods applicable to non-linear PDEs as well as linear ones.
In this context existence and uniqueness results, and theorems concerning the regularity of solutions, are more difficult. Generalizations of results about partial differential equations often lead to statements about function spaces and other topological vector spaces. This image slightly hand-edited for clarity. Dynamical systems as of 2000. Publishing, Teaneck, NJ, 1987.