Cauchy problem for an evolution PDE

A Cauchy problem specifies an evolution PDE and the required initial data. For a first-order time equation it has the schematic form

\partial_tu=\mathcal F(t,x,u,\nabla u,\ldots), \qquad u(t_0,x)=u_0(x).

The spatial domain, coefficient functions, forcing, and class in which the solution is sought are part of the problem.

Higher time order and boundaries

A second-order time equation usually requires both $u(t_0,x)$ and $\partial_tu(t_0,x)$ . On a bounded spatial domain one generally also imposes boundary conditions, giving an initial-boundary-value problem. On the whole space, decay or integrability assumptions often replace boundary conditions.

References

John K. Hunter, Notes on Partial Differential Equations.