Continuation and reflection of solutions of partial differential equations
Fritz John · Bulletin of the American Mathematical Society · 1957
A solution of an ordinary differential equation can be continued as long as its graph stays in the domain, in which the equation is regular. On the other hand a solution of a partial differential equation can have a natural boundary interior to the domain of regularity of the equation. Let R be a closed region and S a portion of the boundary of R. Then S is a natural boundary for a solution u defined in R, if there exists no solution defined in a full neighbourhood of a point of S which agrees with u in R. Examples for the occurrence of such natural boundaries are well known from the theory of harmonic functions. Neither the equation nor the solution has to show any very singular behavior on approaching a natural boundary S.