On reflection principles for parabolic equations in one space variable

Abstract

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equationacross an analytic arc σ: x=s1(t) satisfies the boundary dataWe assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by