On the Continuous Dependence of the Solution of a Linear Parabolic Partial Differential Equation on the Boundary Data and the Solution at an Interior Spatial Point

We consider the equation u _t = (1 + a(x,t)) u_xx +b(x, t)u_x + c(x, t)u + f(x, t), 0 < x < 1, 0 < t ≤ T, subject to the condition u(0,t) = φ(t), u(1,t) = ψ(t), u(ξ, t) = g(t), 0 < t < T_m, T_m ≤ T, where ξ is an irrational number in 0 < x < 1. Under the additional conditions that the C ²+ ^α,1+ ^α /² norm of и is bounded by M, 0 < x < 1, where M is a specified positive constant, we demostrate that и depends continouously upon the data ƒ, φ, ψ, g and M provided that the coefficients a, b, and с tend to zero sufficiently fast as t tends to zero. An interesting subset of the analysis is an estimate of the L^p norm of the theta function for 1 ≤ p ≤ 3.