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

Abstract

We consider the equation ut = (1+a(x, t)) uxx+b(x, t)ux+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 < Tm, Tm ≤ 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 u is bounded by M, 0 < x < 1, where M is a specified positive constant, we demostrate that u depends continouously upon the data ƒ, φ, ψ, g and M provided that the coefficients a, b, and c tend to zero sufficiently fast as t tends to zero. An interesting subset of the analysis is an estimate of the Lp norm of the theta function for 1 ≤ p ≤ 3.

Publication Date

1-1-2017

Publication Title

Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci

Number of Pages

57-68

Document Type

Article; Book Chapter

Personal Identifier

scopus

DOI Link

https://doi.org/10.1201/9780203744369

Socpus ID

17844393189 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/17844393189

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