On a heat potential

Flows of heat and time moving boundary

“…There is a number y > 0 with the following property: For any T 0 < b there is a cp XQ e Q) b such that 0 g cp XQ ^ 1 in R m + 1 , cp XQ = 1 on B n R XQ and (3)(4)(5)(6)(7) |<c/> ro , ff,^>| < y for every [£, T] G B 0 .…”

“…In [11] the assumption that E is of the form D x (T l9 T 2 ) is essential. The case of time moving boundary is considered in [3], [4], [5] but with the restric tion that m = 1 and E is of the form E = {[x 9 1] eR 2 ; te (a 9 b) 9 x > <p(t)} , where (p is a continuous function of bounded variation on an interval <a, b>, or of the form E = {[x, r] eR 2 ; te (a, b) 9 cp x (t) < x < <p 2 (t)} , where <p l9 <p 2 are continuous functions of bounded variation on the interval <a, b>, (p t (t) < (p 2 (t) for te(a, fc>. Only the first boundary value problem is considered in [3], [4].…”

“…The case of time moving boundary is considered in [3], [4], [5] but with the restric tion that m = 1 and E is of the form E = {[x 9 1] eR 2 ; te (a 9 b) 9 x > <p(t)} , where (p is a continuous function of bounded variation on an interval <a, b>, or of the form E = {[x, r] eR 2 ; te (a, b) 9 cp x (t) < x < <p 2 (t)} , where <p l9 <p 2 are continuous functions of bounded variation on the interval <a, b>, (p t (t) < (p 2 (t) for te(a, fc>. Only the first boundary value problem is considered in [3], [4]. In [5] it is shown that in the case of time moving boundary the first and second boundary value problems are not adjoint* to each other in the sense of itnegral equations but the first boundary value problem and some special type of the third boundary value problem are.…”

“…In [5] it is shown that in the case of time moving boundary the first and second boundary value problems are not adjoint* to each other in the sense of itnegral equations but the first boundary value problem and some special type of the third boundary value problem are. The assumption m = 1 is essential in [3], [4], [5].…”

Third boundary value problem for the heat equation. I.

“…Now we shall show another way how to express the term Wf(£, t). For this purpose we shall recall some definitions from [3] and introduce the following notation.…”