Temperature of a semi-infinite rod which radiates both linearly and nonlinearly

Asymptotic behavior of temperature in a nonlinear radiating, linear absorbing rod of finite length

“…In other words, the combined energy radiation at the end x = 0 and the "leakage" (of heat) at x = I dominate the energy absorption due to the simultaneous input h and the absorption along the surface of the rod. Hence the consideration of a finite rod instead of a semi-infinite rod leads to sharper results than those obtained in [4,5].…”

“…In the special case of h(t) = 0 on [Ty , °°) for some 7\ as assumed in [4], all the conditions on h in Theorem 3.1 are satisfied for any y > 0. In this situation, the temperature decays to zero exponentially with a decay rate 5 = I)(t/21)2 -c. On the other hand, the work in [5] requires that /0" h(t) dt < <*>.…”

“…The problem (1.1)-(1.3) and its various generalization have been treated by several authors (cf. [1][2][3][4][5][6][7][8]). The recent work by Keller-Olmstead [5] and Hartka [4] studies the asymptotic behavior of the temperature in a semi-infinite rod for the case c < 0, u0 = 0.…”

“…Several estimates for insuring the diminishing property of u as t -* m are given in [3,5] for the case c -ft = 0 and in [4] for the case ft2 + c < 0. However, when the rod is finite, no matter how great its length may be, it is reasonable to expect that the temperature remains diminishing to zero for certain range of values of c, ft even if both constants are positive.…”

“…We also extend the above investigation to a heat-conduction problem in a two-dimensional rectangular region. The consideration of a finite domain instead of a semi-infinite domain makes it possible to sharpen the results obtained in [3][4][5] and leads to more information about the solution. In fact, we obtain explicit estimates for the rate of decay for the solution c. v. pao in terms of the physical parameters D, c, a, 13, n, I and the boundary source h(t).…”