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On the Stability of Heat Equation
We prove the generalized Hyers-Ulam stability of the heat equation, , in a class of twice continuously differentiable functions under certain conditions.
Let be a normed space and let be an open interval. If for any function satisfying the differential inequality for all and for some , there exists a solution of the differential equation such that for any , where is an expression of only, then we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace and by and , where are functions not depending on and explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability. (This type of stability is sometimes called the Hyers-Ulam-Rassias stability.)
We may apply these terminologies for other differential equations and partial differential equations. For more detailed definitions of the Hyers-Ulam stability and the generalized Hyers-Ulam stability, refer to [1–7].
Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [8, 9]). Here, we will introduce a result of Alsina and Ger (see ). If a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any . This result was generalized by Miura et al. (see [10, 11]).
In 2007, Jung and Lee  proved the Hyers-Ulam stability of the first-order linear partial differential equation where and are constants with . It seems that the first paper dealing with Hyers-Ulam stability of partial differential equations was written by Prástaro and Rassias . For a recent result on this subject, refer to .
In this paper, using an idea from the paper , we investigate the generalized Hyers-Ulam stability of the heat equation in the class of radially symmetric functions, where denotes the Laplace operator, , , and is open. The heat equation plays an important role in a number of fields of science. It is strongly related to the Brownian motion in probability theory. The heat equation is also connected with chemical diffusion and it is sometimes called the diffusion equation.
2. Main Result
For a given integer , denotes the th coordinate of any point in ; that is, . We assume that , , and are constants with and , and we define where .
Due to an idea from [16, Section 2.3.1], we may search for a solution of (4) of the form for some twice continuously differentiable function and constants and . Based on this argument, we define where we set and the constants and will be chosen appropriately.
Theorem 1. Let and be functions such that If a twice continuously differentiable function satisfies for all and , then there exists a solution of the heat equation (4) such that and for all and .
Proof. Since belongs to , there exists a function such that
for any and , where we set . Using this notation, we calculate and :
So we have
for any , and .
If we set and in the previous equality, then we have for all , and . Moreover, from the last equality and (10), it follows that or for all and . In view of (9), we have for any .
We integrate each term of the last inequality from to and take account of the definition of to get or for all .
According to [17, Theorem 1], together with (8), there exists a unique such that for all , or equivalently for all and .
Now, we set for all and . Then it is easy to show that and is a solution of the heat equation (4). Moreover, inequality (11) is an immediate consequence of (22).
Corollary 2. Let and be functions. Assume that , and that there exist constants and such that If a twice continuously differentiable function satisfies for all and , then there exists a solution of the heat equation (4) such that and for all and .
Proof. It follows from (24) that
for all . Moreover, by the previous inequality, it holds that
since the assumption, , implies that .
According to Theorem 1, there exists a solution of the heat equation (4) such that inequality (27) holds, for all and .
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557) and also by 2013 Hongik University Research Fund.
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Copyright © 2013 Balázs Hegyi and Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.