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On the stability of the heat equation with an initial condition
Journal of Inequalities and Applicationsvolume 2013, Article number: 475 (2013)
In this paper, we prove the generalized Hyers-Ulam stability of the heat equation with an initial condition
in a class of twice continuously differentiable functions under certain conditions.
Let X be a normed space, and let I 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 ε only, then we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace ε and by and , where are functions not depending on f 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 to other differential equations and partial differential equations. For more detailed definitions of the Hyers-Ulam stability and the generalized Hyers-Ulam stability, refer to [1–8].
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Here, we introduce the result of Alsina and Ger (see ): If a differentiable function is a solution of the differential inequality , where I is an open subinterval of ℝ, then there exists a solution of the differential equation such that for any . This result was generalized by Miura et al. (see [11, 12]).
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 the Hyers-Ulam stability of partial differential equations was written by Prastaro and Rassias . For a recent result on this subject, refer to .
in the class of radially symmetric functions, where △ denotes the Laplace operator. 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 i th coordinate of any point x in , i.e., . We assume that is a constant with , and we define
Due to an idea from [, Section 2.3.1], we may search for a solution of (1.1) of the form for some function v. Based on this argument, we define
Theorem 2.1 Let and be functions such that
Assume that is a function with . If a twice continuously differentiable function satisfies
then there exist solutions of the heat equation and a real number γ such that
for all and with , where
Proof Since , there exists a function such that
for any and , where we set . Using this notation, we calculate and △u:
So, we have
for all , with . Moreover, from the last equality and (2.3), it follows that
for all and . In view of (2.2), we have
for any .
We integrate each term of the last inequality from r to ∞ and take account of the definition of U to get
for all .
According to [, Theorem 1], together with (2.1), there exists a (unique) constant such that
for all , or equivalently
for all and with , which proves the validity of inequality (2.4), and in view of (2.6), it is not difficult to prove that is a solution of the heat equation, i.e., .
If we replace x with and multiply each term by , and if we integrate all terms in the last inequality over , then we obtain
for all and . If we define as in (2.6) for any and , following the proof of [, Theorem 1 in Section 2.3], we can then easily prove that
for all and with , which shows the validity of inequality (2.5). □
The linearity of solutions of heat equation (1.1) implies that and are also solutions of the heat equation.
As in the proof of [, Theorem 1 in Section 2.3], we can show that
for each .
If a function satisfies
for all and for some , then there exists a positive number A such that
i.e., satisfies condition (2.1).
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).
The author declares that he has no competing interests.
The author declares that this paper is his original paper.