On the Stability of Heat Equation

Abstract

We prove the generalized Hyers-Ulam stability of the heat equation, Δ𝑢=𝑢_𝑡, in a class of twice continuously differentiable functions under certain conditions.

1. Introduction

Let 𝑋 be a normed space and let 𝐼 be an open interval. If for any function 𝑓∶𝐼→𝑋 satisfying the differential inequality 𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)≤𝜀, (1) for all 𝑥∈𝐼 and for some 𝜀≥0, there exists a solution 𝑓₀∶𝐼→𝑋 of the differential equation 𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)=0 (2) such that ‖𝑓(𝑥)−𝑓₀(𝑥)‖≤𝐾(𝜀) for any 𝑥∈𝐼, where 𝐾(𝜀) 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 𝐾(𝜀) by 𝜑(𝑥) and Φ(𝑥), where 𝜑,Φ∶𝐼→[0,∞) are functions not depending on 𝑓 and 𝑓₀ 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 [1]). If a differentiable function 𝑓∶𝐼→ℝ is a solution of the differential inequality |𝑦^′(𝑥)−𝑦(𝑥)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a solution 𝑓₀∶𝐼→ℝ of the differential equation 𝑦^′(𝑥)=𝑦(𝑥) such that |𝑓(𝑥)−𝑓₀(𝑥)|≤3𝜀 for any 𝑥∈𝐼. This result was generalized by Miura et al. (see [10, 11]).

In 2007, Jung and Lee [12] proved the Hyers-Ulam stability of the first-order linear partial differential equation 𝑎𝑢_𝑥𝑥,𝑦+𝑏𝑢_𝑦𝑥,𝑦+𝑐𝑢𝑥,𝑦+𝑑=0, (3) where 𝑎,𝑏∈ℝ and 𝑐,𝑑∈ℂ are constants with ℜ(𝑐)≠0. It seems that the first paper dealing with Hyers-Ulam stability of partial differential equations was written by Prástaro and Rassias [13]. For a recent result on this subject, refer to [14].

In this paper, using an idea from the paper [15], we investigate the generalized Hyers-Ulam stability of the heat equation Δ𝑢(𝑥,𝑡)−𝑢_𝑡(𝑥,𝑡)=0 (4) in the class of radially symmetric functions, where Δ denotes the Laplace operator, 𝑡>0, 𝑥∈𝐼, and 𝐼⊂ℝ^𝑛 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 𝑛≥2, 𝑥_𝑖 denotes the 𝑖th coordinate of any point 𝑥 in ℝ^𝑛; that is, 𝑥=(𝑥₁,…,𝑥_𝑖,…,𝑥_𝑛). We assume that 𝑎, 𝑏, and 𝑡₁ are constants with 0<𝑎<𝑏≤∞ and 0<𝑡₁≤∞, and we define 𝐷=𝑥∈ℝ^𝑛∣𝑎<|𝑥|<𝑏,𝑇=𝑡∈ℝ∣0<𝑡<𝑡₁, (5) where |𝑥|=𝑥²₁+⋯+𝑥²_𝑛.

Due to an idea from [16, Section  2.3.1], we may search for a solution of (4) of the form 𝑢(𝑥,𝑡)=(1/𝑡^𝛼)𝑣(|𝑥|/𝑡^𝛽) for some twice continuously differentiable function 𝑣 and constants 𝛼 and 𝛽>0. Based on this argument, we define ()=1𝑈=𝑢∶𝐷×𝑇⟶ℝ∣𝑢𝑥,𝑡𝑡^𝛼𝑤(𝑟)𝑟∀𝑥∈𝐷,𝑡∈𝑇andforsomefunction𝑤∶₀|𝑥|,∞⟶ℝwith𝑟=𝑡^𝛽andlim_𝑟→∞𝑟^𝑛𝑤(𝑟)=lim_𝑟→∞𝑟^𝑛−1𝑤^′(𝑟)=0, (6) where we set 𝑟₀𝑎∶=𝑡^𝛽₁for0<𝑡₁,0<∞for𝑡₁=∞ (7) and the constants 𝛼 and 𝛽 will be chosen appropriately.

Theorem 1. Let 𝜑∶(𝑟₀,∞)→[0,∞) and 𝜓∶𝑇→[0,∞) be functions such that ^∞_𝑟0𝑒^𝑢2/4𝑢^𝑛−1∞_𝑢𝑠^𝑛−1𝜑(𝑠)𝑑𝑠𝑑𝑢<∞,𝑐∶=inf_𝑡∈𝑇𝑡^𝑛/2+1𝜓(𝑡)>0. (8)(9) If a twice continuously differentiable function 𝑢∈𝑈 satisfies Δ𝑢(𝑥,𝑡)−𝑢_𝑡(|𝑥|𝑥,𝑡)≤𝜑√𝑡𝜓(𝑡) (10) for all 𝑥∈𝐷 and 𝑡∈𝑇, then there exists a solution 𝑢₀∶𝐷×𝑇→ℝ of the heat equation (4) such that 𝑢₀∈𝑈 and 𝑢(𝑥,𝑡)−𝑢₀(≤𝑐𝑥,𝑡)𝑡^𝑛/2𝑒^{√−(|𝑥|/2𝑡)2}×^∞_{√|𝑥|/𝑡}𝑒^𝑢2/4𝑢^𝑛−1∞_𝑢𝑠^𝑛−1𝜑(𝑠)𝑑𝑠𝑑𝑢 (11) for all 𝑥∈𝐷 and 𝑡∈𝑇.

Proof. Since 𝑢(𝑥,𝑡) belongs to 𝑈, there exists a function 𝑤∶(𝑟₀,∞)→ℝ such that 𝑢()=1𝑥,𝑡𝑡^𝛼𝑤(𝑟) (12) for any 𝑥∈𝐷 and 𝑡∈𝑇, where we set 𝑟=|𝑥|/𝑡^𝛽. Using this notation, we calculate 𝑢_𝑡 and Δ𝑢: 𝑢_𝑡()𝛼𝑥,𝑡=−𝑡^𝛼+1𝑤(𝑟)−𝛽𝑡^𝛼+1𝑟𝑤^′(𝑟),𝑢_𝑥𝑖()=1𝑥,𝑡𝑡^𝛼+𝛽𝑤^′(𝑟)𝑥_𝑖|𝑥|,𝑢_{𝑥𝑖𝑥𝑖}()=1𝑥,𝑡𝑡^𝛼+𝛽1𝑡^𝛽𝑤^′′(𝑟)𝑥²_𝑖|𝑥|²+𝑤^′(𝑟)1|𝑥|−𝑥²_𝑖|𝑥|³. (13) So we have Δ𝑢(𝑥,𝑡)−𝑢_𝑡()=1𝑥,𝑡𝑡^𝛼+2𝛽×𝑤^′′(𝑟)+𝑛−1𝑟𝑤^′(𝑟)+1𝑡^𝛼+1𝛽𝑟𝑤^′(𝑟)+𝛼𝑤(𝑟) (14) for any 𝑥∈𝐷, 𝑡∈𝑇 and 𝑟>𝑟₀.
If we set 𝛼=𝑛/2 and 𝛽=1/2 in the previous equality, then we have Δ𝑢(𝑥,𝑡)−𝑢_𝑡()=1𝑥,𝑡𝑡^𝑛/2+1𝑤^′′(𝑟)+𝑛−1𝑟𝑤^′(𝑟)+𝑟2𝑤^′(𝑟)+𝑛2𝑤(𝑟)=1𝑟^𝑛−11𝑡^𝑛/2+1𝑟^𝑛−1𝑤^′′(𝑟)+(𝑛−1)𝑟^𝑛−2𝑤^′(𝑟)+12𝑟^𝑛𝑤^′(𝑟)+𝑛𝑟^𝑛−1𝑤(𝑟)=1𝑟^𝑛−11𝑡^𝑛/2+1𝑟^𝑛−1𝑤^′(𝑟)+𝑟^𝑛2𝑤(𝑟)^′ (15) for all 𝑥∈𝐷, 𝑡∈𝑇 and 𝑟>𝑟₀. Moreover, from the last equality and (10), it follows that Δ𝑢(𝑥,𝑡)−𝑢_𝑡(=1𝑥,𝑡)𝑟^𝑛−11𝑡^𝑛/2+1×𝑟^𝑛−1𝑤^′(𝑟)+𝑟^𝑛2𝑤(𝑟)^′≤𝜑(𝑟)𝜓(𝑡) (16) or 𝑟^𝑛−1𝑤^′(𝑟)+𝑟^𝑛2𝑤(𝑟)^′≤𝑟^𝑛−1𝜑(𝑟)𝑡^𝑛/2+1𝜓(𝑡) (17) for all 𝑟>𝑟₀ and 𝑡∈𝑇. In view of (9), we have −𝑐𝑟^𝑛−1𝜑(𝑟)≤𝑟^𝑛−1𝑤^′(𝑟)+𝑟^𝑛2𝑤(𝑟)^′≤𝑐𝑟^𝑛−1𝜑(𝑟) (18) for any 𝑟>𝑟₀.
We integrate each term of the last inequality from 𝑟 to ∞ and take account of the definition of 𝑈 to get −𝑐^∞_𝑟𝑠^𝑛−1𝜑(𝑠)𝑑𝑠≤−𝑟^𝑛−1𝑤^′(𝑟)−𝑟^𝑛2𝑤(𝑟)≤𝑐^∞_𝑟𝑠^𝑛−1𝜑(𝑠)𝑑𝑠 (19) or 𝑤^′(𝑟)+𝑟2𝑤(𝑟≤𝑐)𝑟^𝑛−1∞_𝑟𝑠^𝑛−1𝜑(𝑠)𝑑𝑠 (20) for all 𝑟>𝑟₀.
According to [17, Theorem 1], together with (8), there exists a unique 𝛾∈ℝ such that 𝑤(𝑟)−𝛾𝑒^−𝑟2/4≤𝑐𝑒^{−𝑟2/4∞}_𝑟𝑒^𝑢2/4𝑢^𝑛−1∞_𝑢𝑠^𝑛−1𝜑(𝑠)𝑑𝑠𝑑𝑢 (21) for all 𝑟>𝑟₀, or equivalently 𝑢()−𝛾𝑥,𝑡𝑡^𝑛/2𝑒^{√−(|𝑥|/2𝑡)2}≤𝑐𝑡^𝑛/2×𝑒^{√−(|𝑥|/2𝑡)2}×^∞_{√|𝑥|/𝑡}𝑒^𝑢2/4𝑢^𝑛−1∞_𝑢𝑠^𝑛−1𝜑(𝑠)𝑑𝑠𝑑𝑢, (22) for all 𝑥∈𝐷 and 𝑡∈𝑇.
Now, we set 𝑢₀()𝛾𝑥,𝑡∶=𝑡^𝑛/2𝑒^{√−(|𝑥|/2𝑡)2} (23) for all 𝑥∈𝐷 and 𝑡∈𝑇. Then it is easy to show that  𝑢₀∈𝑈 and 𝑢₀ is a solution of the heat equation (4). Moreover, inequality (11) is an immediate consequence of (22).

Corollary 2. Let 𝜑∶(𝑟₀,∞)→[0,∞) and 𝜓∶𝑇→[0,∞) be functions. Assume that 𝑛>2, 0<𝑡₁<∞ and that there exist constants 𝑐 and 𝜃 such that 𝜑|𝑥|√𝑡≤𝜃𝑒^{√−(|𝑥|/2𝑡)2}|𝑥|√𝑡^2−𝑛(∀𝑥∈𝐷,𝑡∈𝑇),𝑐∶=inf_𝑡∈𝑇𝑡^𝑛/2+1𝜓(𝑡)>0. (24)(25) If a twice continuously differentiable function 𝑢∈𝑈 satisfies Δ𝑢(𝑥,𝑡)−𝑢_𝑡(|𝑥|𝑥,𝑡)≤𝜑√𝑡𝜓(𝑡) (26) for all 𝑥∈𝐷 and 𝑡∈𝑇, then there exists a solution 𝑢₀∶𝐷×𝑇→ℝ of the heat equation (4) such that 𝑢₀∈𝑈 and 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤2𝑐𝜃|𝑥|𝑛−2𝑡^2−𝑛𝑒^{√−(|𝑥|/2𝑡)2}, (27) for all 𝑥∈𝐷 and 𝑡∈𝑇.

Proof. It follows from (24) that ^∞_𝑢𝑠^𝑛−1𝜑(𝑠)𝑑𝑠≤^∞_𝑢𝑠^𝑛−1𝜃𝑒^−𝑠2/4𝑠^2−𝑛𝑑𝑠=2𝜃𝑒^−𝑢2/4 (28) for all 𝑢>𝑟₀. Moreover, by the previous inequality, it holds that ^∞_𝑟0𝑒^𝑢2/4𝑢^𝑛−1∞_𝑢𝑠^𝑛−1𝜑(𝑠)≤𝑑𝑠𝑑𝑢_∞𝑟0𝑒^𝑢2/4𝑢^𝑛−12𝜃𝑒^−𝑢2/4𝑑𝑢=2𝜃𝑟𝑛−2^02−𝑛<∞, (29) since the assumption, 0<𝑡₁<∞, implies that 𝑟₀>0.
According to Theorem 1, there exists a solution 𝑢₀∈𝑈 of the heat equation (4) such that inequality (27) holds, for all 𝑥∈𝐷 and 𝑡∈𝑇.

Acknowledgments

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.