The Analysis of Hyers–Ulam Stability for Heat Equations with Time-Dependent Coefficient

Abstract

In this paper, we prove the Hyers–Ulam stability and generalized Hyers–Ulam stability of $u_t(x,t)=a\left(t\right)\Delta u(x,t)$ with an initial condition $u(x,0)=f\left(x\right)$ for $x\in R^n$ and $0<t<T$; the corresponding conclusions of the standard heat equation can be also derived as corollaries. All of the above results are proved by using the properties of the fundamental solution of the equation.

1. Introduction

Hyers–Ulam stability was first proposed by D. H. Hyers and S. M. Ulam in 1941 [1]. Th. M. Rassias had generalized the Hyers’ results, gave the conditions for the existence of linear mappings near approximate linear mappings, and proved the Hyers–Ulam stability of linear mappings [2,3]. Th. M. Rassias and S. Jung obtained some results on the Hyers–Ulam stability of functional equations in Banach spaces and the linear functional equation in a single variable $f\left(\psi \right(x\left)\right)=g\left(x\right)·f\left(x\right)$ on a complete metric group, respectively [4,5].

A. Prastaro and Th. M. Rassias first considered the stability of partial differential equations and mainly proved the Hyers–Ulam stability of Navier–Stokes equations [6]. In [7], F. Wang, L. Xue, K. Zhao, and X. M. Zheng have studied the global stabilization of solutions to initial-boundary value problems for the generalized Fisher/KPP equation. M. Takeshi, S. Jung, and S. Takahasi proved the Hyers–Ulam–Rassias stability of the Banach space valued differential equation $y^{\prime }=\lambda y$, where $\lambda$ is a complex constant [8]. In [9], N. Lungu and D. Popa have proved the Hyers–Ulam stability of a linear partial differential equation of the first order. A. E. Hamza and D. Marian studied the Ulam–Hyers stability of first-order partial differential equations and nonlinear second order dynamic equations on time scales [10,11]. In [12], D. Otrocol and V. Ilea studied the Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability for a delay differential equation. A. Zada, S. Shaleena, and T. X. Li investigated different Ulam-type stabilities for nonlinear differential equations with integrable impulses of the fractional type [13]. In [14], D. Popa and I. Rasa derived a result on Hyers–Ulam stability for the linear differential operator of higher order with nonconstant coefficients in Banach spaces. In [15], T. Miura, S. Miyajima, and S. Takahasi gave a necessary and sufficient condition so that the operator $T_h$ ($T_{h}u=u^{{}^{\prime }}+hu$) would have Hyers–Ulam stability. H. Rezaei, S. Jung, and Th. M. Rassias proved that the differential equation $y^{\left(n\right)}+{\sum }_{k=0}^{n-1}{\alpha }_k{y}^{\left(k\right)}\left(t\right)=f\left(t\right)$ has Hyers–Ulam stability by using the Laplace transform method, where ${\alpha }_k$ is a scalar [16].

In [17], Z. Y. Gao and J. R. Wang discussed the stability of time-fractional order heat conduction equations and proved the Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of time-fractional-order heat-conduction equations via a fractional Green function involving the Wright function. S. Jung proved the generalized Hyers–Ulam stability of the homogeneous diffusion equation $u_t(x,t)=\Delta u(x,t)$ and the wave equation $u_{tt}(x,t)=1/c^2\Delta u(x,t)$, where $\Delta$ is the Laplace differential operator ($\Delta u={\sum }_{i=1}^n{u}_{x_i{x}_i}$) [18,19,20]. Moreover, G. Choi, S. Jung, and S. Min proved the generalized Hyers–Ulam stability of the inhomogeneous diffusion equation $u_t(x,t)-\Delta u(x,t)=f(x,t)$ and the wave equation $u_{tt}(x,t)-1/c^2\Delta u(x,t)=f(x,t)$ by using the operator and other methods [21,22,23]. In [24,25], D. Kang improved the results of Jung ’s diffusion equation, proved the generalized Hyers–Ulam stability of the homogeneous heat equation in n-dimensional space for the first time in 2022 by using Fourier transform and $L^2$ norm estimation, and verified it by numerical results. As we all know, many physical and engineering problems require the solution of the parabolic equation with a time-dependent coefficient. However, to the best of our knowledge, the Hyers–Ulam stability of the heat equation with the time-dependent coefficient has not been discussed.

In this paper, we will improve the results of D. Kang and B. Hoewoon in [24] by proving the Hyers–Ulam stability and generalized Hyers–Ulam stability of heat equations with time-dependent coefficient $u_t(x,t)=a\left(t\right)\Delta u(x,t)$ with an initial condition $u(x,0)=f\left(x\right)$, that is,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t(x,t)=a\left(t\right)\Delta u(x,t),x\in R^n,0<t<T,\hfill \\ u(x,0)=f\left(x\right),x\in R^n.\hfill \end{array}\right.\end{array}$$

(1)

where $u(x,t)\in C^{2,1}\left(D_T\right)(D_T=(x,t):x\in R^n,0<t<T)$ and $a\left(t\right)$ is a continuous function on $[0,T]$, satisfying

$$0<a\le a\left(t\right)\le b<\infty .$$

This assumes that the initial condition in the initial value problem (1) is continuous and bounded on $R^n$ and the solution u decays sufficiently fast at infinity.

2. Preliminary Knowledge

In this section, we introduce some fundamental definitions and lemmas. Before analyzing the stability of the equation, we first solve the solution of the equation and prove the uniqueness of the solution. In [26,27], the existence and uniqueness of the Cauchy problem for the heat equation with the Tikhonov condition and the uncertain heat equation have been discussed. Therefore, we discuss the existence and uniqueness of the heat equations with time-dependent coefficient firstly.

Lemma 1

(Existence and uniqueness of solution). When $a\left(t\right)$ is a continuous function on $[0,T]$, satisfying $0<a\le a\left(t\right)\le b<\infty .$ Let $A\left(t\right)={\int }_0^{t}a\left(\tau \right)d\tau$; then there is $A\left(t\right)>0$. The unique solution of the heat equations with the time-dependent coefficient (1) is

$$\begin{array}{c}\hfill u(x,t)={\int }_{R^n}{\left(4\pi A\left(t\right)\right)}^{-(n/2)}{exp(-|x-y|}^2/4A\left(t\right))f\left(y\right)dy.\end{array}$$

Assuming the initial condition in the initial value problem (1) is continuous and bounded on $R^n$ and the solution u decays sufficiently fast at infinity.

Proof. 

Using Fourier transform for Equation (1), we can obtain the transformed equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\widehat{u_t}(\xi ,t)+{\left|\xi \right|}^{2}a\left(t\right)\widehat{u}(\xi ,t)=0,\xi \in R^n,0<t<T,\hfill \\ \widehat{u}(\xi ,0)=\widehat{f}\left(\xi \right),\xi \in R^n,\hfill \end{array}\right.\end{array}$$

(2)

where $\widehat{u}(\xi ,t)$ and $\widehat{f}\left(\xi \right)$ are Fourier transforms of $u(x,t)$ and $f\left(x\right)$, respectively. We can easily obtain the solution of Equation (2) as

$$\widehat{u}(\xi ,t)=\widehat{f}{\left(\xi \right)exp(-|\xi |}^{2}A\left(t\right)).$$

From the inverse Fourier transform, we can obtain

$$\begin{array}{cc}& {\mathcal{F}}^{-1}[exp(-{\left|\xi \right|}^{2}A\left(t\right)\left)\right]\hfill \\ & =\frac{1}{{\left(2\pi \right)}^n}{\int exp(-|\xi |}^{2}A\left(t\right))exp\left(i(x·\xi )\right)d\xi \hfill \\ & ={\left(4\pi A\left(t\right)\right)}^{-(n/2)}{exp(-|x|}^2/4A\left(t\right)).\hfill \end{array}$$

Finally, using the convolution theorem, we can obtain the solution of Equation (1)

$$u(x,t)={\int }_{R^n}{\left(4\pi A\left(t\right)\right)}^{-(n/2)}{exp(-|x-y|}^2/4A\left(t\right))f\left(y\right)dy.$$

Next we prove the uniqueness of the solution. Assume that the equation also has a solution v, satisfying $v_t(x,t)=a\left(t\right)\Delta v(x,t),v(x,0)=f\left(x\right)$. Let $w=u-v$; then, w satisfies the following equation:

$$w_t-a\left(t\right)\Delta w=0,w(x,0)=0.$$

(3)

Using Fourier transform for Equation (3), we can obtain $\widehat{w}(\xi ,t)=0$. So, we have $w(x,t)=0$, that is, $u=v$. The above conclusions show that the solution of Equation (1) exists and is unique, and the stability can be discussed. □

Lemma 2

([24]). (Duhamel’s principle) Assume $v(x,t)$ is the solution of homogeneous heat equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}_t(x,t)-\Delta v(x,t)=0,x\in R^n,0<t<T,\hfill \\ v(x,0)=h(x,\tau ),x\in R^n\hfill \end{array}\right.\end{array}$$

where τ is a parameter. Then, the solution of the nonhomogeneous heat equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t(x,t)-\Delta u(x,t)=h(x,t),x\in R^n,0<t<T,\hfill \\ u(x,0)=0,x\in R^n\hfill \end{array}\right.\end{array}$$

is

$$u(x,t)={\int }_0^{t}v(x,t-\tau ,\tau )d\tau .$$

When $x\in R^n,0<t<T$, we have the following lemmas.

Lemma 3

([24,28]). The fundamental solution of equation $u_t(x,t)=\Delta u(x,t)$ is $k(x,t)={\left(4\pi t\right)}^{-(n/2)}{exp(-|x|}^2/4t)$. For all $\left|x\right|<R,t_0\le t<T$, when $0<t_0<T,R>0,\gamma \ge 1$, there is a constant N about $t_0,T,R,\gamma$ such that $k(x,t)$ satisfies the following inequality:

$$0\le k(x-y,t)\le N/{\left(4\pi t\right)}^{n/2}{exp(-(\left|x\right|}^2/4t\left)\right)exp(-{\left|y\right|}^2/4\gamma t).$$

Lemma 4.

Similar to the proof of Lemma 3, the fundamental solution of equation $u_t(x,t)=a\left(t\right)\Delta u(x,t)(0<a\le a\left(t\right)\le b<\infty )$ is $k(x,t)={\left(4\pi A\left(t\right)\right)}^{-(n/2)}{exp(-|x|}^2/4A\left(t\right))$, where $A\left(t\right)={\int }_0^{t}a\left(\tau \right)d\tau$. For all $\left|x\right|<R,t_0\le t<T$, when $0<t_0<T,R>0,\gamma =b/a\ge 1$, there is a constant N about $t_0,T,R,\gamma$ such that $k(x,t)$ satisfies the following inequality:

$$0\le k(x-y,t)\le N/{\left(4\pi A\left(t\right)\right)}^{n/2}exp(-{\left|x\right|}^2/4A\left(t\right))exp(-{\left|y\right|}^2/4\gamma A\left(t\right)).$$

Definition 1.

If for each function u satisfying

$$|u_t(x,t)-a\left(t\right)\Delta u(x,t)|\le \epsilon ,x\in R^n,0<t<T,$$

(4)

where $\epsilon >0$, there is a solution v of the equation $u_t(x,t)=a\left(t\right)\Delta u(x,t)$ such that

$$|u-v|\le K\left(ϵ\right),$$

for all $x\in R^n,0<t<T$ and $\underset{ϵ\to 0}{lim}K\left(ϵ\right)=0$. Then, we say that the Equation (1) has the Hyers–Ulam stability.

Definition 2.

Let $\alpha (x,t)$ and $\beta (x,t)$ be the non-negative continuous function defined on $D_T(D_T=(x,t):x\in R^n,0<t<T)$, if for each function u satisfying

$$|u_t(x,t)-a\left(t\right)\Delta u(x,t)|\le \alpha (x,t),x\in R^n,0<t<T,$$

(5)

there is a solution v of the equation $u_t(x,t)=a\left(t\right)\Delta u(x,t)$ such that

$$|u-v|\le \beta (x,t),$$

for all $x\in R^n,0<t<T$. Then, we say that the Equation (1) has the generalized Hyers–Ulam stability.

In particular, when $\alpha (x,t)=\epsilon$, $\beta (x,t)=K\left(ϵ\right)$, the generalized Hyers–Ulam stability is also called the Hyers–Ulam stability. Therefore, Hyers–Ulam stability is a special generalized Hyers–Ulam stability.

3. Main Results

Theorem 1.

Let $\epsilon >0$, and $u(x,t)\in C^{2,1}\left(D_T\right)(D_T=(x,t):x\in R^n,0<t<T)$ satisfies the following inequality:

$$|u_t(x,t)-a\left(t\right)\Delta u(x,t)|\le \epsilon ,$$

for all $x\in R^n,0<t<T.$

Then, there is a solution $v(x,t)\in C^{2,1}\left(D_T\right)$ to (1) such that

$$|u-v|\le \epsilon M_n,$$

where $M_n=\frac{NT{\lambda }_n}{2{\left(\pi \right)}^{n/2}}{\gamma }^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right)$, ${\lambda }_n$ is the surface area of the unit sphere in $R^n$, $\mathsf{\Gamma }\left(\frac{n}{2}\right)$ is the gamma function, N is a positive constant, and $\gamma =b/a\ge 1$.

Proof. 

We now consider the following heat equation:

$$v_t-a\left(t\right)\Delta v=0,u_t-a\left(t\right)\Delta u=c\ne 0.$$

Let $w=u-v$; then, w satisfies the following equation:

$$w_t-a\left(t\right)\Delta w=c,w(x,0)=0.$$

(6)

Using the Lemma 2, we obtain the solution w of the initial value problem (6)

$$w(x,t)={\int }_0^t{\int }_{R^n}k(x-y,t-\tau )cdyd\tau .$$

(7)

From the conclusion obtained by Lemma 3, when $|w_t-a\left(t\right)▵w|=|c|\le \epsilon$, for $\forall (x,t)\in D_T$ we can obtain

$$\begin{array}{cc}& \left|w(x,t)\right|=|{\int }_0^t{\int }_{R^n}k(x-y,t-\tau )cdyd\tau |\hfill \\ & \le {\int }_0^t{\int }_{R^n}k(x-y,t-\tau )\left|c\right|dyd\tau \hfill \\ & \le \frac{\epsilon N}{{\left(4\pi \right)}^{\frac{n}{2}}}exp\left(\frac{-{\left|x\right|}^2}{4A\left(t\right)}\right){\int }_0^t{\int }_{R^n}\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\left|y\right|}^2}{4\gamma A(t-\tau )})dyd\tau .\hfill \end{array}$$

The integral on $R^n$ can be written as follows:

$$\begin{array}{cc}& {\int }_0^t{\int }_{R^n}\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\left|y\right|}^2}{4\gamma A(t-\tau )})dyd\tau \hfill \\ & ={\int }_0^t{\int }_S{\int }_0^{\infty }\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )}){\rho }^{n-1}d\rho d\sigma d\tau \hfill \\ & ={\lambda }_n{\int }_0^t{\int }_0^{\infty }\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )}){\rho }^{n-1}d\rho d\tau ,\hfill \end{array}$$

where S is the unit sphere in $R^n$, $d\sigma$ is the surface measure on S, and ${\lambda }_n$ is the surface area of the unit sphere in $R^n$. Then, take a change of variables $s=\frac{\rho }{\sqrt{A(t-\tau )}}$, that is, $\rho =s\sqrt{A(t-\tau )}$, so $d\rho =\sqrt{A(t-\tau )}ds$, we can obtain

$$\begin{array}{cc}& {\lambda }_n{\int }_0^t{\int }_0^{\infty }\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )}){\rho }^{n-1}d\rho d\tau \hfill \\ & ={\lambda }_n{\int }_0^t{\int }_0^{\infty }{\left(\frac{\rho }{\sqrt{A(t-\tau )}}\right)}^n\frac{1}{\rho }exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )})d\rho d\tau \hfill \\ & ={\lambda }_n{\int }_0^t{\int }_0^{\infty }{s}^n\frac{1}{s\sqrt{A(t-\tau )}}exp(-\frac{s^2}{4\gamma })\sqrt{A(t-\tau )}dsd\tau \hfill \\ & ={\lambda }_n{\int }_0^t{\int }_0^{\infty }{s}^{n-1}exp(-\frac{s^2}{4\gamma })dsd\tau .\hfill \end{array}$$

Let $x=\frac{s^2}{4\gamma }$, i.e., $s=\sqrt{4\gamma x}$, and $ds=\sqrt{\gamma }{x}^{-\frac{1}{2}}dx$, and using the Fubini’s Theorem, then we obtain

$$\begin{array}{cc}& {\lambda }_n{\int }_0^t{\int }_0^{\infty }{s}^{n-1}exp(-\frac{s^2}{4\gamma })dsd\tau \hfill \\ & ={\lambda }_n{\int }_0^t{\int }_0^{\infty }{\left(4\gamma x\right)}^{\frac{n-1}{2}}exp(-x)\sqrt{\gamma }{x}^{-\frac{1}{2}}dxd\tau \hfill \\ & =2^{n-1}{\lambda }_n{\int }_0^t{\int }_0^{\infty }{\left(\gamma \right)}^{\frac{n}{2}}exp(-x)x^{\frac{n}{2}-1}dxd\tau \hfill \\ & =2^{n-1}{\lambda }_n{\int }_0^{t}d\tau {\int }_0^{\infty }{\left(\gamma \right)}^{\frac{n}{2}}exp(-x)x^{\frac{n}{2}-1}dx\hfill \\ & =2^{n-1}T{\lambda }_n{\int }_0^{\infty }{\left(\gamma \right)}^{\frac{n}{2}}exp(-x)x^{\frac{n}{2}-1}dx\hfill \\ & =2^{n-1}T{\lambda }_n{\left(\gamma \right)}^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right).\hfill \end{array}$$

From the previous conclusions, we can obtain

$$|u(x,t)-v(x,t)|\le \frac{\epsilon N}{{\left(4\pi \right)}^{\frac{n}{2}}}exp\left(\frac{-{\left|x\right|}^2}{4A\left(t\right)}\right)·2^{n-1}T{\lambda }_n{\left(\gamma \right)}^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right).$$

Let

$$M_n=\frac{NT{\lambda }_n}{2{\left(\pi \right)}^{\frac{n}{2}}}{\gamma }^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right),$$

when

$$0<a\le a\left(t\right)\le b<\infty ,A\left(t\right)={\int }_0^{t}a\left(\tau \right)d\tau >0,$$

then we can obtain the conclusion

$$|u(x,t)-v(x,t)|\le \epsilon M_{n}exp(-\frac{{\left|x\right|}^2}{4A\left(t\right)})\le \epsilon M_n.$$

When n is determined, $M_n$ is a constant. Therefore, the proof of the Theorem 1 is complete. This means that the heat equation with the time-dependent coefficient (1) has the Hyers–Ulam stability. When $a\left(t\right)=1$, we can also obtain the similar conclusion of the standard heat equation. □

Corollary 1.

Let $\epsilon >0$, and $u(x,t)\in C^{2,1}\left(D_T\right)(D_T=(x,t):x\in R^n,0<t<T)$ satisfies the following inequality:

$$|u_t(x,t)-\Delta u(x,t)|\le \epsilon ,$$

for all $x\in R^n,0<t<T$.

Then, there is the solution $v(x,t)\in C^{2,1}\left(D_T\right)$ to $u_t(x,t)=\Delta u(x,t)$ with an initial condition $u(x,0)=f\left(x\right)$ for $x\in R^n$ and $0<t<T$ such that

$$|u(x,t)-v(x,t)|\le \epsilon M_{n}exp(-\frac{{\left|x\right|}^2}{4t})\le \epsilon M_n.$$

where $M_n=\frac{NT{\lambda }_n}{2{\left(\pi \right)}^{n/2}}{\gamma }^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right)$, ${\lambda }_n$ is the surface area of the unit sphere in $R^n$, $\mathsf{\Gamma }\left(\frac{n}{2}\right)$ is the gamma function, N is a positive constant, and $\gamma =b/a\ge 1$. This means that the heat equation $u_t(x,t)=\Delta u(x,t)$ has the Hyers–Ulam stability.

The proof of this corollary is similar to the Theorem 1. Therefore, we skip its proof here.

After discussing the Hyers–Ulam stability, we prove the more general stability of Equation (1), that is, the generalized Hyers–Ulam stability defined by Definition 2.

Theorem 2.

Assume that $\varphi :(0,\infty )\to [0,\infty ),\psi :(0,T)\to [0,\infty )$, and satisfy the following conditions

$$\begin{array}{cc}& {\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds<\infty ,\hfill \\ & c:=\underset{t\in (0,T)}{sup}A\left(t\right)\psi \left(t\right)>0.\hfill \end{array}$$

If there is a function $u(x,t)\in C^{2,1}\left(D_T\right)$ and the initial condition is $u(x,0)=f\left(x\right)$, for $\forall x\in R^n,t_0\le t<T$ satisfies the following inequality:

$$|u(x,t)-a\left(t\right)▵u(x,t)|\le \varphi \left(\frac{\left|x\right|}{\sqrt{A\left(t\right)}}\right)\psi \left(t\right).$$

Then, there is a solution $v(x,t)\in C^{2,1}\left(D_T\right)$ to (1) such that

$$|u-v|\le \frac{c}{b}{M}_{n}exp(-\frac{{\left|x\right|}^2}{4A\left(t\right)})\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right).$$

where $M_n=\frac{N}{{\pi }^{\frac{n}{2}}}{\lambda }_n{\gamma }^{\frac{3n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right)$, ${\lambda }_n$ is the surface area of the unit sphere in $R^n$, $\mathsf{\Gamma }\left(\frac{n}{2}\right)$ is the gamma function, N is a positive constant, and $\gamma =b/a\ge 1$.

Proof. 

We now consider the following heat equation with time-dependent coefficient

$$v_t-a\left(t\right)\Delta v=0,u_t-a\left(t\right)\Delta u=z(x,t)\ne 0.$$

where $z(x,t)$ is continuous and bounded in $R^n$. Let $w=u-v$; then, w satisfies the following equation:

$$w_t-a\left(t\right)\Delta w=z(x,t),w(x,0)=0.$$

(8)

Using Lemma 1, we obtain the solution w of the initial value problem (8)

$$w(x,t)={\int }_0^t{\int }_{R^n}k(x-y,t-\tau )z(y,\tau )dyd\tau .$$

(9)

From the conclusion obtained by Lemma 3, when $|w_t-a\left(t\right)▵w|=|z(x,t)|\le \varphi \left(\frac{\left|x\right|}{\sqrt{A\left(t\right)}}\right)\psi \left(t\right)$, for $\forall (x,t)\in D_T$ we can obtain

$$\begin{array}{cc}& \left|w(x,t)\right|=|{\int }_0^t{\int }_{R^n}k(x-y,t-\tau )z(y,\tau )dyd\tau |\hfill \\ & \le {\int }_0^t{\int }_{R^n}k(x-y,t-\tau )\left|z(y,\tau )\right|dyd\tau \hfill \\ & \le {\int }_0^t{\int }_{R^n}k(x-y,t-\tau )\varphi \left(\frac{\left|y\right|}{\sqrt{A\left(\tau \right)}}\right)\psi \left(\tau \right)dyd\tau \hfill \\ & \le Nexp\left(\frac{-{\left|x\right|}^2}{4A\left(t\right)}\right){\int }_0^t{\int }_{R^n}\frac{1}{{\left(4\pi A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\left|y\right|}^2}{4\gamma A(t-\tau )})\varphi \left(\frac{\left|y\right|}{\sqrt{A\left(\tau \right)}}\right)\psi \left(\tau \right)dyd\tau \hfill \\ & =Nexp\left(\frac{-{\left|x\right|}^2}{4A\left(t\right)}\right){\int }_0^t{\int }_S{\int }_0^{\infty }\frac{1}{{\left(4\pi A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )})\varphi \left(\frac{\rho }{\sqrt{A\left(\tau \right)}}\right)\psi \left(\tau \right){\rho }^{n-1}d\rho d\sigma d\tau ,\hfill \end{array}$$

where S is the unit sphere in $R^n$ and $d\sigma$ is the surface measure on S. Then, take a change of variables $s=\frac{\rho }{\sqrt{A\left(\tau \right)}},$ that is, $\rho =s\sqrt{A\left(\tau \right)}$, so $d\rho =\sqrt{A\left(\tau \right)}ds$, and because $0<a\le a\left(t\right)\le b<\infty$, we can obtain

$$\begin{array}{cc}& {\int }_0^t{\int }_S{\int }_0^{\infty }\frac{1}{{\left(4\pi A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{{\rho }^2}{4\gamma A(t-\tau )})\varphi \left(\frac{\rho }{\sqrt{A\left(\tau \right)}}\right)\psi \left(\tau \right){\rho }^{n-1}d\rho d\sigma d\tau \hfill \\ & =\frac{{\lambda }_n}{{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^t{\int }_0^{\infty }\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{A\left(\tau \right)s^2}{4\gamma A(t-\tau )})\varphi \left(s\right)\psi \left(\tau \right)s^{n-1}{\left(A\left(\tau \right)\right)}^{\frac{n}{2}}dsd\tau \hfill \\ & \le c\frac{{\lambda }_n}{{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^t{\int }_0^{\infty }\frac{1}{{\left(A(t-\tau )\right)}^{\frac{n}{2}}}exp(-\frac{A\left(\tau \right)s^2}{4\gamma A(t-\tau )})\varphi \left(s\right)s^{n-1}{\left(A\left(\tau \right)\right)}^{\frac{n}{2}-1}dsd\tau \hfill \\ & \le c\frac{{\lambda }_n}{{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^t{\int }_0^{\infty }\frac{1}{{\left[a(t-\tau )\right]}^{\frac{n}{2}}}exp(-\frac{s^2}{4\gamma }·\frac{a\tau }{b(t-\tau )})\varphi \left(s\right)s^{n-1}{\left(b\tau \right)}^{\frac{n}{2}-1}dsd\tau \hfill \\ & =c\frac{{\lambda }_n}{{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^t{\int }_0^{\infty }\frac{1}{b}{\left(\frac{b}{a}\right)}^{\frac{n}{2}}{\left(\frac{\tau }{t-\tau }\right)}^{\frac{n}{2}}\frac{1}{\tau }exp(-\frac{s^2}{4{\gamma }^2}·\frac{\tau }{t-\tau })\varphi \left(s\right)s^{n-1}dsd\tau ,\hfill \end{array}$$

let $\nu =\frac{t-\tau }{\tau },$ or $\tau =\frac{t}{\nu +1},$ so $d\tau =-\frac{t}{{(\nu +1)}^2}d\nu$, which then can be equal to

$$\begin{array}{cc}& c\frac{{\lambda }_n}{{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^t{\int }_0^{\infty }\frac{1}{b}{\left(\frac{b}{a}\right)}^{\frac{n}{2}}{\left(\frac{\tau }{t-\tau }\right)}^{\frac{n}{2}}\frac{1}{\tau }exp(-\frac{s^2}{4{\gamma }^2}·\frac{\tau }{t-\tau })\varphi \left(s\right)s^{n-1}dsd\tau \hfill \\ & =c\frac{{\lambda }_n{\gamma }^{\frac{n}{2}}}{b{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^{\infty }{\int }_0^{\infty }\frac{1}{{\nu }^{\frac{n}{2}}}\frac{1}{\tau }exp(-\frac{s^2}{4{\gamma }^2}·\frac{1}{\nu })\varphi \left(s\right)s^{n-1}\frac{t}{{(\nu +1)}^2}dsd\nu \hfill \\ & \le c\frac{{\lambda }_n{\gamma }^{\frac{n}{2}}}{b{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^{\infty }{\int }_0^{\infty }\frac{1}{{\nu }^{\frac{n}{2}+1}}exp(-\frac{s^2}{4{\gamma }^2}·\frac{1}{\nu })\varphi \left(s\right)s^{n-1}dsd\nu .\hfill \end{array}$$

Take another coordinate transformation $x=-\frac{s^2}{4{\gamma }^2\nu },\mathrm{i.e.},\nu =\frac{s^2}{4{\gamma }^{2}x}$, so $d\nu =-\frac{s^2}{4{\gamma }^2{x}^2}dx$, and using Fubini’s Theorem, we can obtain

$$\begin{array}{cc}& c\frac{{\lambda }_n{\gamma }^{\frac{n}{2}}}{b{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^{\infty }{\int }_0^{\infty }\frac{1}{{\nu }^{\frac{n}{2}+1}}exp(-\frac{s^2}{4{\gamma }^2}·\frac{1}{\nu })\varphi \left(s\right)s^{n-1}dsd\nu \hfill \\ & =c\frac{{\lambda }_n{\gamma }^{\frac{n}{2}}}{b{\left(4\pi \right)}^{\frac{n}{2}}}{\int }_0^{\infty }{\int }_0^{\infty }{\left(\frac{4{\gamma }^{2}x}{s^2}\right)}^{\frac{n}{2}+1}exp(-x)\varphi \left(s\right)s^{n-1}\frac{s^2}{4{\gamma }^2{x}^2}dsdx\hfill \\ & =c\frac{{\lambda }_n{\gamma }^{\frac{3n}{2}}}{b{\pi }^{\frac{n}{2}}}{\int }_0^{\infty }{\int }_0^{\infty }exp(-x)\varphi \left(s\right)x^{\frac{n}{2}-1}{s}^{-1}dsdx\hfill \\ & =c\frac{{\lambda }_n{\gamma }^{\frac{3n}{2}}}{b{\pi }^{\frac{n}{2}}}({\int }_0^{\infty }{x}^{\frac{n}{2}-1}exp(-x)dx)\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right)\hfill \\ & =c\frac{{\lambda }_n{\gamma }^{\frac{3n}{2}}}{b{\pi }^{\frac{n}{2}}}\left(\mathsf{\Gamma }\left(\frac{n}{2}\right)\right)\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right).\hfill \end{array}$$

From the previous conclusions, we can obtain

$$|u(x,t)-v(x,t)|\le \frac{c}{b{\pi }^{\frac{n}{2}}}N{\lambda }_n{\gamma }^{\frac{3n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right)exp(-\frac{{\left|x\right|}^2}{4A\left(t\right)})\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right).$$

Let

$$M_n=\frac{N}{{\pi }^{\frac{n}{2}}}{\lambda }_n{\gamma }^{\frac{3n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right),$$

then we can obtain the conclusion:

$$|u(x,t)-v(x,t)|\le \frac{c}{b}{M}_{n}exp(-\frac{{\left|x\right|}^2}{4A\left(t\right)})\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right).$$

Therefore, the proof of the Theorem 2 is complete. This means that the heat equation with initial conditions (1) has generalized Hyers–Ulam stability. When $a\left(t\right)=1$, we can also obtain a similar conclusion of the standard heat equation proved by D. Kang and B. Hoewoon in [24]. □

Corollary 2

([24]). Assume that $\varphi :(0,\infty )\to [0,\infty ),\psi :(0,T)\to [0,\infty )$, and satisfy the following conditions:

$$\begin{array}{cc}& {\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds<\infty ,\hfill \\ & c:=\underset{t\in (0,T)}{sup}t\psi \left(t\right)>0.\hfill \end{array}$$

If there is a function $u(x,t)\in C^{2,1}\left(D_T\right)$ and the initial condition is $u(x,0)=f\left(x\right)$, for $\forall x\in R^n,0<t<T$ satisfies the following inequality:

$$|u(x,t)-▵u(x,t)|\le \varphi \left(\frac{\left|x\right|}{\sqrt{t}}\right)\psi \left(t\right).$$

Then, there is a solution $v(x,t)\in C^{2,1}\left(D_T\right)$ to $u_t(x,t)=\Delta u(x,t)$ with an initial condition $u(x,0)=f\left(x\right)$ for $x\in R^n$ and $0<t<T$ such that

$$|u(x,t)-v(x,t)|\le c{M}_{n}exp(-\frac{{\left|x\right|}^2}{4t})\left({\int }_0^{\infty }\varphi \left(s\right)s^{-1}ds\right).$$

where $M_n=N{\lambda }_n{\left(\frac{\gamma }{\pi }\right)}^{\frac{n}{2}}\mathsf{\Gamma }\left(\frac{n}{2}\right)$, ${\lambda }_n$ is the surface area of the unit sphere in $R^n$, $\mathsf{\Gamma }\left(\frac{n}{2}\right)$ is the gamma function, N is a positive constant, and $\gamma \ge 1$.

The proof of Corollary 2 is in [24]; we skip its proof here.

Finally, we prove that when the control function $\varphi \left(x\right)$ in the Theorem 2 satisfies some inequality, the error between a approximate solution and the true solution of the heat equations with time-dependent coefficient (1) can be bounded by the Gaussian kernel of $k(x,t)$ and the Gamma function. We can obtain a corollary similar to Corollary 2.3 in [24].

Corollary 3.

Assume that $\varphi :(0,\infty )\to [0,\infty ),\psi :(0,T)\to [0,\infty )$, and satisfy the following conditions

$$\begin{array}{cc}\hfill & \varphi \left(s\right)\le s^{\alpha }exp(-s),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & c:=\underset{t\in (0,T)}{sup}A\left(t\right)\psi \left(t\right)>0,\hfill \end{array}$$

(11)

where α is a constant. If a function $u(x,t)\in C^{2,1}\left(D_T\right)(D_T=(x,t):x\in R^n,0<t<T)$ and the initial condition is $u(x,0)=f\left(x\right)$, for $\forall x\in R^n,t_0\le t<T$ satisfies the following inequality:

$$|u(x,t)-a\left(t\right)▵u(x,t)|\le \varphi \left(\frac{\left|x\right|}{\sqrt{A\left(t\right)}}\right)\psi \left(t\right).$$

Then, there is the solution $v(x,t)\in C^{2,1}\left(D_T\right)$ to (1) such that

$$|u-v|\le \frac{c}{b}{M}_{n}exp(-\frac{{\left|x\right|}^2}{4A\left(t\right)})\mathsf{\Gamma }\left(\alpha \right).$$

The constant $M_n$ is the same as the constant in the Theorem 2. The proof of Corollary 3 is similar to the Corollary 2.3 in [24], so we skip its proof here.

4. Conclusions

This paper is devoted to considering the Hyers–Ulam and the generalized Hyers–Ulam stability of heat equations with the time-dependent coefficient. Based on some properties of the fundamental solution of the equation, the conclusions are obtained. We can see that when the space dimension n is determined and the approximate solution of Equation (1) satisfies the condition of the Theorem 1, the error between the approximate solution and the exact solution of Equation (1) is a known constant multiple of $\epsilon$, that is, when $\epsilon$ tends to 0, the error also tends to 0. When the control function $\varphi \left(x\right)$ in Theorem 2 satisfies the condition (10), the error between an approximate solution and the true solution of the heat equations with the time-dependent coefficient (1) can be bounded by the Gaussian kernel of $k(x,t)$ and the Gamma function. It is also worth considering the fractional differential equations of variable-order problems related to thermodynamics, dynamics, resonance, etc.