About mean value theorems for the singular parabolic equation

Abstract

A huge number of physical, geometric, and probabilistic problems lead to the construction and study of parabolic partial differential equations. The emergence of new problems of information propagation and processes with memory leads to the need to consider parabolic type equations with various operators acting on spatial variables. In this article, mean value theorems for the singular parabolic equation were obtained. The singularity is due to the presence of the Laplace–Bessel operator.

Appendices

Appendix

In this section, we give some formulas and calculations used in the article.

Proposition 1

[13] Integral $\int \limits _{S_1^+(n)}\textbf{j}_\gamma (r\theta ,\xi ) \theta ^\gamma \,\textrm{d}S$ is calculated by the formula

$$\begin{aligned} \int \limits _{S_1^+(n)}\textbf{j}_\gamma (x,r\theta ) \theta ^\gamma \,\textrm{d}S=\frac{\prod \limits ^n_{i=1}\Gamma \left( \frac{\gamma _i+1}{2}\right) }{2^{n-1}\Gamma \left( \frac{n+\mid \gamma \mid }{2}\right) }\, j_{\frac{n+\mid \gamma \mid }{2}-1}(r\mid x\mid ), \end{aligned}$$

(16)

where

$$\begin{aligned} S_1^+(n)=\{x\in \overline{\mathbb {R}}\,^n_+:\mid x\mid =1\}\cup \{x\in \overline{\mathbb {R}}\,^n_+:x_i=0,\mid x\mid {\le } r,i=1,\ldots ,n\} \end{aligned}$$

is the part of the unit sphere belonging to $\mathbb {R}^n_+$.

Proposition 2

The next formula is valid

$$\begin{aligned} (\textbf{F}_\gamma ^{-1})_\xi \{e^{-a^2\mid \xi \mid ^2t}\}(x)=\frac{2^{-\mid \gamma \mid }}{a^{n+\mid \gamma \mid } t^{\frac{n+\mid \gamma \mid }{2}}\Gamma \left( \frac{n+\mid \gamma \mid }{2}\right) \prod \limits _{j=1}^n\, \Gamma \left( \frac{\gamma _j{+}1}{2}\right) }\, e^{-\frac{\mid x\mid ^2}{4 a^2 t}}. \end{aligned}$$

(17)

Proof

We have

$$\begin{aligned} (\textbf{F}_\gamma ^{-1})_\xi \{e^{-a^2\mid \xi \mid ^2t}\}(x)= & {} \frac{2^{n-\mid \gamma \mid }}{\prod \limits _{j=1}^n\, \Gamma ^2\left( \frac{\gamma _j{+}1}{2}\right) }\int \limits _{\mathbb {R}^n_+} \textbf{j}_\gamma (x,\xi )e^{-a^2\mid \xi \mid ^2t}\xi ^\gamma \,\textrm{d}\xi =\{\xi =r\theta \}= \\= & {} \frac{2^{n-\mid \gamma \mid }}{\prod \limits _{j=1}^n\, \Gamma ^2\left( \frac{\gamma _j{+}1}{2}\right) }\int \limits _0^\infty e^{-a^2r^2t}r^{n+\mid \gamma \mid -1}dr \int \limits _{S^+_1(n)} \textbf{j}_\gamma (x,r\theta )\theta ^\gamma \textrm{d}S. \end{aligned}$$

Using formula (16), we obtain

$$\begin{aligned}{} & {} (\textbf{F}_\gamma ^{-1})_\xi \{e^{-a^2\mid \xi \mid ^2t}\}(x) \\{} & {} \quad =\frac{2^{1-\mid \gamma \mid }}{\Gamma \left( \frac{n+\mid \gamma \mid }{2}\right) \prod \limits _{j=1}^n\, \Gamma \left( \frac{\gamma _j{+}1}{2}\right) }\,\int \limits _0^\infty e^{-a^2r^2t}j_{\frac{n+\mid \gamma \mid }{2}-1}(r\mid x\mid )r^{n+\mid \gamma \mid -1}dr \\{} & {} \quad =\frac{2^{\frac{n-\mid \gamma \mid }{2}}}{\Gamma \left( \frac{n+\mid \gamma \mid }{2}\right) \prod \limits _{j=1}^n\, \Gamma \left( \frac{\gamma _j{+}1}{2}\right) }\, \frac{1}{\mid x\mid ^{\frac{n+\mid \gamma \mid }{2}-1}} \int \limits _0^\infty e^{-a^2r^2t}J_{\frac{n+\mid \gamma \mid }{2}-1}(r\mid x\mid )r^{\frac{n+\mid \gamma \mid }{2}}dr \\{} & {} \quad =\frac{2^{-\mid \gamma \mid }}{a^{n+\mid \gamma \mid } t^{\frac{n+\mid \gamma \mid }{2}}\Gamma \left( \frac{n+\mid \gamma \mid }{2}\right) \prod \limits _{j=1}^n\, \Gamma \left( \frac{\gamma _j{+}1}{2}\right) }\, e^{-\frac{\mid x\mid ^2}{4 a^2 t}}. \end{aligned}$$

$\square$

Proposition 3

The next formula is valid

$$\begin{aligned} \,^\gamma \textbf{T}^y_x e^{-\frac{\mid x\mid ^2}{4 a^2 t}}=e^{-\frac{\mid x\mid ^2+\mid y\mid ^2}{4a^2t}}\textbf{i}_{\gamma }\left( x,\frac{y}{2a^2t}\right) , \end{aligned}$$

(18)

where $\textbf{i}_\gamma$ is defined by (6).

Proof

We have

$$\begin{aligned} \,^\gamma \textbf{T}^y_x e^{-\frac{\mid x\mid ^2}{4 a^2 t}}=\prod \limits _{k=1}^n \,^\gamma {T}^{y_k}_{x_k} e^{-\frac{1}{4 a^2 t}x_k^2}. \end{aligned}$$

Using the formula 3.154 from [13], we obtain

$$\begin{aligned}{} & {} \,^\gamma {T}^{y_k}_{x_k} e^{-\frac{1}{4 a^2 t}x_k^2}= \\{} & {} \quad =\frac{2^{\gamma _k} C(\gamma _k)}{(4x_ky_k)^{\gamma _k-1}}\int \limits _{\mid x_k-y_k\mid }^{x_k+y_k}z e^{-\frac{1}{4 a^2 t}z^2}[(z^2-(x_k-y_k)^2)((x_k+y_k)^2-z^2)]^{\frac{\gamma _k}{2}-1}dz. \end{aligned}$$

Find the integral

$$\begin{aligned} I= & {} \int \limits _{\mid x_k-y_k\mid }^{x_k+y_k}z e^{-\frac{1}{4 a^2 t}z^2}[(z^2-(x_k-y_k)^2)((x_k+y_k)^2-z^2)]^{\frac{\gamma _k}{2}-1}dz =\{z^2=\zeta \}= \\= & {} \frac{1}{2}\int \limits _{(x_k-y_k)^2}^{(x_k+y_k)^2} e^{-\frac{1}{4 a^2 t}\zeta }[(\zeta -(x_k-y_k)^2)((x_k+y_k)^2-\zeta )]^{\frac{\gamma _k}{2}-1}d\zeta \\= & {} \{\zeta -(x_k-y_k)^2=w\}= \\= & {} \frac{1}{2}e^{-\frac{(x_k-y_k)^2}{4 a^2 t}}\int \limits _{0}^{4x_ky_k} e^{-\frac{1}{4 a^2 t}w}[w(4x_ky_k-w)]^{\frac{\gamma _k}{2}-1}dw. \end{aligned}$$

Applying formula 2.3.6.2 from [14] of the form

$$\begin{aligned}{} & {} \int \limits _0^a x^{\alpha -1}(a-x)^{\alpha -1}e^{-px}\textrm{d}x=\sqrt{\pi }\Gamma (\alpha )\left( \frac{a}{p}\right) ^{\alpha -1/2}e^{-ap/2} I_{\alpha -1/2}(ap/2), \nonumber \\{} & {} \text {Re}\,\alpha >0, \end{aligned}$$

(19)

we get

$$\begin{aligned}{} & {} \int \limits _{0}^{4x_ky_k} e^{-\frac{1}{4 a^2 t}w}[w(4x_ky_k-w)]^{\frac{\gamma _k}{2}-1}dw= \\{} & {} \quad =(4a)^{\gamma _k-1}\sqrt{\pi }\Gamma \left( \frac{\gamma _k}{2}\right) \,e^{-\frac{x_ky_k}{2a^2t}} (tx_ky_k)^{\frac{\gamma _k-1}{2}}I_{\frac{\gamma _k-1}{2}}\left( \frac{x_ky_k}{2a^2t}\right) \end{aligned}$$

and

$$\begin{aligned} I=2^{2\gamma _k-3}a^{\gamma _k-1}\sqrt{\pi }\Gamma \left( \frac{\gamma _k}{2}\right) \,e^{-\frac{x_k^2+y_k^2}{4a^2t}} (tx_ky_k)^{\frac{\gamma _k-1}{2}}I_{\frac{\gamma _k-1}{2}}\left( \frac{x_ky_k}{2a^2t}\right) . \end{aligned}$$

Then, substituting the resulting formula into the product $\prod \limits _{k=1}^n \,^\gamma {T}^{y_k}_{x_k} e^{-\frac{1}{4 a^2 t}x_k^2}$ after simplification, we get (18). $\square$

Conclusion

Parabolic equations are the main sources of diffusion problems and the theory of stochastic processes. If diffusion is slowed down or accelerated, or if the process has memory, then instead of the Laplace operator, other operators appear in the heat equation. The main attention in the article was given to a detailed study of the properties of a parabolic equation with a Bessel operator acting on all spatial variables. Maximum and minimum principle for the singular parabolic equation as well as the uniqueness of its solution were given. Using the form of fundamental solution of the singular parabolic equation, mean value theorems were obtained.