Singular heat and wave equations on the Euclidien space $\R^n$

TL;DR

This paper derives explicit solutions for singular generalized heat and wave equations on Euclidean space, providing valuable formulas for understanding these complex differential equations.

Contribution

It presents the first explicit formulas for solutions to singular generalized heat and wave equations on Euclidean space.

Findings

Explicit formulas for solutions derived

Enhanced understanding of singular PDEs achieved

Potential applications in mathematical physics explored

Abstract

In this paper we give the explicit formulas for the solution of the singular generalized heat and wave equations on the Euclidian space $\R^n$.

Equations

$$\left\{\begin{array}[]{cc}\left(\frac{\partial}{\partial t}+\frac{k}{t}\right)u(t,X)=\Delta u(t,X);(t,X)\in I\!\!R^{\ast}_{+}\times R^{n}\\ u(0,X)=f(X);f\in C^{\infty}(I\!\!R)\end{array}\right.$$

$$\left\{\begin{array}[]{cc}\left(\frac{\partial}{\partial t}+\frac{k}{t}\right)\left(\frac{\partial}{\partial t}-\frac{k}{t}\right)w(t,X)=\Delta w(t,X);(t,X)\in I\!\!R^{\ast}_{+}\times I\!\!R\\ w(0,X)=0,w_{t}(0,X)=g(X),g\in C^{\infty}(I\!\!R)\end{array}\right.$$

$$\Delta =\frac{{\partial }^2}{\partial X_1^2}+\frac{{\partial }^2}{\partial X_2^2}+...+\frac{{\partial }^2}{\partial X_n^2}$$

$$B{t}^{-n/2}\exp \left(-\frac{|X{|}^2}{4t}\right)U\left(k,\frac{n}{2},\frac{|X{|}^2}{4t}\right)$$

$${}_1{F}_1(a;c;z)=\sum _{k=0}\frac{(a{)}_k}{(c{)}_{k}k!}{z}^{k}c\ne 0,-1,-2,.....$$

$$U(a,c,z)=\frac{\pi }{\sin \pi c}\left[\frac{{}_1{F}_1(a;c;z)}{\Gamma (c)\Gamma (1+a-c)}-z^{1-c}\frac{{}_1{F}_1(a+1-c;2-c;z)}{\Gamma (a)\Gamma (2-c)}\right]$$

$$(a{)}_n=\frac{\Gamma (a+n)}{\Gamma (a)}$$

$$\left[4y({\partial }^2/\partial y^2)+2n(\partial /\partial y)\right]\Psi (t,y)=\left[(\partial /\partial t)+(k/t)\right]\Psi (t,y)$$

$$\Psi (t,y)=t^{-n/2}\Phi (t,y);z=-y/4t$$

$$\left[z(d^2/d{z}^2)+((n/2)-z)(d/dz)\right]\Phi (z)-((n/2)-k)\Phi (z)=0$$

$$u(t,X)={\int }_{R^n}{H}_n^k(t,X,Y)f(Y)dm(Y)$$

$$H_n^k(t,X,Y)=\Gamma (k)(4\pi t{)}^{-n/2}\exp \left(-\frac{|X-Y{|}^2}{4t}\right)U\left(k,\frac{n}{2},\frac{|X-Y{|}^2}{4t}\right)$$

$$U(a,c,z)=z^{-a}+O(|z{|}^{-a-1})$$

$$U(a,c,z)=(1/\Gamma (a))\left[\log z+\psi (a)-2\gamma \right]+O(|z\log z|),c=1$$

$$U(a,c,z)=(\Gamma (c-1)/\Gamma (a))z^{1-c}+O(1),1<\Re c<2$$

$$U(a,c,z)=(\Gamma (c-1)/\Gamma (a))z^{1-c}+O(|\log z|),c=2$$

$$U(a,c,z)=(\Gamma (c-1)/\Gamma (a))z^{1-c}+O(|z{|}^{\Re c-2}),\Re c\ge 2,c\ne 2$$

$$u(t,X)=(\Gamma (k)/2{\pi }^{n/2}){\int }_0^{\infty }\exp (-z)U(k,n/2,z)z^{(n/2)-1}{f}_X^{\#}(\sqrt{4tz})dz$$

$$f_X^{\#}(r)={\int }_{S^{n-1}}f(X+r\omega )d\omega$$

$${\lim }_{t⟶0}u(t,X)=c_n{f}_X^{\#}(0){\int }_0^{\infty }\exp (-z)U(k,n/2,z)z^{(n/2)-1}dz$$

$${\int }_0^{\infty }\exp (-z)U(a,c,z)z^{c-1}dz=-\exp (-z)z^{c}U(a,c+1,z)$$

$${\lim }_{t\to 0}u(t,X)={\left[-\frac{\Gamma (k)}{{\pi }^{n/2}}{f}_X^{\#}(0)\exp (z)U(k,n/2,z)z^{(n/2)}\right]}_0^{\infty }$$

$${\lim }_{t\to 0}w(t,X)=\frac{\Gamma (k)}{{\pi }^{n/2}}\frac{2{\pi }^{n/2}}{\Gamma (n/2)}\frac{\Gamma (n/2)}{\Gamma (k)}{f}_X^{\#}(0)=f(X)$$

$$w(t,X,Y)=A{t}^{1-n}{}_2{F}_1\left(\frac{n-k}{2},\frac{n-1+k}{2},\frac{n+1}{2},1-\frac{|X-Y{|}^2}{t^2}\right)+$$

$$B{\left(t^2-|X-Y{|}^2\right)}^{(1-n)/2}{}_2{F}_1\left(\frac{1-k}{2},\frac{k}{2},\frac{3-n}{2},1-\frac{|X-Y{|}^2}{t^2}\right)$$

$$F(a,b,c;z)=\sum _{n=0}^{\infty }\frac{(a{)}_n(b{)}_n}{(c{)}_{n}n!}{z}^n,|z|<1,$$

$$\left[4x({\partial }^2/\partial x^2)+2(\partial /\partial x)+(k(1-k)/x)\right]\Psi (x,y)$$

$$\Psi (x,y)=x^{-(n-1)/2}\Phi (x,y);z=y/x$$

$$z(1-z)\frac{d^2}{d{z}^2}\Phi (z)+[n/2-(n+1/2)z]\frac{d}{dz}\Phi (z)-(n-k)(n-1+k)/4\Phi (z)=0$$

$${\Phi }_1(z)=F((n-k)/2,(n-1+k)/2;(n+1)/2,1-z)$$

Full text

Math. Subj. Classification 2010 : 35JO5, 35JO8, 35K08.

1 Introduction

In this paper we discuss the explicit formulas for the solution of the following singular generalized heat and wave equations on the Euclidian space $R^{n}$:

[TABLE]

[TABLE]

where

[TABLE]

is the usual n-dimensional Euclidian Laplacian on $R^{n}$ and $k$ is a real number.

The mathematical interest in these equations, however, comes mainly from the fact that the time inverse potential $\frac{k}{t}$ (resp. the time inverse square $\frac{k(1-k)}{t^{2}}$) is homogeneous of degree -1 (resp. -2) and therefore scales exactly the same as $\partial/\partial t$ (resp. $\partial^{2}/\partial t^{2}$).

An inconvenient of the time dependent potential is the absence of the relation between the semi-groups of the Schrödinger equation and the spectral properties of the operator. The space inverse potential $k/x$ is called Coulomb potential and is widely studied in physical and mathematical literature$[1]$.

The space inverse square potential $k(1-k)/x^{2}$ arises in several contexts, one of them is the Schrödinger equation in non relativistic quantum mechanics (Reed and Simon $[7]$) . For example, the Hamiltonian for a spinzero particle in Coulomb field gives rise to a Schrödinger operator involving the space inverse square potential (Case $[2]$). The Cauchy problem for the wave equation with the space inverse square potential in Euclidean space $I\!\!R^{n}$ is extensively studied (Cheeger and Taylor $[3]$), (Planchon et al$[6]$). The cases considered frequentely are $k=0$, the equations in $(1.1)$ and $(1.2)$ then turn into the classical heat and wave equation on the Euclidean spaces $I\!\!R^{n}$ and these equations appear in several branches of mathematics and physics (Folland $[4],p.143,171$). Our main objective of this paper is to solve the Cauchy problems $(1.1)$ and $(1.2).$

2 Singular heat equation

Theorem 2.1 The generalized singular heat equation in $(1.1)$ has the following general solution

$\varphi(t,X)=At^{-n/2}{}_{1}F_{1}\left(\frac{n}{2}-k,\frac{n}{2},\frac{|X|^{2}}{4t}\right)+$

[TABLE]

with $A$ and $B$ are complex constants and ${}_{1}F_{1}\left(a,c,z\right)$ and $U(a,c,z)$ are the confluent hypergeometric functions of the first and the second kind given respectively by ([5],p.263):

[TABLE]

[TABLE]

where as usual $(a)_{n}$ is the Pochhamer symbol defined by

[TABLE]

and $\Gamma$ is the classical Euler function.

Proof Using the geodesic polar coordinates centred at $X$, $Y=X+r\omega,r>0;\omega\in S^{n-1}$ with $S^{n-1}$ is the sphere of dimension $n-1$ , and setting $y=r^{2}$ in the generalized singular heat equation in $(1.1)$ we obtain

[TABLE]

by the change of function and the change of the variable below.

[TABLE]

the equation $(2.5)$ is transformed into the following confluent hypergeometric equation

[TABLE]

with parameters $a=n/2-k;c=n/2$,($[5]$ p.$268$). An appropriate independent solutions of this equation are: ($[5]$ p.$270$) ${}_{1}F_{1}\left(a,c,z\right)$ and $\exp(z)U(c-a,c,-z)$. From the formulas $(2.5),(2.6)$ and $(2.7)$ we conlude that the function $\varphi$ in $(2.1)$ is the general solution of the generalized singular heat equation in $(1.1)$.

Theorem 2.2 For $n\geq 2$ and $k\neq 0,-1,-2,...$, the Cauchy problem for the singular generalized heat equation $(1.1)$ has the unique solution given by

[TABLE]

where

[TABLE]

and $U(a,c,z)$ is the confluent hypergeometric function of the second kind given in $(2.3)$.

**Proof ** In view of the proposition 2.1, to finish the proof of the theorem it remains to show the limit condition in $(1.1)$, for this we recall the asymptotic behavior of the degenerate confluent hypergeometric function $U(a,c,z)$, ($[5]$ p.$288-289$),

for $z\rightarrow+\infty$

[TABLE]

and for $z\longrightarrow 0$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using the geodesic polar coordinates centred at $X$, and by setting $y=r^{2};z=y/4t$ in $(1.1)$ we get

[TABLE]

with

[TABLE]

Taking the limit in $(2.15)$ in view of the formulas $(2.10)-(2.14)$ we can reverse the limit and the integral and we obtain

[TABLE]

and by the formula ($[1]$ p.$266$):

[TABLE]

[TABLE]

using again the formulas $(2.10)$ and $(2.14)$ we have

[TABLE]

The unequeness is clear from the properties of the confluent hypergeometric equation ($[5]$ p.268-270).

3 The generalized singular wave equation on $I\!\!R^{n}$

Theorem 3.1 The generalized singular wave equation in $(1.2)$ has the following general solution

[TABLE]

[TABLE]

with $A$ and $B$ are complex constants and ${}_{2}F_{1}\left(a,b,c,z\right)$ is the Gauss hypergeometric function given by

[TABLE]

Proof Using the geodesic polar coordinates centred at $X$, $Y=X+r\omega,r>0;\omega\in S^{n-1}$, and setting $y=r^{2}$ and $x=t^{2}$ in the generalized singular wave equation in $(1.2)$ we obtain

$\left[4y(\partial^{2}/\partial y^{2})+2n(\partial/\partial y)\right]\Psi(x,y)=$

[TABLE]

setting

[TABLE]

we obtain the following Gauss hypergeometric equation

[TABLE]

with parameters: $a=(n-k)/2;b=(n-1+k)/2;c=n/2$.

The hypergeometric equation $(3.5)$ has the following system of solutions $([5],p.42-43)$

[TABLE]

and

[TABLE]

hence the following functions satisfy the generalized singular wave equation in $(1.2)$

$\varphi_{1}^{k}(t,X,Y)=t^{1-n}\times$

[TABLE]

$\varphi_{2}^{k}(t,X,Y)=\left(t^{2}-|X-Y|^{2}\right)^{(1-n)/2}\times$

[TABLE]

and the proof of the theorem 3.1 is finished.

In the remainder of this section we present several lemmas.

Lemma 3.2 For $X,Y\in I\!\!R^{n}$ and $t\in I\!\!R^{+}$ set

$W_{2}(t,X,Y)=$

[TABLE]

and

[TABLE]

for $n$ even $n\geq 4$

$W_{n}(t,X,Y)=$

[TABLE]

and

[TABLE]

then for

[TABLE]

the following formulas hold

i)

[TABLE]

ii)

[TABLE]

iii)For $g\in C_{0}^{\infty}(I\!\!R^{n})$ we have

$A_{x}^{\frac{n-3}{2}}A_{x}^{\frac{n-5}{2}}...A_{x}^{\frac{1}{2}}\left[x^{1/2}g(\sqrt{xz})\right]=$

[TABLE]

with $b_{i}$ ; $i=1,2,...,(n-2)/2$ are real constants.

**Proof **: To show $i)$ we use the formula $([5]$ p.$41)$

[TABLE]

$ii)$ comes from $i)$.

iii) we can demonstrate iii) by induction over even $n\geq 4$.

Lemma 3.3 For $z\longrightarrow 0$ we have:

i) $F((1+k)/2,1-k/2;(n+1)/2,1-z)=$

[TABLE]

ii) $F((1-k)/2,k/2;1/2,1-z)=$

[TABLE]

$k\neq 0,-2,-4,....$

Proof i) is easily seen from the formula $([5],p.47)$

$F(a,b,c,z)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-b)\Gamma(c-a)}F(a,b,a+b-c+1,1-z)+$

[TABLE]

ii) is a consequence of the formula $([5]$ p.$44)$

$F(a,b;a+b,z)=\left(\Gamma(a+b)/\Gamma(a)\Gamma(b)\right)\sum^{\infty}_{n=0}\times$

[TABLE]

$\arg(1-z)<\pi$;$|1-z|<1$

4 Cauchy problem for the singular wave equation on $I\!\!R^{n}$, $n$ odd

Theorem 4.1 Suppose $n$ is odd and

[TABLE]

with

$C_{n}=\frac{\Gamma(n/2)k(k-1)}{2\pi^{n/2}(n-1)}\times$

[TABLE]

If $g\in C_{0}^{\infty}(I\!\!R^{n})$, the function

[TABLE]

solves the Cauchy problem for the generalized singular wave equation$(1.2)$

Proof In view of the theorem 3.1, we see that the kernel in $(4.1)$ satisfies the generalized singular wave equation in $(1.2)$ and hence the function $w(t,X)$ in $(4.3)$ satisfies the same equation. To complete the proof of the theorem 4.1 it remains to show the limit conditions. Using the geodesic polar coordinates and setting $y=r^{2};x=t^{2},z=y/x$ in $(4.3)$ we have

$w(t,X)=C_{n}\int^{1}_{0}\frac{1}{2}g_{X}^{\#}(t\sqrt{z})z^{(n-2)/2}\times$

[TABLE]

with $g_{X}^{\#}(r)$ is as in $(2.16)$. By the formula $([5]p.47)$.

[TABLE]

we can write

[TABLE]

hence by taking the limit in $(4.6)$ using the formula $(3.19)$ we can reverse the integral and the limit to obtain

[TABLE]

For the second condition we derive the expression in $(4.6)$, using again $(3.19)$ we can derive under the integral sign to obtain

$\frac{\partial}{\partial t}w(t,X)=$

[TABLE]

Hence by taking the limit of $(4.8)$ in view of $(3.19)$ we can reverse the limit and the integral to write

[TABLE]

[TABLE]

In view of the formula ($[5]$ p.$41$)

[TABLE]

we obtain

$\mathop{\rm lim}_{t\rightarrow 0}\frac{\partial}{\partial t}w(t,X)=-C_{n}(1/2)g_{X}^{\#}(0)\frac{n-1}{k(k-1)}\times$

[TABLE]

that is

$\mathop{\rm lim}_{t\rightarrow 0}\frac{\partial}{\partial t}w(t,X)=-C_{n}(1/2)g_{X}^{\#}(0)\frac{n-1}{k(k-1)}\times$

[TABLE]

And by the formula $(4.2)$ and ($[5]$ p.$40$):

[TABLE]

$c\neq 0,-1,-2,-3,...$ .

we obtain $\mathop{\rm lim}_{t\rightarrow 0}\frac{\partial}{\partial t}w(t,X)=g(X)$.

5 Cauchy problem for the singular wave equation on the Euclidien plane $I\!\!R^{2}$

Theorem 5.1 Suppose $n=2$ and

[TABLE]

with

[TABLE]

If $g\in C^{\infty}_{0}(R^{2})$, the function

[TABLE]

solves the Cauchy problem $(1.2)$.

Proof From the theorem 3.1 we see that the functions $w(t,X)$ in $(5.3)$ satisfies the generalized singular wave equation in $(1.2)$ .

Now to show the limit conditions, by the geodesic polar coordinates and the change of variables $y=r^{2};x=t^{2},z=y/x$ in $(5.3)$, we have for for $n=2$

[TABLE]

By taking the limit in $(5.4)$ we can use the formula $(3.20)$ to reverse the limit and the integral and to obtain

[TABLE]

Now to show the second condition we derive the expression $(5.4)$ and in view of the formula $(3.20)$ we can derive under the integral sign to obtain

$\frac{\partial}{\partial t}w(t,X)=c_{2}\frac{1}{2}\int^{1}_{0}(1-z)^{-1/2}\times$

[TABLE]

Using again the formula $(3.20)$ we can reverse the limit and the integral and we have

[TABLE]

[TABLE]

we have by the formula $(3.18)$

[TABLE]

and from $(4.14)$

[TABLE]

6 Cauchy problem for the singular wave equation on $I\!\!R^{n}$, $n\geq 4$ even

Theorem 6.1 Suppose $n$ is even and $n\geq 4$, let $W_{2}^{k}(t,X,Y)$ is as in theorem 5.1 and $A^{a}_{x}$ is as in $(3.11)$ and

[TABLE]

If $g\in C^{\infty}_{0}(R^{n})$, the function

[TABLE]

solves the Cauchy problem $(1.2)$.

Proof In view of the theorem 3.1, we see that the functions $w(t,X)$ in $(6.2)$ satisfies the generalized singular wave equation in $(1.2)$ .

To finish the proof of the theorem we show the limit condition in the even case $n\geq 4$: using the geodesic polar coordinates and setting $y=r^{2};x=t^{2};z=y/x$ in $(6.2)$; we have:

for $n$ even $n\geq 4$:

[TABLE]

with

[TABLE]

Using the formula iii) of lemma 3.2 we have

$w(t,X)=C_{n}\frac{t}{2}\int^{1}_{0}(1-z)^{-1/2}F((1-k)/2,k/2;1/2,1-z)g_{X}^{\#}(t\sqrt{z})dz+$

[TABLE]

with

[TABLE]

Taking the limit of the expression $(6.3)$ and using the formula $(3.20)$ we can reverse the integral and the limit to obtain

[TABLE]

For the second condition we derive the expression $(6.4)$ and by $(3.17)$ we can derive under the integral sign

$\frac{\partial}{\partial t}w(t,X)=c_{n}c_{2}2^{-(n-2)/2}\frac{(n-2)/2!}{(n-3)!!}\frac{1}{2}\int^{1}_{0}(1-z)^{-1/2}\times$

[TABLE]

Taking now the limit in $(6.6)$ and using again $(3.20)$ we can reverse the limit and the integral sign

$\mathop{\rm lim}_{t\longrightarrow 0}\frac{\partial}{\partial t}w(t,X)==c_{n}c_{2}2^{-(n-2)/2}\frac{(n-2)/2!}{(n-3)!!}\frac{1}{2}g_{X}^{\#}(0)\times$

[TABLE]

$\mathop{\rm lim}_{t\longrightarrow 0}\frac{\partial}{\partial t}w(t,X)=$

[TABLE]

by the formula $(3.18)$ we have

$\mathop{\rm lim}_{t\longrightarrow 0}\frac{\partial}{\partial t}w(t,X)==c_{n}c_{2}2^{-(n-2)/2}\frac{(n-2)/2!}{(n-3)!!}\frac{1}{2}\times$

[TABLE]

using the formula $(4.14)$ we have

$\mathop{\rm lim}_{t\rightarrow 0}\frac{\partial}{\partial t}w(t,X)=c_{n}c_{2}2^{-(n-2)/2}\frac{(n-2)/2!}{(n-3)!!}\times$

[TABLE]

7 Applications

Remark 7.1: we have

[TABLE]

where

[TABLE]

is the classical heat kernel on $R^{n}$.

Corollary 7.2 The generalized Cauchy problem for the heat equation on $I\!\!R^{n}$:

[TABLE]

has the unique solution given by

[TABLE]

where

[TABLE]

Proof The proof of this corollary is simple and is omitted.

Corollary 7.3 We have

[TABLE]

with

[TABLE]

is the classical wave kernel on $I\!\!R^{n}$ $[4]$

Proof The proof of this corollary is simple and is left to the reader.