The Liouville Theorem of a Torsion System and its Application to the Symmetry Group of a Porous Medium Type Equation on Symmetric Spaces

X.-P. Chen; S.-Z. Du; T.-P. Guo

doi:10.5802/jolt.1177

X.-P. Chen; S.-Z. Du; T.-P. Guo

Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 393-411

Abstract

We first prove a Liouville theorem to the torsion system $$ \begin{cases} \displaystyle \xi^i_i=\lambda(x)\pm\frac{2x^k\xi^k}{|x|^2+1}, \forall i=1,2,\cdots,n\\ \xi^i_j+\xi^j_i=0, \forall i\not=j \end{cases} $$ for $(\xi,\lambda)\in C^\infty({\mathbb{R}}^n,{\mathbb{R}}^n\times{\mathbb{R}})$. As an application, complete resolutions of symmetry groups to the porous medium equation $$ u_t-\triangle_g(u^m)=u^p, \ \ \forall(x,t)\in M\times{\mathbb{R}} $$ of Fujita type are obtained, where $M$ is the sphere ${\mathbb{S}}^n\subset{\mathbb{R}}^{n+1}$ or hyperbolic space ${\mathbb{H}}^n$ with canonical metric $g$.

arXiv Zbl

DOI: 10.5802/jolt.1177

Classification: 53C35, 35K59, 35K65
Keywords: Porous medium equation, prolongation formula

@article{JOLT_2021_31_2_a5,
     author = {X.-P. Chen and S.-Z. Du and T.-P. Guo},
     title = {The {Liouville} {Theorem} of a {Torsion} {System} and its {Application} to the {Symmetry} {Group} of a {Porous} {Medium} {Type} {Equation} on {Symmetric} {Spaces}},
     journal = {Journal of Lie Theory},
     pages = {393--411},
     year = {2021},
     volume = {31},
     number = {2},
     doi = {10.5802/jolt.1177},
     zbl = {1480.53067},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1177/}
}

TY  - JOUR
AU  - X.-P. Chen
AU  - S.-Z. Du
AU  - T.-P. Guo
TI  - The Liouville Theorem of a Torsion System and its Application to the Symmetry Group of a Porous Medium Type Equation on Symmetric Spaces
JO  - Journal of Lie Theory
PY  - 2021
SP  - 393
EP  - 411
VL  - 31
IS  - 2
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1177/
DO  - 10.5802/jolt.1177
ID  - JOLT_2021_31_2_a5
ER  -

%0 Journal Article
%A X.-P. Chen
%A S.-Z. Du
%A T.-P. Guo
%T The Liouville Theorem of a Torsion System and its Application to the Symmetry Group of a Porous Medium Type Equation on Symmetric Spaces
%J Journal of Lie Theory
%D 2021
%P 393-411
%V 31
%N 2
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1177/
%R 10.5802/jolt.1177
%F JOLT_2021_31_2_a5

X.-P. Chen; S.-Z. Du; T.-P. Guo. The Liouville Theorem of a Torsion System and its Application to the Symmetry Group of a Porous Medium Type Equation on Symmetric Spaces. Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 393-411. doi: 10.5802/jolt.1177

Cited by Sources: