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

Xiao-Peng Chen¹ ; Shi-Zhong Du¹ ; Tian-Pei Guo¹

¹Dept. of Mathematics, Shantou University, Shantou 515063, P. R. China

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

Résumé

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 $g$.

Détail

Classification : 53C35, 35K59, 35K65
Mots-clés : Porous medium equation, prolongation formula