Classification of radially symmetric self-similar solutions of \(u_t=\Delta \log u\) in higher dimensions. (English) Zbl 1212.35150

Hsu, Shu-Yu

Differ. Integral Equ. 18, No. 10, 1175-1192 (2005).

Summary: We give a complete classification of radially symmetric self-similar solutions of the equation \(u_t=\Delta \log u\), \(u>0\), in higher dimensions. For any \(n\geq 2\), \(\eta >0\), \(\alpha , \beta \in \mathbb R\), we prove that there exists a radially symmetric solution for the corresponding elliptic equation \(\Delta \log v+\alpha v+\beta x\cdot \nabla v=0\), \(v>0\), in \(\mathbb R^n\), \(v(0)=\eta \), if and only if either \(\alpha \geq 0\) or \(\beta >0\). For \(n\geq 3\), we prove that \(\lim _{r\to \infty }r^2v(r)=2(n-2)/(\alpha -2\beta)\) if \(\alpha >\max (2\beta ,0)\) and \(\lim _{r\to \infty }r^2v(r)/\log r=2(n-2)/\beta \) if \(\alpha =2\beta >0\). For \(n\geq 2\) and \(2\beta >\max (\alpha ,0)\), we prove that \(\lim _{r\to \infty }r^{\alpha /\beta }v(r)=A\) for some constant \(A>0\).

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B40	Asymptotic behavior of solutions to PDEs
34B40	Boundary value problems on infinite intervals for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations

Keywords:

singular diffusion equation