Rate of convergence to a singular steady state of a supercritical parabolic equation
The authors study global positive solutions to the Cauchy problem for a supercritical parabolic equation of the type \[ \begin{cases} u_t=\Delta u+ u^p & x\in \mathbb R^N,\;t>0,\\ u(x,0)=u_0(x) & x\in \mathbb R^N, \end{cases} \] which converge to a steady state that is singular at \(x=0\). Here \(u_0\) is a nonnegative continuous function in \(\mathbb R^N\) and the exponent \(p\) is greater than the critical value \[ {{(N-2)^2-4N+8\sqrt{N-1}}\over {(N-2)(N-10)}},\quad N>10. \] The rate of convergence to the singular steady state is determined in \(L^\infty({\mathbb{R}}^{N} \setminus B_{\nu}(0))\) where \(B_\nu(0)\) is an \(N\)-dimensional ball centered at the origin and of radius \(\nu\).
scientific article; zbMATH DE number 5562657