Self-similar solutions for a nonlinear degenerate parabolic equation
This paper concerns the self-similar solution of the doubly nonlinear degenerate parabolic equation in \(\mathbb R^n\) \[ \frac{\partial u}{\partial t} = \text{div}(| x| ^\alpha| \nabla(| u| ^{m-1} u)| ^{N-1}\nabla(| u| ^{m-1}u)). \] The author proves the existence of the nonnegative self-similar solution of the form \[ u(x,t) = u_0 t^{\gamma}y(c| x| t^{-\beta}), \] where \(y \in C^1([0,+\infty))\), \(m >1,\;N >1\), \(\gamma > 0\), \(u_0 > 0\), \(\beta= \frac{\gamma(mN-1)+1}{N-\alpha+1}\), \(c^{N+1}=u_0^{mn-1}\beta,\) and \( \frac {\gamma}{\beta} \geq -n.\)
scientific article; zbMATH DE number 5141822