Self-similar solutions for a class of non-divergence form equations
This paper deals with self-similar solutions for degenerate and singular parabolic equations in non divergence form of the type \[ \frac{\partial u}{\partial t}= u^{m}\mathrm{div}(|\nabla u|^{p-2}\nabla u),\tag{1} \] where \( m\geq 1 \) and \( p>1 \). It may be verified that if \( u(x,t)\not\equiv 0 \) is a solution of (1), then \[ u_{\lambda}(x,t)= \lambda^{\alpha}u( \lambda^{-\beta}x,\lambda t),\quad \beta\geq 0, \alpha=\frac{1+\beta p}{m+p-2}, \] for \( \lambda > 0\), are solutions of (1). In particular, if \( \lambda=1/t\), then \( u_{\lambda}(x,t)= t^{-\alpha}f( t^{\beta}x )\) is called a self-similar solution of (1). Generally, the study of self-similar solutions is restricted to the forward self-similar solution \[ u(x,t)=(t+\tau)^{-\alpha}f(\eta), \] where \( \eta=(t+\tau)^{\beta}x \) and to the backward self-similar solution \[ u(x,t)=(\tau -t)^{-\alpha}f(\eta),\eta=(\tau -t)^{\beta}x. \] In this paper, a family of forward self-similar solutions of (1), radially symmetric in \(x\), of the form \( u(x,t)=(t+1)^{-\alpha}f(\eta) \) are obtained, where \( \eta =(t+1)^{\beta}|x|^{2}, \beta\geq 0, \alpha=\frac{2+p\beta}{2(m+p-2)}. \) It is verified that \( u(x,t) \) is a self-similar solution of (1) if and only if \( f \) satisfies the equation \[ 2^{p}\eta^{p/2}f^{m}(|f'|^{p-2}f')'+(N+p-2)2^{p-1}\eta^{(p-2)/2}f^{m}|f'|^{p-2}f'-\beta\eta f'+\alpha f=0. \] For \( h=\eta^{p/2}|f'|^{p-2}f', \) (2) is transformed into \[ f'=\eta^{-\frac{p}{2(p-1)}}|h|^{q-2}h, h'=\frac{2-N}{2\eta}h + 2^{-p}\beta \eta^{(p-2)/2(p-1)}f^{-m}|h|^{q-2}h-\frac{\alpha}{2^{p}}f^{1-m}, \] where \( q \) is the dual number of the constant \( p \), that is, \( 1/p + 1/q = 1. \) Thus, the system (3) is considered with the initial value conditions \[ f(0)=A, \quad A>0, h(0)=0. \] In this paper, the solutions with compact support of (3) and (4) are obtained. The authors discuss convergence speeds of the solutions going to zero. Furthermore, they investigate the singularities of the reciprocal of solutions and compare them with the Dirac function in some sense as \(t\rightarrow\infty \). In addition, the nonexistence of self-similar solutions without compact support is also discussed.
scientific article; zbMATH DE number 6189421