On nonexistence of similarity solutions
Consider the semilinear parabolic equation \[ (1)\quad u_ t=\Delta u- u^ p,\quad x\in R^ N,\quad t>0, \] where \(N\geq 1\), \(0<p<1\). Let u(x,t) be a nonnegative solution of (1) for which the initial value u(x,0) is uniformly bounded in \(R^ N\) by a constant M. Let \(T=\sup \{t>0:\quad u(x,t)=0\}\leq \{M/(1-p)\}^{1-p}.\) Then (1) may be changed into the form \[ (2)\quad \Delta v-\eta \cdot \nabla v+(v/(1-p))-v^ p=0. \] This paper proves that (2) has no radially symmetric solutions in \(R^ N\) which are nonnegative and bounded, except \(v(\eta)=0\), \(v(\eta)=k=(1-p)^{1/(1-p)}.\)
scientific article; zbMATH DE number 4101714
scientific article