Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p (1,2) and q>1, self-similar singular solutions of the equation vt=div( vp−2 v)−vq in Rn×(0, ∞); here by self-similar we mean that v takes the form v(x,t)=t−αw(xt−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and limt 0 v(x,t)=0 for all x≠0. That is, we consider the ODE problem We show that this ODE problem has a non-trivial solution if and only if q