Peaking self-similar solutions and continuation after blow-up for a quasilinear Hardy-Hénon equation

Abstract

We consider the quasilinear Hardy-Hénon equation $$u_t=\Delta u^m+|x{|}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,T),$$ in the range of exponents and dimension $$N\ge 3,\sigma \in (-2,\infty ),m>1,p>p_S(\sigma )=\frac{m(N+2\sigma +2)}{N-2}.$$ We establish sharp ranges of exponents $(m,p,\sigma )$ guaranteeing existence of \emph{peaking solutions} in backward self-similar form $$u(x,t)=(T-t{)}^{-\alpha }f(|x|(T-t{)}^{-\beta }).$$ By peaking solutions we understand solutions blowing up at a finite time $T\in (0,\infty )$ but allowing for a continuation after the blow-up time. As a remarkable outcome, a number of ranges of exponents $\sigma$ and $m$ for which peaking solutions exist for any $p>p_S(\sigma )$ are deduced, contrasting to the previously considered exponent $\sigma =0$, where a Lepin-type critical exponent $p_L$ provides an upper bound for $p$.