Boundedness and evolution rates for a quasilinear reaction-diffusion equation

We consider the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and in the range of exponents $1<p<m$ and $-\infty<\sigma<-2$. We prove that, for initial conditions $u_0$ satisfying $$ u_0\geq0, \quad u_0\not\equiv0, \quad \lim\limits_{|x|\to\infty}|x|^{-(\sigma+2)/(m-p)}u_0(x)=0, $$ the solution $u$ to the corresponding Cauchy problem remains uniformly bounded from above and below: $$ C_1\leq \|u(t)\|_{\infty}\leq C_2, \quad t\in(0,\infty), $$ for some positive constants $C_1$ and $C_2$. Under suitable conditions on $p$, we also establish the rate of expansion of the upper limit $R(t)$ of the positivity set for compactly supported data, that is, $$ At^{\beta}\leq R(t)\leq Bt^{\beta}, \quad \beta=-\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ and a \emph{different time scale in outer sets}, that is $$ D_1t^{-\alpha}\leq u(x,t)\leq D_2t^{-\alpha}, \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad {\rm if} \ |x|\geq Ct^{\beta}. $$ The boundedness is in striking contrast with the property of grow-up as $t\to\infty$ established in previous works by the authors for $\sigma>-2$, illustrating the character of threshold of the exponent $\sigma=-2$.