Boundedness and evolution rates for a quasilinear reaction-diffusion equation

Abstract

We consider the following quasilinear reaction-diffusion equation $${\partial }_{t}u=\hat{I}”u^m+(1+|x|{)}^{\ddot{I}}ƒu^p,(x,t)\in {\mathbb{R}}^N\times (0,\infty ),$$ in dimension $N\ge 3$ and in the range of exponents $1<p<m$ and $-\infty <\ddot{I}ƒ<-2$. We prove that, for initial conditions $u_0$ satisfying $$u_0\ge 0,u_0\not\equiv 0,\underset{|x|\to \infty }{\lim }|x{|}^{-(\ddot{I}ƒ+2)/(m-p)}{u}_0(x)=0,$$ the solution $u$ to the corresponding Cauchy problem remains uniformly bounded from above and below: $$C_1\le \Vert u(t){\Vert }_{\infty }\le C_2,t\in (0,\infty ),$$ 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, $$A{t}^2\le R(t)\le B{t}^2,{\hat{I}}^2=-\frac{m-p}{\ddot{I}ƒ(m-1)+2(p-1)},$$ and a \emph{different time scale in outer sets}, that is $$D_1{t}^{-\hat{I}\pm }\le u(x,t)\le D_2{t}^{-\hat{I}\pm },\hat{I}\pm =\frac{\ddot{I}ƒ+2}{\ddot{I}ƒ(m-1)+2(p-1)},\mathrm{if}|x|\ge C{t}^2.$$ The boundedness is in striking contrast with the property of grow-up as $t\to \infty$ established in previous works by the authors for $\ddot{I}ƒ>-2$, illustrating the character of threshold of the exponent $\ddot{I}ƒ=-2$.