On the Cauchy problem for reaction-diffusion equations

Abstract:

The simplest model of the Cauchy problem considered in this paper is the following $(\ast )$ \[ \begin {array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & {x \in {R^n},t > 0,u \geq 0,p > 1,} \hfill \\ {u{|_{t = 0}} = \phi \in {C_B}({R^n}),} \hfill & {\phi \geq 0,\phi \not \equiv 0.} \hfill \\ \end {array} \;\] It is well known that when $1 < p \leq (n + 2)/n$, the local solution of $(\ast )$ blows up in finite time as long as the initial value $\phi$ is nontrivial; and when $p > (n + 2)/n$, if $\phi$ is "small", $(\ast )$ has a global classical solution decaying to zero as $t \to + \infty$, while if $\phi$ is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on $\phi$ for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if $n \geq 3$, $p > n/(n - 2)$, \[ 0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left ( {\frac {{2(n - 2)}} {{{{(p - 1)}^2}}}\left ( {p - \frac {n} {{n - 2}}} \right )} \right )^{1/(p - 1)}}|x{|^{ - 2/(p - 1)}}\] (${u_s}$ is a singular equilibrium of $(\ast )$) where $0 < \lambda < 1$, then $(\ast )$ has a (unique) global classical solution $u$ with $0 \leq u \leq \lambda {u_s}$ and \[ u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.\] (This result implies that ${u_0} \equiv 0$ is stable w.r.t. to a weighted ${L^\infty }$ topology when $n \geq 3$ and $p > n/(n - 2)$.) We also obtain some sufficient conditions on $\phi$ for global nonexistence and those conditions, when combined with our global existence result, indicate that for $\phi$ around ${u_s}$, we are in a delicate situation, and when $p$ is fixed, ${u_0} \equiv 0$ is "increasingly stable" as the dimension $n \uparrow + \infty$. A slightly more general version of $(\ast )$ is also considered and similar results are obtained.