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On the Cauchy problem for reaction-diffusion equations


Author: Xuefeng Wang
Journal: Trans. Amer. Math. Soc. 337 (1993), 549-590
MSC: Primary 35K57; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9947-1993-1153016-5
MathSciNet review: 1153016
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Abstract: The simplest model of the Cauchy problem considered in this paper is the following $ (\ast)$

\begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & ... ... & {\phi \geq 0,\phi \,\not\equiv\,0.} \hfill \\ \end{array} \;\end{displaymath}

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)$,

$\displaystyle 0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left( {\frac{{2... ... \frac{n} {{n - 2}}} \right)} \right)^{1/(p - 1)}}\vert x{\vert^{ - 2/(p - 1)}}$

($ {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

$\displaystyle u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.$

(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.

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DOI: https://doi.org/10.1090/S0002-9947-1993-1153016-5
Article copyright: © Copyright 1993 American Mathematical Society

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