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Existence and nonexistence of global solutions for ${u}_{t}={\Delta }u+a\left(x\right){u}^{p}$ in ${ℝ}^{d}$. (English) Zbl 0876.35048

The paper deals with the nonnegative solutions to the Cauchy problem

$\phantom{\rule{2.em}{0ex}}\left(*\right)\phantom{\rule{2.em}{0ex}}{\mathrm{u}}_{\mathrm{t}}{\Delta }\mathrm{u}+\mathrm{a}\left(\mathrm{x}\right){\mathrm{u}}^{\mathrm{p}},\phantom{\rule{1.em}{0ex}}\mathrm{x}\in {ℝ}^{\mathrm{d}},\phantom{\rule{4pt}{0ex}}\mathrm{t}>0,\phantom{\rule{4pt}{0ex}}\mathrm{p}>1,\phantom{\rule{1.em}{0ex}}\mathrm{u}\left(\mathrm{x},0\right)={\mathrm{u}}_{0}\left(\mathrm{x}\right)\ge 0,\phantom{\rule{1.em}{0ex}}\mathrm{u}¬\equiv 0,$

where $a\left(x\right)\in {C}^{\alpha }\left({ℝ}^{d}\right),$ $a\left(x\right)\ge 0,$ $a\left(x\right)¬\equiv 0,$ and $a\left(x\right)$ is of order ${|x|}^{m}$ for $m\in \left(-2,\infty \right),$ or $a\left(x\right)\le {C|x|}^{-2}·$ Extending the classical result of H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.34002)] and more recent results of C. Bandle and H. A. Levine [Trans. Am. Math. Soc. 316, No. 2, 595-622 (1989; Zbl 0693.35081)] and H. A. Levine and P. Meier [Arch. Ration. Mech. Anal. 109, No. 1, 73-80 (1990; Zbl 0702.35131)], the author finds a critical exponent ${p}^{*}={p}^{*}\left(m,d\right)$ such that if $1 then there exist no solutions to $\left(*\right)$ that are global in time, while if $p>{p}^{*}$ then there exist both global and nonglobal solutions.

MSC:
 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions of PDE 35K15 Second order parabolic equations, initial value problems
Keywords:
nonnegative solutions