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Existence and nonexistence of global solutions for $u\sb t = \Delta u + a(x)u\sp p$ in $\bbfR\sp d$. (English) Zbl 0876.35048
The paper deals with the nonnegative solutions to the Cauchy problem $$ \leqno{(*)}\qquad u_{t} \Delta u+a(x)u^{p},\quad x\in {\bbfR}^{d},\ t>0,\ p>1,\quad u(x,0)=u_{0}(x)\geq 0,\quad u\not\equiv 0,$$ where $a(x)\in C^{\alpha}({\bbfR}^{d}),$ $a(x)\geq 0,$ $a(x)\not\equiv 0,$ and $a(x)$ is of order $|x|^{m}$ for $m\in (-2,\infty),$ or $a(x)\leq C|x|^{-2}.$ Extending the classical result of {\it H. Fujita} [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.34002)] and more recent results of {\it C. Bandle} and {\it H. A. Levine} [Trans. Am. Math. Soc. 316, No. 2, 595-622 (1989; Zbl 0693.35081)] and {\it H. A. Levine} and {\it P. Meier} [Arch. Ration. Mech. Anal. 109, No. 1, 73-80 (1990; Zbl 0702.35131)], the author finds a critical exponent $p^{*}=p^{*}(m,d)$ such that if $1<p\leq p^{*}$ then there exist no solutions to $(*)$ that are global in time, while if $p>p^{*}$ then there exist both global and nonglobal solutions.

35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35K15Second order parabolic equations, initial value problems
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