2002 年 54 巻 4 号 p. 747-792
We consider nonnegative solutions to the Cauchy problem for the quasi-linear parabolic equations ut=Δ um+K(x)up where x∈ \bm{R}N, 1≤ m<p and K(x)≥q 0 has the following properties: K(x)∼|x|σ (-∞≤σ<∞) as |x|→∞ in some cone D and K(x)=0 in the complement of D, where for σ=-∞ we define that K(x) has a compact support. We find a critical exponent pm, σ*=pm, σ*(N) such that if p≤ pm, σ*, then every nontrivial nonnegative solution is not global in time, whereas if p>pm, σ* then there exits a global solution. We also find a second critical exponent, which is another critical exponent on the growth order α of the initial data u0(x) such that u0(x)∼|x|-α as |x|→∞ in some cone D^{'} and u0(x)=0 in the complement of D^{'}.
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