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October, 2002 Existence and nonexistence of global solutions of quasilinear parabolic equations
Ryuichi SUZUKI
J. Math. Soc. Japan 54(4): 747-792 (October, 2002). DOI: 10.2969/jmsj/1191591992


We consider nonnegative solutions to the Cauchy problem for the quasilinear parabolic equations ut=Δum+K(x)up where xRN, 1m<p and K(x)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 ppm,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|-' as |x| in some cone D and u0(x)=0 in the complement of D.


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Ryuichi SUZUKI. "Existence and nonexistence of global solutions of quasilinear parabolic equations." J. Math. Soc. Japan 54 (4) 747 - 792, October, 2002.


Published: October, 2002
First available in Project Euclid: 5 October 2007

MathSciNet: MR1921087
zbMATH: 1036.35085
Digital Object Identifier: 10.2969/jmsj/1191591992

Primary: 35B33
Secondary: 35B05 , 35B40 , 35K15 , 35K55 , 35K65

Keywords: Cauchy problem , Critical exponent , global existence , nonexistence , quasilinear parabolic equation , second critical exponent

Rights: Copyright © 2002 Mathematical Society of Japan


Vol.54 • No. 4 • October, 2002
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