Existence and nonexistence of global solutions of quasilinear parabolic equations

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概要

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• We consider nonnegative solutions to the Cauchy problem for the quasi-linear parabolic equations u<SUB>t</SUB>=Δ u<SUP>m</SUP>+K(x)u<SUP>p</SUP> where x∈ \bm{R}<SUP>N</SUP>, 1≤ m<p and K(x)≥q 0 has the following properties: K(x)∼|x|<SUP>σ</SUP> (-∞≤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 p<SUB>m, σ</SUB><SUP>*</SUP>=p<SUB>m, σ</SUB><SUP>*</SUP>(N) such that if p≤ p<SUB>m, σ</SUB><SUP>*</SUP>, then every nontrivial nonnegative solution is not global in time, whereas if p>p<SUB>m, σ</SUB><SUP>*</SUP> 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 u<SUB>0</SUB>(x) such that u<SUB>0</SUB>(x)∼|x|<SUP>-α</SUP> as |x|→∞ in some cone D^{} and u<SUB>0</SUB>(x)=0 in the complement of D^{}.

• 社団法人 日本数学会の論文
• 2002-10-01

著者

• Suzuki Ryuichi

  Department Of Mathematics Faculty Of Engineering Kokushikan University

関連論文

• Existence and nonexistence of global solutions of quasilinear parabolic equations
• Boundedness of global solutions of one dimensional quasilinear degenerate parabolic equations
• Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in R_N

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