This paper considers a fast diffusion equation with potential ut= um V （x）um＋upin Rn×（0,T）, where 1 2αm＋n m ≤ 1, p 1, n ≥ 2, V （x） ～ω｜x｜2with ω ≥ 0 as ｜x｜ → ∞,and α is the positive root of ...This paper considers a fast diffusion equation with potential ut= um V （x）um＋upin Rn×（0,T）, where 1 2αm＋n m ≤ 1, p 1, n ≥ 2, V （x） ～ω｜x｜2with ω ≥ 0 as ｜x｜ → ∞,and α is the positive root of αm（αm ＋ n 2） ω = 0. The critical Fujita exponent was determined as pc= m ＋2αm＋nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p pcvia the critical decay rates of the initial data.With u0（x） ～ ｜x｜ aas ｜x｜ → ∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.展开更多
This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is establishe...This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4≤ max（ p＋1/pq-1, q＋1/pq-1 ）. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No. 11171048)
文摘This paper considers a fast diffusion equation with potential ut= um V （x）um＋upin Rn×（0,T）, where 1 2αm＋n m ≤ 1, p 1, n ≥ 2, V （x） ～ω｜x｜2with ω ≥ 0 as ｜x｜ → ∞,and α is the positive root of αm（αm ＋ n 2） ω = 0. The critical Fujita exponent was determined as pc= m ＋2αm＋nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p pcvia the critical decay rates of the initial data.With u0（x） ～ ｜x｜ aas ｜x｜ → ∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.
基金supported by the National Natural Science Foundation of China (Nos. 10771024,11171048)the Fundamental Research Funds for the Central Universities (No. 851011)
文摘This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4≤ max（ p＋1/pq-1, q＋1/pq-1 ）. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.
