Cambridge,UK:Cambridge University Press,2002,466页,ISBN 0-521-80524-4。经济活动的空间分布有很强的规律性。一个最普遍最明显的特征便是集聚。集聚现象在空间的各个层次(包括全球、国家、区域和城市)上发生。直观地说,集聚现象的...Cambridge,UK:Cambridge University Press,2002,466页,ISBN 0-521-80524-4。经济活动的空间分布有很强的规律性。一个最普遍最明显的特征便是集聚。集聚现象在空间的各个层次(包括全球、国家、区域和城市)上发生。直观地说,集聚现象的形成与两股相反的经济力量——集聚(或向心)力和扩散(或离心)力有关。经济活动的空间分布特征就是这两种力之间平衡的结果。不可否认集聚现象有自然条件方面的原因,但要寻找更为普遍的经济原因。展开更多
This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first fin...This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.展开更多
This paper studies heat equation with variable exponent ut = △u ＋ Up（x） 4- Uq in RN × （0, T）, where p（x） is a nonnegative continuous, bounded function, 0 〈 p- = infp（x） ≤ p（x） ≤ supp（x） = p＋. It...This paper studies heat equation with variable exponent ut = △u ＋ Up（x） 4- Uq in RN × （0, T）, where p（x） is a nonnegative continuous, bounded function, 0 〈 p- = infp（x） ≤ p（x） ≤ supp（x） = p＋. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max{p＋,q} ≤1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 〈 q ≤ 1 with p＋ 〉 1, or 1 〈 q 〈 1 ＋2/N. In addition, if q 〉 1 ＋2/N, then （i） all solutions blow up in finite time with 0 〈 p- ≤ p＋ ≤ 1 ＋2/N; （ii） there are both global and nonglobal solutions for p- ≤ 1 ＋ 2/N; and （iii） there are functions p（x） such that all solutions blow up in finite time, and also functions p（x） such that the problem possesses global solutions when p＋〈 1＋2/N 〈 p＋.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
文摘Cambridge,UK:Cambridge University Press,2002,466页,ISBN 0-521-80524-4。经济活动的空间分布有很强的规律性。一个最普遍最明显的特征便是集聚。集聚现象在空间的各个层次(包括全球、国家、区域和城市)上发生。直观地说,集聚现象的形成与两股相反的经济力量——集聚(或向心)力和扩散(或离心)力有关。经济活动的空间分布特征就是这两种力之间平衡的结果。不可否认集聚现象有自然条件方面的原因,但要寻找更为普遍的经济原因。
基金supported by National Natural Science Foundation of China(Grant Nos.11171048 and 11201047)the Doctor Startup Foundation of Liaoning Province(Grant No.20121025)the Fundamental Research Funds for the Central Universities
文摘This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.
基金Supported by the National Natural Science Foundation of China(No.11171048)
文摘This paper studies heat equation with variable exponent ut = △u ＋ Up（x） 4- Uq in RN × （0, T）, where p（x） is a nonnegative continuous, bounded function, 0 〈 p- = infp（x） ≤ p（x） ≤ supp（x） = p＋. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max{p＋,q} ≤1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 〈 q ≤ 1 with p＋ 〉 1, or 1 〈 q 〈 1 ＋2/N. In addition, if q 〉 1 ＋2/N, then （i） all solutions blow up in finite time with 0 〈 p- ≤ p＋ ≤ 1 ＋2/N; （ii） there are both global and nonglobal solutions for p- ≤ 1 ＋ 2/N; and （iii） there are functions p（x） such that all solutions blow up in finite time, and also functions p（x） such that the problem possesses global solutions when p＋〈 1＋2/N 〈 p＋.
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.
