In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic system ut = △u ＋ v^P（x0, t） - au^τ, x ∈ Ω, t 〉 0, △u ＋ v^P（x0, t...In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic system ut = △u ＋ v^P（x0, t） - au^τ, x ∈ Ω, t 〉 0, △u ＋ v^P（x0, t） - bu^τ, x ∈ Ω, t 〉 0 subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈ Ω is a fixed point, p, q, r, s ≥ 1 and a, b 〉 0 are constants. In the situation when nonnegative solution （u, v） of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1, lim t→T＊（T＊-t）^p＋1/pq-1u（x,t）=（p＋1）^1/pq-1（q＋1）^p/pq-1（pq-1）^-p＋1/pq-1, lim t→T＊（T＊-t）^q＋1/pq-1u（x,t）=（p＋1）^1/pq-1（q＋1）^p/pq-1（pq-1）^-p＋1/pq-1 are obtained uniformly on compact subsets of/2, where T＊ is the blow-up time.展开更多
We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We ob...We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
基金This study is supported partially by the research program of natural science of universities in Jiangsu province(05KJB110144 and 05KJB110063)the natural science foundation of Yancheng normal institute.
文摘In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic system ut = △u ＋ v^P（x0, t） - au^τ, x ∈ Ω, t 〉 0, △u ＋ v^P（x0, t） - bu^τ, x ∈ Ω, t 〉 0 subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈ Ω is a fixed point, p, q, r, s ≥ 1 and a, b 〉 0 are constants. In the situation when nonnegative solution （u, v） of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1, lim t→T＊（T＊-t）^p＋1/pq-1u（x,t）=（p＋1）^1/pq-1（q＋1）^p/pq-1（pq-1）^-p＋1/pq-1, lim t→T＊（T＊-t）^q＋1/pq-1u（x,t）=（p＋1）^1/pq-1（q＋1）^p/pq-1（pq-1）^-p＋1/pq-1 are obtained uniformly on compact subsets of/2, where T＊ is the blow-up time.
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.
