This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals.For the problem in a bounded inter-val,it is ...This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals.For the problem in a bounded inter-val,it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong.Whereas in the case that the degeneracy is strong enough,the nontrivial solu-tion must blow up in a finite time.For the problem in an unbounded interval,blowing-up theorems of Fujita type are established.It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity,and it may be equal to one or infinity.Furthermore,the critical case is proved to belong to the blowing-up case.展开更多
The authors study the singular diffusion equationwhere Ω(?)Rn is a bounded domain with appropriately smooth boundary δΩ, ρ(x) = dist(x,δΩ), and prove that if α≥p-1, the equation admits a unique solution subjec...The authors study the singular diffusion equationwhere Ω(?)Rn is a bounded domain with appropriately smooth boundary δΩ, ρ(x) = dist(x,δΩ), and prove that if α≥p-1, the equation admits a unique solution subject only to a given initial datum without any boundary value condition, while if 0 <α< p - 1, for a given initial datum, the equation admits different solutions for different boundary value conditions.展开更多
基金This work was supported by the National Key R&D Program of China(Grant No.2020YFA0714i01)by the National Natural Science Foundation of China(Grant No.11925105).
文摘This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals.For the problem in a bounded inter-val,it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong.Whereas in the case that the degeneracy is strong enough,the nontrivial solu-tion must blow up in a finite time.For the problem in an unbounded interval,blowing-up theorems of Fujita type are established.It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity,and it may be equal to one or infinity.Furthermore,the critical case is proved to belong to the blowing-up case.
基金Project supported by the 973 Project of the Ministry of Science and Technology of China, the Outstanding Youth Foundation of China (No.10125107)the Department of Mathematics of Jilin University.
文摘The authors study the singular diffusion equationwhere Ω(?)Rn is a bounded domain with appropriately smooth boundary δΩ, ρ(x) = dist(x,δΩ), and prove that if α≥p-1, the equation admits a unique solution subject only to a given initial datum without any boundary value condition, while if 0 <α< p - 1, for a given initial datum, the equation admits different solutions for different boundary value conditions.
