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Volume 39, Issue 1
Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy

Xinxin Jing, Chunpeng Wang & Mingjun Zhou

DOI: 10.4208/cmr.2021-0108

Commun. Math. Res., 39 (2023), pp. 54-78.

Published online: 2022-10

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  • Abstract

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 interval, 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 solution 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.

  • Keywords

Asymptotic behavior, boundary degeneracy, blowing-up.

  • AMS Subject Headings

35K65, 35D30, 35B33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-54, author = {Jing , XinxinWang , Chunpeng and Zhou , Mingjun}, title = {Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {39}, number = {1}, pages = {54--78}, abstract = {

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 interval, 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 solution 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0108}, url = {http://global-sci.org/intro/article_detail/cmr/21078.html} }
TY - JOUR T1 - Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy AU - Jing , Xinxin AU - Wang , Chunpeng AU - Zhou , Mingjun JO - Communications in Mathematical Research VL - 1 SP - 54 EP - 78 PY - 2022 DA - 2022/10 SN - 39 DO - http://doi.org/10.4208/cmr.2021-0108 UR - https://global-sci.org/intro/article_detail/cmr/21078.html KW - Asymptotic behavior, boundary degeneracy, blowing-up. AB -

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 interval, 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 solution 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.

Jing , XinxinWang , Chunpeng and Zhou , Mingjun. (2022). Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy. Communications in Mathematical Research . 39 (1). 54-78. doi:10.4208/cmr.2021-0108
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