Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy (Communications in Mathematical Research, 2022)
This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy bothin bounded and unbounded intervals. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initialdata 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 criticalFujita exponent depends on the degeneracy of the equation and the asymptoticbehavior of the diffusion coefficient at infinity, and it may be equal to one orinfinity. Furthermore, the critical case is proved to belong to the blowing-upcase.
Keywords: Asymptotic behavior; boundary degeneracy; blowing-up.