Asymptotic behavior of solutions to porous medium equations with boundary degeneracy

Abstract

This article concerns the asymptotic behavior of solutions to a class of one-dimensional porous medium equations with boundary degeneracy on bounded and unbounded intervals. It is proved that the degree of degeneracy, the exponents of the nonlinear diffusion, and the nonlinear source affect the asymptotic behavior of solutions. It is shown that on a bounded interval, the problem admits both nontrivial global and blowing-up solutions if the degeneracy is not strong; while any nontrivial solution must blow up if the degeneracy is strong enough. For the problem on an unbounded interval, the blowing-up theorems of Fujita type  are established. The critical Fujita exponent is finite if the degeneracy is not strong, while infinite if the degeneracy is strong enough. Furthermore, the critical case is proved to be the blowing-up case if it is finite. For more information see https://ejde.math.txstate.edu/Volumes/2021/96/abstr.html