Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary

We consider the Cauchy problem of the porous medium type reaction-diffusion equation partial derivative(t)rho t rho = Delta rho(m) + rho g(rho), (rho ) , (x, t) ) is an element of R(n)x n x R + , n >= 2 , m > 1, where g is the given monotonic decreasing function with the density critical threshold rho M > 0 satisfying g (rho(M) ) = 0. We prove that the pressure P := m m - 1 rho m - 1 in L infinity loc ( R n ) tends to the pressure critical threshold P M := m - 1 m ( rho M ) m - 1 at the time decay rate (1 + t ) - 1 . If the initial density rho(x, ( x, 0) is compactly supported, we justify that the support { x : rho(x, ( x, t ) > 0} } of the density rho expands exponentially in time. Furthermore, we show that there exists a time T 0 > 0 such that the pressure P is Lipschitz continuous for t > T 0 , which is the optimal (sharp) regularity of the pressure, and the free surface partial derivative {( x, t ) : rho ( x, t ) > 0} } boolean AND { t > T-0 } is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary 8{(x, t) {( x, t ) : rho(x, ( x, t ) > 0} } boolean AND { t > T 0 } is a local C-1,C-alpha surface. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.