Large time behavior of ODE type solutions to higher-order semilinear parabolic equations with small initial data

Let u be a solution to the Cauchy problem for semilinear parabolic equations {partial derivative(t)u + (-Delta)(m)u = vertical bar u vertical bar(alpha) in R-N x (0, infinity), u(x, 0) - lambda + phi(x) > 0 in R-N, (P-m) where m = 1, 2, ..., 0 < alpha < 1, N >= 1, lambda > 0, phi is an element of BC(R-N) boolean AND L-r(R-N) with 1 <= r < infinity. In the case of m = 1, by the comparison principle, one can easily show that the solution u to problem (P-m) behaves like a positive solution to ODE zeta' = zeta(alpha) in (0, infinity) as t -> infinity. In the case of m >= 2, because of the lack of the comparison principle, it is not obvious that the solution u to problem (P-m) behaves like a solution to the ODE as t -> infinity. In this paper, by imposing a smallness condition parallel to phi parallel to(L infinity(RN)) < c with a certain constant c dependent on m, alpha and lambda, we prove that the solution u behaves like a solution to the ODE as t -> infinity. Furthermore, we obtain the precise description of the large time behavior of the solution u and reveal the relationship between the diffusion effect (P-m) has. (C) 2022 Elsevier Inc. All rights reserved.