This paper investigates the behavior of solutions to the semilinear diffusion equation for large times t. The equation is given by ∂u/∂t = ∂²u/∂x² + f(u), where f(0) = f(1) = 0 and u(x,t) is in [0,1]. The study considers two problems: the initial value problem in the half-space R × R⁺ and the initial-boundary value problem in the quarter-space R⁺ × R⁺. The equation arises in various applications, including population genetics, combustion, and nerve pulse propagation. In population genetics, the equation models the distribution of genotypes in a one-dimensional habitat. The population consists of three genotypes: aa, aA, and AA. The densities of these genotypes are denoted by ρ₁(x,t), ρ₂(x,t), and ρ₃(x,t), respectively. The population mates randomly, producing offspring with a birth rate r, and diffuses through the habitat with a diffusion constant l. The death rates of the genotypes depend on their viability, denoted by τ₁, τ₂, and τ₃. These death rates may differ slightly, and reproduction can be incorporated by adding negative quantities to the death rates. The paper does not assume the signs of the τᵢ. The study aims to understand the long-term behavior of solutions to the semilinear diffusion equation, considering different forms of the function f(u) that arise from various applications. The analysis focuses on the dynamics of solutions in different spatial domains and their evolution over time. The results provide insights into the behavior of populations, combustion processes, and nerve pulse propagation, highlighting the importance of nonlinear diffusion in these contexts.This paper investigates the behavior of solutions to the semilinear diffusion equation for large times t. The equation is given by ∂u/∂t = ∂²u/∂x² + f(u), where f(0) = f(1) = 0 and u(x,t) is in [0,1]. The study considers two problems: the initial value problem in the half-space R × R⁺ and the initial-boundary value problem in the quarter-space R⁺ × R⁺. The equation arises in various applications, including population genetics, combustion, and nerve pulse propagation. In population genetics, the equation models the distribution of genotypes in a one-dimensional habitat. The population consists of three genotypes: aa, aA, and AA. The densities of these genotypes are denoted by ρ₁(x,t), ρ₂(x,t), and ρ₃(x,t), respectively. The population mates randomly, producing offspring with a birth rate r, and diffuses through the habitat with a diffusion constant l. The death rates of the genotypes depend on their viability, denoted by τ₁, τ₂, and τ₃. These death rates may differ slightly, and reproduction can be incorporated by adding negative quantities to the death rates. The paper does not assume the signs of the τᵢ. The study aims to understand the long-term behavior of solutions to the semilinear diffusion equation, considering different forms of the function f(u) that arise from various applications. The analysis focuses on the dynamics of solutions in different spatial domains and their evolution over time. The results provide insights into the behavior of populations, combustion processes, and nerve pulse propagation, highlighting the importance of nonlinear diffusion in these contexts.