Reaction–diffusion and traveling fronts

On the real line, consider uₜ=D uₓₓ+r u(1−u), with constants D>0 and r>0. A right-moving front has u(x,t)=U(z), z=x−ct, with U approaching 1 as z tends to negative infinity and 0 as z tends to positive infinity. Substitution gives D U″+cU′+rU(1−U)=0. The profile is decreasing for the usual invasion orientation, and c is positive for motion to the right. Changing the moving-coordinate convention changes the sign in the ODE.

The scalar comparison principle keeps a solution between zero and one when its initial data lie in that range and the whole-line solution is in the appropriate bounded classical or mild class. The reaction vanishes at both endpoints and is positive between them. This invariant range helps analyze the nonlinear equation but does not determine a wave speed by itself. A wave is also translation invariant: replacing U(z) by U(z−z₀) gives the same shape at another location, so a normalization such as a chosen profile value at z=0 is additional data.

Near the leading edge U=0, replacing the reaction by rU and testing a positive exponential tail exp(−λz) gives Dλ²−cλ+r=0. Real positive tail rates require c at least 2√(Dr). For the logistic Fisher equation, the classical traveling-wave theorem indeed gives monotone fronts precisely for speeds at or above that value. The quadratic calculation motivates the threshold; existence of the nonlinear connection and exclusion of inadmissible profiles require a separate phase-plane or PDE argument.

Wave existence is not the same question as speed selection from an initial profile. For nontrivial continuous compactly supported initial data between zero and one, Fisher–KPP spreading has the minimal asymptotic speed. Other tail behaviors can lead to different selection, so a pre-existing faster traveling wave is not inconsistent with that compact-data result. At the minimal speed the tail has a repeated-root structure, and its detailed asymptotics differ from a generic faster wave. A leading exponential alone does not capture every prefactor or transient location shift.

For D=r=1, a particular profile is U(z)=(1+exp(z/√6))⁻² with speed 5/√6. It has the correct end states and is strictly decreasing. Differentiating and substituting verifies the ODE everywhere, giving an exact solution rather than a plot-based guess. Its speed is slightly greater than two, so this explicit profile is not the minimal-speed wave. The fact that a formula is available at one speed does not establish uniqueness or a closed formula for all other speeds.

Front models require attention to domain and mechanism. A finite interval with prescribed endpoints is a different problem from a connection at spatial infinities. Spatially varying coefficients, bistable reactions, or nonlocal diffusion can change thresholds and qualitative propagation. Linear heat smoothing alone has no logistic saturation, and a sigmoid gate alone has no spatial evolution. Use a profile instrument to inspect shape and an actual PDE theorem to justify motion; neither a smooth animation nor a linear tail fit proves long-time selection.