Quenching behavior and critical speed for a parabolic problem due to a moving nonlinear source

This article investigates an initial–boundary value problem on the semi-infinite interval for a parabolic equation with a moving nonlinear source. The study presents criteria for both finite-time quenching and global existence of the solution. It is shown that there exists a critical speed v ∗ for the nonlinear source, ensuring global existence of the solution when the speed of the moving nonlinear source is greater than or equal to v ∗ , while finite-time quenching occurs when the speed is smaller than v ∗ . The formula to calculate the critical speed v ∗ is also provided for a special case.