Article · Discrete and Continuous Dynamical Systems - S · September 30, 2026

Global existence and blow-up for a semilinear heat equation with a moving boundary

Abstract

In this paper, we establish sufficient conditions for global existence and finite-time blow-up for a semilinear heat equation with homogeneous Dirichlet conditions on a prescribed moving interval of bounded, nondecreasing length. Under a sufficient condition on the boundary velocities, stable data below a uniform potential-well threshold generate global solutions, with exponential decay under a stronger stability condition. An additional bound on the relative expansion rate ensures blow-up for data in the unstable wells, including data of positive initial energy. The standard critical stable level and the modified critical stable and unstable levels are also treated. The standard threshold equals the classical well depth on an interval of limiting length, divided by that length. The results are formulated for the original moving-domain problem and illustrated by numerical experiments.

Stability and Controllability of Differential EquationsNonlinear Partial Differential EquationsAdvanced Mathematical Physics Problemssemilinear heat equationmoving boundary problemglobal existence

Global existence and blow-up for a semilinear heat equation with a moving boundary

Abstract

More by Nurbol Koshkarbayev