ASYMPTOTIC BEHAVIOR OF A FRONT PROPAGATION MODEL: CASE OF A ONE-DIMENSIONAL FREE BOUNDARY PROBLEM
Keywords:
free boundary problem, heat equation, traveling wave solution, Laplace transform, asymptotic behaviorDOI:
https://doi.org/10.17654/0975045225003Abstract
We consider a physical model of straight flame front propagation in a homogeneous solid medium in one space dimension governed by a free boundary problem. The position of the flame front at time $t$ is $\xi(t)$ and $u(t, x)$ is the temperature at time $t$ in the domain bounded by the front. We are particularly interested in the evolution over time of the pair $(\xi, u)$. We define a notion of traveling wave solution or stationary wave solution, the couple $(c, u)$ with $c>0$ a front propagation velocity and $u=u(x)$ a stationary temperature. We show that the solution of our problem converges to a wave solution when $t \rightarrow \infty$. Finally, the theoretical results are confirmed by various numerical tests.
Received: November 6, 2024
Revised: December 4, 2024
Accepted: December 14, 2024
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