Stability of strong detonation travelling waves to combustion model (Q1922920)

The one-dimensional combustion model: \[ (u+qz)_t+ f(u)_x= u_{xx},\quad z_t=dz_{xx}-BH(u)z \] is studied under the main assumptions: (H.1) \(0<\gamma_0\leq f''(u)\leq M_1\), \(f'(u)>0\), \(f'''(u)\leq 0\) for \(u\in(-\infty,+\infty)\), (H.2) \(0\leq H'(u)\leq M_1\), \(H(u)=0\) for \(u\leq 0\) and \(H(u)=1\) for \(u\geq \eta_0>0\). The existence of the travelling waves and their stabilities have been shown under additional assumptions in several articles. The authors obtain a similar result on the stability of the strong detonation waves for sufficiently small \(B\) by deriving the asymptotic estimates in terms of the weighted \(L_\infty\) norm provided the initial perturbation is sufficiently small.