The Quenching of Solutions of Semilinear Hyperbolic Equations

We consider the problem uₜₜ = uₓₓ + φ (u(x,t)),0 < x < L,t > 0;u(0,t) = u(L,t) = 0;u(x,0) = uₜ (x,0) = 0. Assume that φ :( - ∞ ,A) → (0,∞ ) is continuously differentiable, monotone increasing, convex, and satisfies lim u → A^ - φ (u) = + ∞. We prove that there exist numbers L₁ and L₂, 0 < L₁ L such that if L > L₂, then a weak solution u (to be defined) quenches in the sense that u reaches A in finite time; if L < L₁, then u does not quench. We also investigate the behavior of the weak solution for small L and establish the local (in time) existence of u.