QUENCHING FOR A PARABOLIC PROBLEM DUE TO A CONCENTRATED NONLINEAR SOURCE ON A SEMI-INFINITE INTERVAL

Let α, b, and T be positive numbers, D = (0,∞), D¯ = [ 0, ∞ ), and Ω = D × ( 0, T ]. This article studies the first initial-boundary value problem with a concentrated nonlinear source situated at b,

                                                 u_t − u_xx = αδ( x − b ) f( u(x, t) )  in Ω,

                                                 u( x, 0 ) = 0 on D¯, 

                                                 u( 0, t ) = 0 and u( x, t ) → 0 as x → ∞  for  0 < t ≤ T,

where δ (x) is the Dirac delta function and f is a given function such that lim_u→c− f (u) = ∞ for some positive constant c, and f(u) and its derivatives f´(u) and f''(u) are positive for 0 ≤ u < c. The problem has a unique continuous solution u before sup {u (x, t) : 0 ≤ x < ∞} reaches c⁻, and u is a strictly increasing function of t in Ω. It is shown that if

                                                    sup { u (x, t) : 0 ≤ x < ∞ }

reaches c⁻, then u attains the value c in a finite time only at the point b. A criterion for u to exist globally and a criterion for u to quench in a finite time are given. It is also shown that there exists a critical position b^∗ for the nonlinear source to be placed such that for b ≤ b^∗ , u exists for 0 ≤ t < ∞,  and for b > b^∗ , u quenches in a finite time. This also implies that u does not quench in infinite time. The formula for computing b^∗ is also derived.