Abstract |
This paper studies the existence and non-existence of the solution $T(x,t)$ of the nonlinear parabolic problem: [ egin{array}{c} D frac{partial T}{partial t}(x,t)-frac{partial^2T}{partial x^2}(x,t)=delta(x-x_0)F(T(x,t)), 0<x<infty, t>0, T(x,0)=widehat{T}geq0, 0<x<infty, T(0,t)=0, end{array} ] where $delta(x-x_0) $ is the Dirac delta distribution, $F(T)$ is a given function with $F(T)>0,F'(T)>0,F'(T)>0$ and $D lim_{T ightarrowinfty}F(T)=infty$, and $widehat{T}(0)=0, widehat{T}(x) ightarrow 0$ as $x ightarrow infty$. The blow-up behavior of the solution will be studied, the effects of the initial position and the velocity of the source related with the blow-up properties will be given. |