Single blow-up point and critical speed for a parabolic problem with a moving nonlinear source on a semi-infinite interval

Let $v$ and $T$ be positive numbers, $D=\left ( 0,\infty \right )$, $\Omega =D\times \left ( 0,T\right ]$, and $\bar {D}$ be the closure of $D$. This article studies the first initial-boundary value problem, \[ \begin {array}{l} u_{t}-u_{xx}=\delta (x-vt)f\left ( u(x,t)\right ) \text { in }\Omega , \\ u(x,0)=\psi (x)\text { on }\bar {D}, \\ u(0,t)=0,u(x,t)\rightarrow 0\text { as }x\rightarrow \infty \text { for } 0<t\leq T, \end {array} \] where $\delta \left ( x\right )$ is the Dirac delta function, and $f$ and $\psi$ are given functions. It is shown that if the solution $u$ blows up in a finite time $t_{b}$, then it blows up only at the point $x=vt_{b}$. A criterion for $u$ to exist globally and a criterion for $u$ to blow up in a finite time are given. Furthermore, the problem is shown to have a critical speed $v^{\ast }$ of the moving nonlinear source such that no blowup occurs for $v\geq v^{\ast }$ and blowup occurs in a finite time for $v<v^{\ast }$. The formula for computing $v^{\ast }$ is also derived.