Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

Let $\alpha$, $b$ and $T$ be positive numbers, $D=\left ( 0,\infty \right )$, $\bar {D}=\left [0,\infty \right )$, and $\Omega =D\times \left (0,T\right ]$. This article studies the first initial-boundary value problem, \[ \begin {array} [c]{c} u_{t}-u_{xx}=\alpha \delta (x-b)f\left ( u(x,t)\right ) \text { in }\Omega ,\\ u(x,0)=\psi (x)\text { on }\bar {D},\\ u(0,t)=0=\lim _{x\rightarrow \infty }u(x,t)\text { for }0<t\leq T, \end {array} \] where $\delta \left ( x\right )$ is the Dirac delta function, and $f$ and $\psi$ are given functions. We assume that $f\left ( 0\right ) \geq 0$, $f(u)$ and its derivatives $f^{\prime }(u)$ and $f^{\prime \prime }\left ( u\right )$ are positive for $u>0$, and $\psi (x)$ is nontrivial, nonnegative and continuous such that $\psi \left ( 0\right ) =0=\lim _{x\rightarrow \infty }\psi \left ( x\right )$, and \[ \psi ^{\prime \prime }+\alpha \delta (x-b)f\left ( \psi \right ) \geq 0\text { in } D\text {.} \] It is shown that if $u$ blows up, then it blows up in a finite time at the single point $b$ only. A criterion for $u$ to blow up in a finite time and a criterion for $u$ to exist globally are given. It is also shown that there exists a critical position $b^{\ast }$ for the nonlinear source to be placed such that no blowup occurs for $b\leq b^{\ast }$, and $u$ blows up in a finite time for $b>b^{\ast }$. This also implies that $u$ does not blow up in infinite time. The formula for computing $b^{\ast }$ is also derived. For illustrations, two examples are given.