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,

where $\delta\left( x\right)$ is the Dirac delta function, and $f$ and $\psi$ are given functions. We assume that $f\left( 0\right) \geq0$, $f(u)$ 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

$$\psi^{\prime\prime}+\alpha\delta(x-b)f\left( \psi\right) \geq0$$ in  $$D$$ .

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.