We investigate the critical exponent of non‐global solutions to the following inhomogeneous pseudo‐parabolic equation with a space‐time forcing term: ut−kΔut=Δu+|u|p+tσω(x)forx∈ℝn,t>0,u(x,0)=u0(x)forx∈ℝn,$$ \left\{\begin{array}{lll}{u}_t-k\Delta {u}_t=\Delta u+{\left|u\right|}^p+{t}^{\sigma}\omega (x)\kern0.30em & & \mathrm{for}\kern0.30em x\in {\mathrm{\mathbb{R}}}^n,t>0,\\ {}u\left(x,0\right)={u}_0(x)& & \mathrm{for}\kern0.30em x\in {\mathrm{\mathbb{R}}}^n,\end{array}\right. $$where n≥1$$ n\ge 1 $$ is an integer; k>0$$ k>0 $$, p>1$$ p>1 $$, and σ>−1$$ \sigma >-1 $$ are three constants; and u0,ω∈C0(ℝn)$$ {u}_0,\omega \in {C}_0\left({\mathrm{\mathbb{R}}}^n\right) $$. By obtaining a priori estimate for the solutions and the contradiction argument, we show that there exists a critical exponent: pc(σ):=2σ−12σ+1,ifn=1andσ∈−1,−12,∞,ifn=1andσ∈−12,∞,1−1σ,ifn=2andσ∈(−1,0),∞,ifn=2andσ∈[0,∞),2σ−n2σ−n+2,ifn>2andσ∈(−1,0],∞,ifn>2andσ∈(0,∞),$$ {p}_c\left(\sigma \right):= \left\{\begin{array}{rcl}& \frac{2\sigma -1}{2\sigma +1},& \mathrm{if}\kern0.30em n=1\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left(-1,-\frac{1}{2}\right),\\ {}& \infty, & \mathrm{if}\kern0.30em n=1\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left[-\frac{1}{2},\infty \right),\\ {}& 1-\frac{1}{\sigma },& \mathrm{if}\kern0.30em n=2\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left(-1,0\right),\\ {}& \infty, & \mathrm{if}\kern0.30em n=2\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left[0,\infty \right),\\ {}& \frac{2\sigma -n}{2\sigma -n+2},& \mathrm{if}\kern0.30em n>2\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left(-1,0\right],\\ {}& \infty, & \mathrm{if}\kern0.30em n>2\kern0.30em \mathrm{and}\kern0.30em \sigma \in \left(0,\infty \right),\end{array}\right. $$such that the problem does not admit any global solutions when p0$$ {\int}_{{\mathrm{\mathbb{R}}}^n}\omega (x) dx>0 $$. Our obtained results show that the forcing term induces an interesting phenomenon of continuity/discontinuity of the critical exponent pc(σ)$$ {p}_c\left(\sigma \right) $$ depending on the dimension n$$ n $$. Namely, we found that when n=1$$ n=1 $$, limσ→−12−pc(σ)=limσ→−12+pc(σ)=∞$$ \underset{\sigma \to -{\frac{1}{2}}^{-}}{\lim }{p}_c\left(\sigma \right)=\underset{\sigma \to -{\frac{1}{2}}^{+}}{\lim }{p}_c\left(\sigma \right)=\infty $$; when n=2$$ n=2 $$limσ→0−pc(σ)=limσ→0+pc(σ)=∞$$ \underset{\sigma \to {0}^{-}}{\lim }{p}_c\left(\sigma \right)=\underset{\sigma \to {0}^{+}}{\lim }{p}_c\left(\sigma \right)=\infty $$; and when n≥3$$ n\ge 3 $$limσ→0−pc(σ)=nn−2<∞$$ \underset{\sigma \to {0}^{-}}{\lim }{p}_c\left(\sigma \right)=\frac{n}{n-2}<\infty $$, limσ→0+pc(σ)=∞$$ \underset{\sigma \to {0}^{+}}{\lim }{p}_c\left(\sigma \right)=\infty $$. Furthermore, limσ→κ−pc(σ)$$ \underset{\sigma \to {\kappa}^{-}}{\lim }{p}_c\left(\sigma \right) $$ with κ=−12$$ \kappa =-\frac{1}{2} $$ when n=1$$ n=1 $$ and κ=0$$ \kappa =0 $$ when n≥2$$ n\ge 2 $$ coincides with the critical exponent of the above problem with σ=0$$ \sigma =0 $$. [ABSTRACT FROM AUTHOR]