Consider the equation
$$\frac{\partial u}{\partial t} +\frac{\partial}{\partial x}(u^2+uf(x,t)) =0$$
where $f(x,t)$ is a suitably well behaved function. Given the initial condition $u(x,t=0)$, for which family of functions $f(x,t)$ will shocks form?
Note, for $f=0$, we return to the inviscid Burgers' equation, which admits shock solutions.
Take one spatial derivative of the equation. You get something like
$$ u_{xt}+u_x^2+uu_{xx}+(u_xf+uf_x)_x=0 $$
Write $v=u_x$. Then,
$$ v_{t}+v^2+(u+f)u_{xx}+2vf_x+uf_{xx}=0. $$
We know that $$ \max_x |u(x,t)|\leq \max_x|u_0(x)|, $$ where $u_0$ is the initial data. Then, we have that if we evaluate the eqution for $v$ at the point where $v$ reaches its minimum, we have $$ \frac{d}{dt}y+y^2+2yf_x+uf_{xx}=0, $$ where $y(t)=\min_x v(x,t)=v(X(t),t)$.
If a shock is formed is because $\liminf_{t\rightarrow T} y(t)=-\infty$, for certain $T$.
So, now you can see for which functions $f$ (and initial datas $y(0)$) the equation for $y$ does NOT blow up. Those are the functions $f$ that prevent the shocks.
Hopefully this helps.
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Let me try to give you a family of functions such that there is NO shock. The same idea may be used to show some families of f leading to shocks.
The idea is to get some assumptions on $u_0$ and $f$ such that we prove that $y'>0$. Then, as $y(0)<0$, we obtain a global bound for the minimum of the derivative (so, no shocks are formed).
Assume
1) $$ u_0< 0 $$
2) $$ f\leq0 $$
(so, $u(t)$ remains non-positive for all times where the solution exists). To see that notice that the evolution of $$ U(t)=\max_x u(x,t)=u(\tilde{x}(t),t) $$ reads $$ U'(t)=-f_x(\tilde{x}(t),t)U(t). $$
3) Assume also that $\infty>c\geq f_x(x)\geq C>0$ and $f_{xx}(x)\geq 0$ for all $x$.
Then the previous ODE for $y$ implies $$ y'\geq -y(y+2f_x(X(t)). $$ We need $$ y(t)+2f_x(X(t))>y(t)+2C>0. $$ Thus, if $u_0$ is such that $y(0)+2C>0$ (or $0>\min_s (u_0)_x(s)>-2C$, which is a 'smallness' requirement), we have $y'(h)>0$ (for $h$ close to zero), which means $y(t)>y(0)>-2C$ (so, our assumptions on the initial data propagates in time).
