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We consider a question posed in [GSZZ22], namely the blow-up of the PDE $$u_t+ (b(t,x) u^{1+k})_x = u_{xx}$$ when $b$ is uniformly bounded, Lipschitz and $k= 2$. We give a complete answer to the behavior of solutions when $b$ belongsto the Lorentz spaces $b \in L^{p,\infty}$, $p \in (2,\infty]$, or $b_x \inL^{p,\infty}$, $p \in (1,\infty]$.

Existence and Blow-up for Non-autonomous Scalar Conservation Laws With Viscosity