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

Abstract

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 $b$ belongs to the Lorentz spaces $b\in L^{p,\infty }$, $p\in (2,\infty ]$, or $b_x\in L^{p,\infty }$, $p\in (1,\infty ]$.