In this paper, we study the asymptotic stability of viscous shock waves for Burgers equation with fast diffusion \(u_t+f(u)_x=\mu (u^m)_{xx}\) on \(\mathbb {R} \times (0, +\infty )\) when \(0<m<1\) . For the proposed constant states \(u_->u_+=0\) , the equation with fast diffusion \((u^m)_{xx}=m\left( \frac{u_x}{u^{1-m}}\right) _x\) possesses a singularity at \(u_+=0\) , which causes some difficulty when studying the stability of viscous shocks. We observe that, there exists two different types of viscous shocks, one which is a non-degenerate shock satisfying Lax’s entropy condition \(f'(u_+)<s<f'(u_-)\) , where the wave speed is determined by the Rankine-Hugoniot condition \(s=[f(u_-)-f(u_+)]/(u_--u_+)\) , and another which is a degenerate viscous shock \(f'(u_+)=s<f'(u_-)\) . The non-degenerate shock decays to the singular state \(u_+=0\) in the algebraical form faster than the degenerate shock, which implies that the singularity of \(\left( \frac{u_x}{u^{1-m}}\right) _x\) for the non-degenerate shock is stronger than that of the degenerate shock. In order to overcome the singularity at \(u_+=0\) , we use the weighted energy method technique and develop a new strategy where the weights depending on the non-degenerate/degenerate shock waves are carefully selected. Numerical simulations are also carried out in different cases to illustrate and validate our theoretical results. In particular, we numerically approximate the solution for different value of \(0<m<1\) , and find that the shapes of shock waves become steeper when the singularity \(\left( \frac{u_x}{u^{1-m}}\right) _x\) is stronger as \(m\rightarrow 0\) , thus indicating that the effect of singular fast diffusion on the solution is essential.
