Pointwise estimates on the Green's function for a scalar linear convection-diffusion equation (Q1300096)

The author studies the equations of the form \(v_t+(a(x)v)_x=(b(x)v_x)_x\), \(v,x,t \in {\mathbb{R}}, t>0\), and \(v_t+a(x)v_x=b(x)v_{xx}\), \(v,x,t \in {\mathbb{R}}, t>0\) with \(a,b\in C^K({\mathbb{R}})\) for some \(K\geq 1\), \(a(x),b(x)\) asymptotically constant at \(x=\pm \infty\), and \(b(x)\geq b_0>0\). The author obtains Gaussian type estimates from above for the fundamental solutions of these equations and their derivatives. The Gaussian factor in the estimates is centered around asymptotic values of the convection function \(a\).