Existence and decay rates of solutions to the generalized Burgers equation. (Q1406530)

The authors examine the generalized Burgers equation, which is the mathematical model of the propagation of the finite-amplitude sound waves in variable-area ducts. They demonstrate the existence and uniqueness of classical solutions to the initial value problem of \(u_t+(u^2/2)_x=f(t)u_{xx};\) \(f(t)>0 \) for \( t>0\) with rough initial data belonging to \(L^\infty(\mathbb{R})\). In addition, they obtain that the decay rates of \(u(.)\) in some \(L^p\) norms are of algebraical order.