Large-time behavior of smooth solutions to a nonuniformly parabolic equation
The authors of this interesting paper study the large-time existence of smooth solutions with periodic behavior to nonuniformly parabolic equations, which have the form, \[ u_t+\biggl( {u^2\over 2} \biggr)_x=tu_{xx}, \;\;x\in \mathbb R^1, \;\;t>0, \] with the bounded initial data \(u(x,0)=u_0(x)\). These equations are used for the mathematical model of the propagation of finite-amplitude sound waves of duct with variable-area, where the dependent \(u\) is an acoustic variable. The same equations serve to study some hyperbolic partial differential equations of first order, which are certain conservation laws. The main result is that there exist solutions with decay rate in \(L^p\)-norm (\(p\in [1,\infty ]\)). Moreover, these solutions converge to a self-similar solution, which only depends on the behavior of the initial function at infinity. Of special interest, here is the case of periodic initial data. Then there exist solutions with exponential decay in \(L^2\)-norm. The periodic behavior of the solutions is discussed as well.
finite-amplitude sound waves periodic behavior
scientific article; zbMATH DE number 2075465