Complete classification of shape functions of self-similar solutions
Let \(q>0\), \(q\neq1\), and consider a generalized KPZ equation of the type \(u_t=u_{xx}-| u_x| ^q\), \(x>0\), \(t>0\), complemented by the Neumann boundary condition \(u_x(0,t)=0\). Self-similar solutions of this problem are of the form \(u(x,t)=t^{-\alpha}g(\xi)\), where \(\alpha=(2-q)/(2q-2)\), \(\xi=x/\sqrt{t}\) and \(g\) solves the ODE \(g''+{\xi\over2}g'+\alpha g=| g'| ^q\). The authors study the asymptotic behavior of solutions of this ODE subject to the initial conditions \(g'(0)=0\), \(g(0)=\lambda>0\). In particular, they are interested in the existence of fast orbits (satisfying \(\lim_{\xi\to\infty}\xi^{2\alpha}g(\xi)=0\)) which correspond to very singular solutions \(u\) of the PDE. The results depend on the exponent \(q\); the critical values of \(q\) are \(1,3/2\) and \(2\).
scientific article; zbMATH DE number 5149537