Renormalization group calculation of anomalous exponents for nonlinear diffusion
Gunduz Caginalp · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · 1996
We consider the heat equation with nonlinearity, ${\mathit{u}}_{\mathit{t}}$=${\mathit{u}}_{\mathit{x}\mathit{x}}$+\ensuremath{\epsilon}f(x,u,${\mathit{u}}_{\mathit{x}}$,${\mathit{u}}_{\mathit{x}\mathit{x}}$), where \ensuremath{\epsilon} is a small parameter. Using a renormalization group approach, we calculate that for large space and time, the solution is characterized by u(x,t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$${\mathit{u}}^{\mathrm{*}}$(${\mathit{xt}}^{\mathrm{\ensuremath{-}}1/2}$,1), where \ensuremath{\alpha} is a simple function of the powers of x, u, ${\mathit{u}}_{\mathit{x}}$, and ${\mathit{u}}_{\mathit{x}\mathit{x}}$ in f. The same approach can be used to calculate the exponent and coefficient for finite time blow-up of equations such as ${\mathit{u}}_{\mathit{t}}$=${\mathit{u}}_{\mathit{x}\mathit{x}}$+${\mathit{u}}^{\mathit{r}}$, where r>1. In both cases the calculations can be performed within the standards of asymptotic analysis. \textcopyright{} 1996 The American Physical Society.