Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption
The paper deals with the Cauchy problem for the reaction-diffusion equation \[ u_t - (u^m)_{xx} + bu^{\beta} = 0, \quad u(x,0) = u_0(x) \qquad (x \in \mathbb{R}, 0<t<T). \] The short-time behaviour of the interface function \(\eta(t) = \sup\{x: u(x,t) > 0 \}\) and of the local solution near the interface are studied in the case \(0<m<1\) (fast diffusion). This paper continues earlier investigations of the author together with \textit{J. R. King} [SIAM J. Math. Anal. 32, 235-260 (2000; Zbl 0964.35086)], where the case \(m>1\) (slow diffusion) was treated. The methods used are based on the results concerning the general theory of initial-boundary value problems for reaction-diffusion equations, obtained previously by the author [J. Differ. Equations 164, 321-354 (2000; Zbl 0956.35069)].
nonlinear degenerate parabolic equations
scientific article; zbMATH DE number 1788029
scientific article