On the behaviour of blow-up interfaces for an inhomogeneous filtration equation

Abstract

We study the asymptotic behaviour of blow-up interfaces of the solutions to the one-dimensional nonlinear filtration equation in inhomogeneous media p(x)u 1 = (u m ) xx in Q= R x R + , where m > 1 is a constant and p(x) = |x| -α (for |x| ≥ 1, with α > 2) is a bounded, positive, smooth, and symmetric function. The initial data are assumed to be smooth, bounded, compactly supported, symmetric, and monotone. It is known that due to the fast decay of the density p(x) as |x| → ∞ the support of the solution increases unboundedly in a finite time T. We prove that as t → T - the interface behaves like O((T - t) -b ), where the exponent b > 0 (which depends on m and α only) is given by a unique self-similar solution of the second kind satisfying the equation |x| -α u t = (u m ) xx . The corresponding rescaled profiles also converge. We establish the stability of the self-similar solution of the second kind for the exponential density p(x) = e -|x| for |x| ≥ 1. We give a formal asymptotic analysis of the blow-up behaviour for the non-self-similar density p(x) = e -|x|2 Several exact self-similar solutions and their corresponding asymptotics are constructed.