ON THE BEHAVIOR OF BLOW-UP INTERFACES FOR AN INHOMOGENEOUS FILTRATIONEQUATION

We study the asymptotic behaviour of blow-up interfaces of the solutio ns to the one-dimensional nonlinear filtration equation in inhomogeneo us media rho(x)u(t) = (u(m))(xx) in Q = R x R(+), where m > 1 is a con stant and rho(x) = \x\(-alpha) (for \x\ greater than or equal to 1, wi th alpha > 2) is a bounded, positive, smooth, and symmetric function. The initial data are assumed to be smooth, bounded, compactly supporte d, symmetric, and monotone. It is known that due to the fast decay of the density rho(x) as \x\ --> infinity the support of the solution inc reases unboundedly in a finite time T. We prove that as t --> T- the i nterface behaves like O((T - t)(-b)), where the exponent b > O (which depends on m and alpha only) is given by a unique self-similar solutio n of the second kind satisfying the equation \x\(-alpha) u(t) = (u(m)) (xx). The corresponding rescaled profiles also converge. We establish the stability of the self-similar solution of the second kind for the exponential density rho(x) = e(-\x\) for \x\ greater than or equal to 1. We give a formal asymptotic analysis of the blow-up behaviour for t he non-self-similar density rho(x)= e(-\x\2). Several exact self-simil ar solutions and their corresponding asymptotics are constructed.