ASYMPTOTIC CONVERGENCE TO DIPOLE SOLUTIONS IN NONLINEAR PARABOLIC EQUATIONS

We study the asymptotic behaviour as t --> infinity of the solution u = u(x,t) greater than or equal to 0 to the quasilinear heat equation w ith absorption u(t) = (u(m))(xx) - f(u) posed for t > 0 in a half-line I = {0 < x < infinity}. Far definiteness, we take f(u) = u(p) but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p > m. We impose u = 0 on the l ateral boundary {x = 0, t > 0}, and consider a non-negative, integrabl e and compactly supported function u(0)(x) as initial data. This probl em is equivalent to solving the corresponding equation in the whole li ne with antisymmetric initial data, u(0)(-x) = -u(0)(x). The behaviour of the solutions depends on the values of m and p. Possibly the most interesting case occurs when p = m + 1 (critical absorption). Then u(x ,t) behaves for large t like some fixed dipole-type self-similar solut ion to the porous medium equation u(t) = (u(m))(xx), after additional rescaling in the variables x and u with factors which depend logarithm ically on t. These scaling factors reflect the influence of the absorp tion term at the asymptotic level. Similar interesting behaviour occur s for the semilinear heat equation with m = 1 and p = 2. The situation for noncritical exponents is different. For p > m + 1, the effect of the absorption term becomes negligible for large times and we have con vergence to a dipole solution of the porous medium equation without an y logarithmic rescaling. On the contrary, for in < p < in + 1 the solu tion u(x,t) converges to a unique self-similar solution of a very sing ular type. In this case both diffusion and absorption appear explicitl y at the asymptotic level.