Asymptotic behaviour of a generalized Burgers' equation

We consider the generalized Burgers equation: (GBE) u(t) = Delta(u(m)) - partial derivative/partial derivative(x1)(u(q)), with exponents m > 1 and q = m + (1/N). We study the large-time behaviour o f nonnegative weak solutions of the Cauchy problem posed in Q = R-N x (0, i nfinity) with integrable and nonnegative data. We construct a uni-parametri c family {U-M} of source-type solutions of (GBE) such that: U-M(.,t) --> M delta(x) in D'(R-N) as t --> 0, and prove that they give the asymptotic behaviour of all solutions of the C auchy problem. These special solutions have the following self-similar form : U(x, t) = t(-alpha) F(xt(-beta)), with alpha = 1/((m - 1) + (2/N)) and N beta = alpha. The criterion to choose the right member of the family is the following mass equality: M = integral(RN) u(0) dx. The construction of the family {U-M} and the proof of the asymptotic convergence in this nonlinear , several dimensional setting needs a new method of asymptotic analysis. Th e results are then extended to equations of the form u(t) = Delta Phi(u) - del . F(u), where Phi and F resemble the preceding power functions as u --> 0. In this more general case the asymptotic behaviour is described by the same family U-M mentioned above. (C) Elsevier, Paris.