Asymptotic behaviour of a generalized Burgers' equation

AbstractWe consider the generalized Burgers equation: (GBE)ut=Δ(um)−∂∂x1(uq),with exponents m>1 and q=m+(1/N). We study the large-time behaviour of nonnegative weak solutions of the Cauchy problem posed in Q=RN×(0,∞) with integrable and nonnegative data. We construct a uni-parametric family {UM} of source-type solutions of (GBE) such that:UM(·,t)→Mδ(x)inD′(RN)ast→0,and prove that they give the asymptotic behaviour of all solutions of the Cauchy problem. These special solutions have the following self-similar form: U(x,t)=t−αF(xt−β), with α=1/((m−1)+(2/N)) and Nβ=α. The criterion to choose the right member of the family is the following mass equality: M=∫RNu0dx. The construction of the family {UM} and the proof of the asymptotic convergence in this nonlinear, several dimensional setting needs a new method of asymptotic analysis. The results are then extended to equations of the formut=ΔΦ(u)−∇·F(u),where Φ and F resemble the preceding power functions as u→0. In this more general case the asymptotic behaviour is described by the same family UM mentioned above