Resumen de On a class of quasilinear Barenblatt equations

Caroline Bauzet, Jacques Giacomoni, Guy Vallet

• We first investigate the following quasilinear parabolic equation of Barenblatt type, f(·, ∂tu) −
  pu − ǫ
  (∂tu) = g in Q = ×]0, T[ u = 0 on 􀀀 = ∂, u(x, 0) = u0(x) in where is a bounded domain with Lipschitz boundary, denoted by 􀀀 in Rd with d ≥ 1, 2d d+2 < p < ∞, ǫ ≥ 0, 0 < T < +∞, u0 ∈ W1,p 0 ( ) and f is a Carath´eodory function which satisfies suitable growth conditions and g ∈ L2(Q). We prove the existence of a weak solution (see definition 1.1) and give some related regularity results. Next, we analyse further the case p = 2 with ǫ = 0. Precisely, we are concerned by the study of a Barenblatt problem involving a stochastic perturbation:

  f ∂t(u − Z t 0 hdw) −
  u = 0 in Q ×  u = 0 on ∂ , u(x, 0) = u0(x) in where R t 0 hdw denotes the Itˆo integral of h and f : R → R is an increasing bi- Lipschitz continuous function. (,F, P) is the probability space. Under these conditions, we prove the existence and the uniqueness of the weak solution (see definition 1.4).