On a class of quasilinear Barenblatt equations
- Autores: Caroline Bauzet, Jacques Giacomoni, Guy Vallet
- Localización: Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza, ISSN 1132-6360, Nº. 38, 2012 (Ejemplar dedicado a: Monique Madaune-Tort), págs. 35-51
- Idioma: inglés
- Enlaces
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Resumen
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).
