Resumen de Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with L1/measure data

Mohammed Abdellaoui

• We study the possibility to give a formulation to the degenerate parabolic problems modeled by (P1b) {ut−div[|∇u|p−2∇u)/(1+|u|)θ(p−1)]=μ in (0,T)×Ω,u(0,x)=u0(x) in Ω,u(t,x)=0 on (0,T)×∂Ω, where θ>0, u0∈L1(Ω) and μ is a general (nonnegative) Radon measure. We also investigate the strong stability of solutions for noncoercive absorption problems whose model (P2b) {ut−div[|∇u|p−2∇u)/(1+|u|)θ(p−1)]+|u|q−1u=f in (0,T)×Ω,u(0,x)=0 in Ω,u(t,x)=0 on (0,T)×∂Ω where q>r(p−1)[1+θ(p−1)]/(r−p) and f∈L1loc(Q∖K) with K is a compact subset of Q of zero r-capacity (or, is a measure concentrated on a set of r-capacity zero). We prove the convergence of approximate solutions un (related to a regular approximation μn of μ) towards a renormalized solutions u of (P1b), and we extend the previous known-results on the nonstability of entropy solutions for problems (P2b).