T02-P-018 Check up of the changes of triglyceride levels in the first 24 hours following the myocardial infarction

Abstract

We prove that the distribution solutions of the very fast diffusion equation ∂u/∂t=Δ(um/m), u>0, in Rn×(0,∞), u(x,0)=u0(x) in Rn, where m<0, n≥2, constructed in [P. Daskalopoulos, M.A. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal. 137 (1997) 363–380] are actually classical maximal solutions of the problem. Under the additional assumption that u0⁄∈L1(Rn), 0≤u0∈Llocp(Rn) for some constant p>n/2, and u0(x)≥ε/|x|2α for any |x|≥R1 where ε>0, R1>0, m0<0, α<min(1/(1−m0),1/|m0|), are constants satisfying p>(1−m0)n/2, we prove that the solution of the above problem will converge uniformly on every compact subset of Rn×(0,∞) to the maximal solution of the equation vt=Δlogv, v(x,0)=u0(x), as m↗0−. For any smooth bounded domain Ω⊂Rn, m0<0, m∈[m0,0)∪(0,1), and 0≤u0∈Lp(Ω) for some constant p>(1−m0)max(1,n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem ∂u/∂t=Δ(um/m), u>0, in Ω×(0,∞), u=u0 in Ω, u=g on ∂Ω×(0,∞) with either finite or infinite positive boundary value g. We also prove a similar convergence result for the solutions of the above Dirichlet problem as m→0.