Some Properties Of The Solutions For A Class Of Semilinear Heat Equations With Strong Nonlinear Sources

This paper is devoted to the study of the initial-boundary problem for a class of semilinear heat equations with strong nonlinear sources of the form u_t = Δu+min{(?)^-1,u^p} on Q = Ω× (0, T), where (?)> 0,p > 1, T > 0, Ω (?) Rⁿ is a bounded domain. The initial data u(x,0) = u₀(x) ∈ C¹(Ω|—) with u₀(x) ≥ 0. Let u^? denote the solution of this problem, and let the truncation function T_K(r) = min{K, max{r, -K}}, (?) K > 0. We obtain that T_K(u^?) is uniformly bounded in W₂^1,1 (Q) with respect to (?), and that there exists a subsequence of {u^?}, still indexed by (?), such that u^?→ u a.e. in Q as (?) → 0, where u is a measurable function defined on Q.