Large solutions of parabolic logistic equation with spatial and temporal degeneracies

There is studied asymptotic behavior as t → T of arbitrary solution of equation P0(u) := ut - Δu = a(t; x)u -b(t; x)|u|p-1u in [0; T) × Ω where is smooth bounded domain in ℝN, 0 < T < 1,∞ p > 1, a(·) is continuous, b(·) is continuous nonnegative function, satisfying condition: b(t; x) ≥ a1(t)g1(d(x)), d(x) := dist(x; ∂Ω). Here g1(s) is arbitrary nondecreasing positive for all s > 0 function and a1(t) satisfies: a1(t) ≥ c0 exp(-ω (T - t)(T - t)-1) ∀t〈 T; c0 = const〉 0 with some continuous nondecreasing function ω(T) ≥ 0 ∀T > 0. Under additional condition: →(·) → ω0 = const > 0 as T → 0 it is proved that there exist constant κ : 0 < κ < ∞, such that all solutions of mentioned equation (particularly, solutions, satisfying initial-boundary condition u|τ = ∞, where τ = (0; T) × ∂ω∪[ {0} ×Ω ) stay uniformly bounded in Ω0 := {x ϵ Ω : d(x) > κ∂0 2 } as t → T. Method of investigation is based on local energy estimates and is applicable for wide class of equations. So in the paper there are obtained similar suficient conditions of localization of singularity set of solutions near to the boundary of domain for equation with main part P0(u) = (|u|λ-1u)t - ∑ i N=1(|∇xu|q-1uxi )xi if 0 < λ ≤ q < p.