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We consider a system of semilinear parabolic equations ut = Δг + с1(x,t)vp, vt = Δv + c2(x,t)uq, (x,t) ∈ Ω × (0,+∞) with nonlinear nonlocal boundary conditions ∂u/∂η = ∫k1(x,y,t)um(y,t)dy, ∂v/∂η = ∫Ωk2(x,y,t)vn(y,t)dy, (x,t) ∈ ∂Ω × (0,+∞) and initial data u(x,0) = u0(x), v(x,0) = v0(x), x ∈ Ω, where p, q, m, n are positive constants, Ω is bounded domain in RN(N ≥ 1) with a smooth boundary ∂Ω, η is unit outward normal on ∂Ω. Nonnegative locally Hӧlder continuous functions ci(x,t),i = 1,2, are defined for x ∈ Ω, t ≥ 0; nonnegative continuous functions ki(x,y,t),i = 1,2 are defined for x ∈ ∂Ω, y ∈ Ω, t ≥ 0; nonnegative continuous functions u0(x),v0(x) are defined for x ∈ Ω and satisfy the conditions ∂u0(x)/∂η = ∫k1(x,y,0)um0(y)dy, ∂v0(x)/∂η = ∫Ωk2(x,y,0)vn0(y)dy for x ∈ ∂Ω. In the paper blow-up set of classical solutions is investigated. It is established that blow-up of the solutions can occur only on the boundary ∂Ω if max(p,q) ≤ 1, max(m,n)> 1 and under certain conditions for the coefficients ki(x,y,t),i = 1,2.

We consider a nonlinear nonlocal parabolic equation ut = Δu + a(x,t)ur∫Ωup(y,t)dy - b(x,t)uq for (x,t) ∈ Ω × (0,+∞) with nonlinear nonlocal boundary condition u(x,t)|∂Ω × (0,+∞) = ∫Ωk(x,y,t)ul(y,t)dy and initial data u(x,0) = u0(x), x ∈ Ω, where r, p, q, l are positive constants; Ω is a bounded domain in Rn with smooth boundary ∂Ω. Nonnegative functions a(x,t) and b(x,t) are defined for x ∈ Ω, t ≥ 0 and local Hӧlder continuous, nonnegative continuous function k(x,y,t) is defined for x ∈ ∂Ω, y ∈ Ω, t ≥ 0, nonnegative continuous function u0(x) is defined for x ∈ Ω and satisfies the condition u0(x) = ∫Ωk(x,y,0)u0t(y)dy for x ∈ ∂Ω. In this paper we study classical solutions. To prove the existence of a local maximal solution, we consider the regularization of the original problem. We establish the existence of a local solution of the regularized problem and the convergence of solutions of this problem to a local maximal solution of the original problem. We introduce definitions of a supersolution and a subsolution. It is shown that a supersolution is not less than a subsolution. We establish the positiveness of solutions of the problem with nontrivial initial data under certain conditions on the data of the problem. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem.

The results of investigation of iron-containing superconductors LаFeAsO(1 – х)Fх (х = 0,11…0,15) are presented, based on joint analysis of Mössbauer spectroscopy, XRD, magnetic and electrophysical methods. A full suppression of magnetic order in these samples with the appearance of the Kondo effect is shown at x = 0.15 and T < 40 K, as well as the transformation of Kondo effect into high-temperature superconductivity at х = 0.11…0.14 and T < 25 K. The revealed effects allow assuming a significant contribution of interaction between magnetic moments of conducting electrons and impurities in the Cooper pair formation for iron-containing superconductors.

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