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Bounds of global solutions of parabolic problems with nonlinear boundary conditions

We prove a priori bounds for global solutions of the linear equation ∂ t u = Δu - au in Ω × R + complemented by the nonlinear boundary condition ∂ v u = |u| p-1 u and the initial condition u(x, 0) = u 0 (x). Here Ω is a smoothly bounded domain in R n , a > 0, p > 1, p(n - 2) < n. Our bounds are uniform for bounded sets of initial conditions. In the case of positive solutions these bounds do not depend on initial conditions at all.