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where u0(x) are assumed to be continuous on Ω and μ is a constant. The function f(z, y) is defined for z ∈ ∂Ω and y ∈ Ω and continuous functions in y ∈ Ω for each z ∈ ∂Ω. Since for each z ∈ ∂Ω, f(z, y) plays the weight of integration in (1.1), the function f(z, y) is called weights throughout this paper. Denote DT = Ω × (0, T ) and DT ∪ ΓT = Ω × [0, T ). The variable z stands for a generic point of boundary ∂Ω. The large time behavior of the solution u to Problem (1.1) is studied by taking an upper bound for u in Section 2. It is shown that the solution of Problem (1.1) with large weights f(z, ·) for each z ∈ ∂Ω has an exponential lower bound in Section 3. Moreover, it is shown that the difference between maximum and minimum value on the boundary ∂Ω can decrease if the weights are zero on a nonempty subset of ∂Ω in z and zero on the boundary in y, that is, f(z, y) = 0, z ∈ Γ ⊂ ∂Ω for all y ∈ Ω and f(z, y) = 0, y ∈ ∂Ω.

Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions