The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

AbstractWe consider the nonlinear parabolic equation ut = (k(u)ux)x + b(u)x, where u = u(x, t, x ϵ R1, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation ut = (unux)x + (uβ)x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations