The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

Abstract

A degenerate parabolic equation of the form (|v| β-1 v) t = div(b(x, t)|∇v| p(x,t)-2 ∇v) + ∇ g • ∇ γ (v) is considered, where g = {g i (x, t)}, γ (v) = {γ i (v)}. If the diffusion coefficient b(x, t) ≥ 0 is degenerate on the boundary, by adding some restrictions on b(x, t) and g, the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.

On the Boundary Value Condition of an Isotropic Parabolic Equation

“…The second one is that we have not used boundary value condition (3) throughout this paper; in other words, condition (9) may replace boundary value condition (3) in some way. Moreover, using some techniques developed by the second author in his work [10], in which the wellposedness of weak solutions to equation,…”

“…which arises in the phenomena of electrorheological fluids [6,7]. The existence of solutions of the initial-boundary value problem to this equation can be found in [8][9][10][11]. Also, one can refer to [12][13][14][15][16][17][18] for some other related works.…”