Existence And Gradient Estimates For Two Degenerate Parabolic Equations With A Nonlinear Convection Term

This dissertation is devoted to the existence and gradient estimates of the solutions or periodic solutions to two degenerate parabolic equations with a nonlinear convection term.Firstly, we are concerned with the initial-boundary value problem of the mean curvature type equations with a nonlinear convection termu_t - div {σ(|â–½u|²)â–½u} + b(u) ·Vu =0 x ∈ Ω, t >0u(x,0) = u₀(x) x∈Ω; u(x,t)=0 x0where Ω is a bounded domain in Rⁿ with some smooth boundary (?)Ω. σ(|â–½u|²) is a function like , and b(u) is a nonlinear vector field such that |b(u)| ≤ k|u|^β with some k >0 andβ≥ 0. For the initial data u₀ we only assume u₀ ∈ L^q(Ω). Utilizing the theory of degenerate parabolic equation, Galiardo-Nirenberg inequality, Moser technique and Aubin technique, we obtain the existence and gradient estimates of the solutions.Secondly, by Leray-Schauder fixed-point theorem and Moser technique, we discuss the existence and gradient estimates of the periodic solutions to the evolution m-Lalacian equations with a nonlinear convection termu_t-div{|Vu|^mâ–½u}+b(u)· â–½u = f(x,t)u^a in Ω×R¹u(x,t) = 0 on (?)Ω×R¹u(x,t + ω) = u(x,t) in Ω×R¹where Ω is a bounded domain in Rⁿ with smooth boundary (?)Ω and ω > 0, m > 0; f(x, t) > 0 is periodic in t with period ω > 0.