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Existence and Gradient Estimates for Two Degenerate Parabolic Equations with a Nonlinear Convection Term

Author: LiuWenJun
Tutor: GuanPing
School: Southeast University
Course: Applied Mathematics
Keywords: Mean curvature type m-Laplacian type Nonlinear Convection Periodic solution Existence Gradient estimate Moser iteration Degenerate parabolic equation
CLC: O175.26
Type: Master's thesis
Year: 2005
Downloads: 33
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Abstract


This paper studies two categories with nonlinear convection degenerate parabolic equations , the existence of periodic solutions and the corresponding solution L of ∞ estimates. First , we consider a class of nonlinear convection mean curvature equation Dirichlet from Boundary Value u t - div { σ ( | ▽ u ??| the 2 ) ▽ u } b (u) · ▽ u = 0 x ∈ Ω, t> 0 u (x, 0) = u 0 (x) x ∈ Ω; u (x, t) = 0 x ∈ (?) Ω, t> 0 where Ω R n bounded domain with smooth boundary ; σ (| ▽ u | 2 ) to form a class 1 / ( 1 | ▽ u | 2 ) 1 /2 function ; b (u) as a vector-valued function , the meet | b ( u ) | ≤ κ | u | β , κ β is the one to determine the number and κ ≥ 0 , β ≥ 0 ; the initial value u 0 ∈ L q ( Ω ). Canonical theory of degenerate parabolic equations Galiardo - Nirenberg inequalities , Moser iterative technique and the Aubin compactness lemma we get to understand the existence and gradient estimation . Secondly, the Leray-Schauder fixed point theorem the Moser iterative techniques we discussed Another type satisfy the Dirichlet boundary conditions with nonlinear convective term m-Laplacian type development the equation u t < / sub> - div {| ▽ u | m ▽ u} b (u) · ▽ u = f (x, t) μ α in Ω × R 1 u (x, t) = 0 on (?) Ω × R 1 u (x, t ω) = u (x, t) in Ω × R 1 periodic solution existence gradient estimate . Bounded domain with smooth boundary where Ω R n ω > 0 , m > 0 ; f ( x, t ) > 0 on t cycle as a function of ω .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Parabolic equation
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