scholarly journals The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Juan Wang ◽  
Jinlin Yang ◽  
Xinzhi Liu
Keyword(s):  
Boundary Value Problem ◽  
Nonlinear Elliptic Equation ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Type Problem ◽  
Neumann Boundary Value Problem ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Neumann Type ◽  
Monge Ampere

We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.

scholarly journals POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

2012 ◽  
Vol 86 (2) ◽  
pp. 244-253 ◽  
Author(s):  
YANG-WEN ZHANG ◽  
HONG-XU LI
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Existence And Uniqueness Theorem ◽  
Neumann Boundary Value Problem ◽  
Image Position ◽  
Continuous Dependence Of Solutions

AbstractIn this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

2021 ◽  
pp. 1-35
Author(s):  
Nakao Hayashi ◽  
Elena I. Kaikina ◽  
Pavel I. Naumkin ◽  
Takayoshi Ogawa
Keyword(s):  
Boundary Value Problem ◽  
Large Time ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Asymptotic Behavior Of Solutions ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Neumann Boundary Value Problem ◽  
Time Existence ◽  
Time Asymptotic Behavior

We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) = γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0.

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