scholarly journals Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

1968 ◽  
Vol 13 (6) ◽  
pp. 466-477
Author(s):  
Věnceslava Šťastnová ◽  
Otto Vejvoda
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Periodic Solutions ◽  
Boundary Value ◽  
Nonlinear Heat Equation ◽  
Weakly Nonlinear

scholarly journals Periodic solutions of the boundary value problem for the nonlinear heat equation

1984 ◽  
Vol 30 (1) ◽  
pp. 99-110 ◽  
Author(s):  
M. N. Nkashama ◽  
M. Willem
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Periodic Solutions ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Nonlinear Heat Equation

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauder's techniques and coincidence degree.

On a nonlinear heat equation with a time delay

2002 ◽  
Vol 13 (3) ◽  
pp. 321-335 ◽  
Author(s):  
YUNKANG LIU
Keyword(s):  
Time Delay ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Heat Equation ◽  
Turbulent Shear ◽  
Heat And Mass Exchange ◽  
Initial Boundary Value

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

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