Time periodic smooth solutions of hyperbolic quasi-linear equations with dissipation term and their approximation by parabolic equations

1985 ◽  
Vol 140 (1) ◽  
pp. 331-344 ◽  
Author(s):  
Albert J. Milani
Keyword(s):  
Parabolic Equations ◽  
Linear Equations ◽  
Smooth Solutions ◽  
Dissipation Term ◽  
Time Periodic

Solutions in the large for certain nonlinear parabolic equations and applications

1996 ◽  
Vol 126 (1) ◽  
pp. 19-45 ◽  
Author(s):  
Huijiang Zhao ◽  
Changjiang Zhu
Keyword(s):  
Parabolic Equations ◽  
Hyperbolic Conservation Laws ◽  
Parabolic Systems ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Parabolic Systems ◽  
Smooth Solutions ◽  
Positive Matrix ◽  
Hyperbolic Conservation ◽  
Nonlinear Parabolic ◽  
Nonlinear Hyperbolic Conservation Laws

We prove some results on the global existence of smooth solutions for certain nonlinear parabolic systems of the form Ut + A(U)Ux = DUxx. Here U is a vector and A(U), D are matrices with D a constant, positive matrix. We show how to use our results to study the global continuous (or generalised) solutions to the corresponding nonlinear hyperbolic conservation laws and a conjecture is given.

scholarly journals On the maximum principle for a class of nonlinear parabolicequations

2017 ◽  
Vol 21 (6) ◽  
pp. 89-92
Author(s):  
A.A. Kon’kov
Keyword(s):  
Maximum Principle ◽  
Nonlinear Equations ◽  
Wide Class ◽  
Lower Order ◽  
Parabolic Equations ◽  
Linear Equations ◽  
Nonlinear Parabolic Equations ◽  
Spatial Variables ◽  
The Maximum Principle ◽  
Nonlinear Parabolic

In this paper, we consider solutions of nonlinear parabolic equations in the half-space.It is well-known that, in the case of linear equations, one needs to impose additional conditions on solutions for the validity of the maximum principle. The most famous of them are the conditions of Tikhonov and T¨acklind. We show that such restrictions are not needed for a wide class of nonlinear equations. In so doing, the coefficients of lower-order derivatives can grow arbitrarily as the spatial variables tend to infinity.We give an example which demonstrates an application of the obtained re- sults for nonlinearities of the Emden - Fowler type.

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