Weak solutions to quasilinear wave equations of Klein-Gordon or Sine-Gordon type and relaxation to reaction-diffusion equations

1997 ◽  
Vol 4 (4) ◽  
pp. 439-457 ◽  
Author(s):  
Bruno Rubino
Keyword(s):  
Weak Solutions ◽  
Wave Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Klein Gordon ◽  
Quasilinear Wave Equations

scholarly journals NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

2013 ◽  
Vol 54 (3) ◽  
pp. 153-170 ◽  
Author(s):  
RUNZHANG XU ◽  
YANBING YANG ◽  
SHAOHUA CHEN ◽  
JIA SU ◽  
JIHONG SHEN ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Wave Equations ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Wave Equations ◽  
Initial Boundary Value

AbstractThis paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

scholarly journals Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Mark O. Gluzman ◽  
Nataliia V. Gorban ◽  
Pavlo O. Kasyanov
Keyword(s):  
Weak Solutions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Balance Model ◽  
Time Behavior ◽  
Differential Reaction ◽  
Regularity Properties ◽  
Trajectory Attractors

AbstractIn this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.

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