Existence and uniqueness of entire solutions for a reaction-diffusion equation
By Xinfu Chen, Jong-Shenq Guo. 2005
114 citations recorded by the source corpus.
Abstract We consider entire solutions of u t = u xx - f ( u ) , i.e. solutions that exist for all ( x , t ) ∈ R 2 , where f ( 0 ) = f ( 1 ) = 0 f ′ ( 0 ) . In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f ′ ( 1 ) > 0 , we show that such entire solution exists and is unique up to space–time translations. In the case f ′ ( 1 ) 0 , we derive two families of such entire solutions. In the first family, one cannot be any space–time translation of the other. Yet all entire solutions in the second family only differ by a space–time translation.
Subfields: Applied Mathematics, Control and Systems Engineering, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/7eec9d95940b734fabc38cc98b52cdf7d66341b7
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In Applied Mathematics
In Control and Systems Engineering
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Citations in the corpus
- Entire solutions in reaction-advection-diffusion equations in cylinders (2008, unknown): “We should also mention that Morita and Ninomiya [25] obtained some novel entire solutions which are completely different from these observed in [10,11,16,18,19,37].”
- Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity (2008, background): “In particular, Chen and Guo [6], Chen et al. [7], Fukao et al. [10], Guo and Morita [13], Hamel and Nadirashvili [14,15] and Morita and Ninomiya [20] have shown that the study of entire solutions is essential for a full understanding of the transient dynamics and the structures of the global attract”