Existence of Traveling Waves of General Gray-Scott Models |
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Authors: | Zhi Zheng Xinfu Chen Yuanwei Qi Shulin Zhou |
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Institution: | 1.School of Mathematics,Peking University,Bejing,China;2.Department of Mathematics,University of Pittsburgh,Pittsburgh,USA;3.Department of Mathematics,University of Central Florida,Orlando,USA;4.School of Mathematical Sciences,Peking University,Bejing,China |
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Abstract: | This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m (\(m > 1\)) between two chemical species, a reactant A and an auto-catalyst B, \( A + m B \rightarrow (m+1) B\), and a super-linear decay of order \( n > 1\), \( B \rightarrow C\), where \( 1< n < m\). Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value \(u_0\) at \( - \infty \). The explicit bound is also derived in terms of \(u_0\) and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases. |
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