Traveling wave solutions of reaction-diffusion equations with x-dependent combustion type nonlinearities
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Date
2016-03
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Abstract
We investigate the existence and uniqueness of traveling wave solutions of
the reaction-diffusion equation in periodic heterogeneous media. The reaction-diffusion
equation is considered in nondivergence form with no first order term.
Our traveling wave problem is considered in similar form in [1] by Xin in
the special case that the reaction-term is given by a combustion nonlinearity
ƒ = ƒ(u).
We prove the existence of traveling wave solutions in case of a class of nonlinearities
ƒ = ƒ(x, u), which are a generalization of a combustion nonlinearity.
In particular, ƒ is allowed to depend explicitly on x. In case of an additional
assumption on ƒ, we also prove a monotonicity result and a uniqueness result.
References
[1] X. Xin. Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity. Indiana Univ. Math. J., 40(3):985–1008, 1991.
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Keywords
Traveling wave, Reaction-diffusion equation, Combustion nonlinearity