Traveling wave solutions for a nonlocal phase-field system
The authors study a nonlocal phase-field system which couples the temperature \(\theta \) and an order parameter \(\varphi \) according to the evolution equations: \[ \partial \varphi /\partial t=J*\varphi -\varphi +f\left( \varphi \right) +\theta, \qquad \partial \left( \theta +l\varphi \right) /\partial t=\Delta \theta. \] They first justify this model giving the expression of an associated free energy. The existence and uniqueness results have already been proved in a previous work by the authors in \textit{P.W. Bates} (ed.) et al. [Proceedings of the US-Chinese conference: Differential equations and applications, Hangzhou, China, 1996. Cambridge, MA: International Press, 14--21 (1997; Zbl 0938.35035)]. In the present paper, the authors prove the existence of traveling wave solutions, with a positive constant and unknown speed \(c\), thus restricting the problem to the one-dimensional case. Their main theorem proves the existence of such traveling waves provided that the parameter \(l\), which represents the latent temperature, is small enough. Solving the second above equation in this framework and taking \(l=0\), the authors first prove the existence of solutions \(\left( \varphi _0,c_0\right) \) of the first equation which can be written as : \(c\varphi '+J*\varphi -\varphi +f\left( \varphi \right) +\theta \left( -\infty \right) \), where \(\varphi _0\) is smooth if \(c_0\) is positive. Then they study the linear operator which associates to every continuously differentiable function \(\psi \) vanishing at \(\pm \infty \) together with its first-order derivative, the function \(c\psi '+J*\psi -\psi +f'\left( \varphi _0\right) \psi \). The proof is then based on Fredholm theory and the implicit function theorem. The end of the paper presents some numerical results for different expressions of the function \(f\), which is supposed to be bistable, and different values of the parameter \(l\).
scientific article; zbMATH DE number 1791238