Heteroclinic travelling waves of gradient diffusion systems

Alikakos, Nicholas; Katzourakis, Nikolaos

doi:10.1090/S0002-9947-2010-04987-6

Heteroclinic travelling waves of gradient diffusion systems
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by Nicholas D. Alikakos and Nikolaos I. Katzourakis PDF

Trans. Amer. Math. Soc. 363 (2011), 1365-1397 Request permission

Abstract:

We establish the existence of travelling waves to the gradient system $u_t = u_{zz} - \nabla W(u)$ connecting two minima of $W$ when $u : \mathbb {R} \times (0,\infty ) \longrightarrow \mathbb {R}^N$; that is, we establish the existence of a pair $(U,c) \in [C^2(\mathbb {R})]^N \times (0,\infty )$, satisfying \[ \left \{\begin {array}{l} U_{xx} - \nabla W ( U ) = - c\; U_x,\\ U(\pm \infty ) = a^{\pm }, \end {array}\right . \] where $a^{\pm }$ are local minima of the potential $W \in C_{\textrm {loc}}^2(\mathbb {R}^N)$ with $W(a^-)< W(a^+)=0$ and $N \geq 1$. Our method is variational and based on the minimization of the functional $E_c (U) = \int _{\mathbb {R}}\Big \{ \frac {1}{2}|U_x|^2 + W( U ) \Big \}e^{cx} dx$ in the appropriate space setup. Following Alikakos and Fusco (2008), we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed.

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