Advances in Differential Equations

Traveling fronts of a conservation law with hyper-dissipation

Jerry L. Bona and Fred B. Weissler

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Studied here are traveling-front solutions $\phi_{\epsilon}(x - {c} t)$ of a conservation law with hyper-dissipation appended. The evolution equation in question is a simple conservation law with a fourth-order dissipative term, namely $$u_t + 2uu_x + {\epsilon} u_{xxxx} = 0 , $$ where ${\epsilon} > 0$. The traveling front is restricted by the asymptotic conditions $\phi_{\epsilon}(x) \to L_\pm$ as $x \to \pm\infty$, where $L_+ < L_-$, and the symmetry condition $\phi_{\epsilon}(x) + \phi_{\epsilon}(-x) = L_- + L_+$ for all $x \in \mathbb R$. Such fronts are shown to exist and proven to be unique. Unlike the corresponding fronts for the Burgers' equation, they do not decay monotonically to their asymptotic states, but oscillate infinitely often around them. Despite this oscillation, it is also shown that $\phi_{\epsilon}(x) \to L_+$ as ${\epsilon} \to 0$, for all $x > 0$, and $\phi_{\epsilon}(x) \to L_-$ as ${\epsilon} \to 0$, for all $x < 0$.

Article information

Adv. Differential Equations Volume 16, Number 9/10 (2011), 917-935.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C37: Homoclinic and heteroclinic solutions 34D05: Asymptotic properties 35A24: Methods of ordinary differential equations 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35C07: Traveling wave solutions 35F21: Hamilton-Jacobi equations 35K25: Higher-order parabolic equations 35K55: Nonlinear parabolic equations


Bona, Jerry L.; Weissler, Fred B. Traveling fronts of a conservation law with hyper-dissipation. Adv. Differential Equations 16 (2011), no. 9/10, 917--935.

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