The bistable reaction-diffusion-convection equation is considered.
Stationary traveling waves of above equation are proved to exist when the nonlinear flux function is symmetric and the reaction term is antisymmetric about the zero state.
Solutions of initial value problems tends to almost piecewise constant functions within small time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts are studied by asymptotic expansion. The equation for the motion of the front is obtained.
In the case of Burgers' flux with specific bistable reaction, the front motion equation takes more explicit form. The front's speed is shown to be a sum of the mean curvature of the front and a delicate term depending on the width of the planar traveling in the normal direction of the front, which contribute to the shrinkage of closed curves. An ellipse in two-dimensional case is found to preserve its shape while shrinking.
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