Non-planar Traveling Wave Solutions Of Bistable Reaction-Diffusion Equations

The theory of nonlinear parabolic equations is one of the most important topics of the modern mathematics research. As a kind of typical nonlinear parabolic equa-tions, reaction-diffusion equations can be used to explain the natural phenomenon found in many disciplines, such as the heat conduction in physics, the material con-centration change in chemical reaction, the species invasion in biology and so on. Since1937, planar traveling wave solutions of reaction-diffusion equations have been intensively studied. Recently, the study of multidimensional traveling wave solutions (non-planar traveling wave solutions) of reaction-diffusion equations in multidimen-sional space has attracted much attention, because many practical problems from the field of physics, chemistry and ecology are high-dimensional problems. Compared with the planar ones, the profiles of non-planar traveling wave solutions become more complicated, and there are various of new types of traveling wave solutions in multidimensional space. Therefore, it is very challenging and valuable to study non-planar traveling wave solutions of reaction-diffusion equations and characterize their qualitative properties. On the other hand, since some of the environments in reality change periodically, for instance, the seasonal variation in nature, con-sequently, it is necessary and important to establish traveling wave solutions for the non-autonomous reaction-diffusion equations, especially for the time-periodic reaction-diffusion equations. This thesis is mainly concerned with the stability of non-planar traveling wave solutions for autonomous reaction-diffusion equations and non-planar traveling wave solutions for the time-periodic case. Here, it should be pointed out that the level sets of non-planar traveling wave solutions are no longer parallel hyperplanes, which make a striking contrast with the planar ones. A typ-ical example is the two-dimensional V-shaped traveling fronts whose level sets are V-shaped curves.First of all. we deal with the global exponential stability of V-shaped traveling fronts. By establishing the comparison of solutions for the corresponding initial value problem and using the squeezing technique combined with the comparison principle, we show that such a V-shaped traveling front is globally exponential stable.Secondly, we consider the multidimensional stability of V-shaped traveling fronts. We prove that the V-shaped traveling front is asymptotically stable un-der the initial perturbations decaying at infinity. In particular, if the initial values are some perturbations of the planar traveling front in L1∩L∞, then the V-shaped traveling front is algebraically stable, and the convergence rate is optimal in a cer-tain sense. Furthermore, we show that the V-shaped traveling front is not stable under general bounded perturbations. From the perspective of dynamical systems, this means that the solutions of the initial value problem contain at least two trans-lation of V-shaped traveling fronts in the topology of Lloc∞.Finally, we study the periodic pyramidal traveling front of bistable reaction-diffusion equations with time periodic nonlinearity. By constructing a mollified pyramid and various types of supersolutions and subsolutions, we obtain the exis-tence of the three dimensional periodic pyramidal traveling front. We also show that the periodic pyramidal traveling front is asymptotically stable by using the known stability results for the two dimensional V-shaped traveling front and finding a two dimensional V-shaped traveling front on the edge of the pyramid. Moreover, we extent the existence results of the three dimensional periodic pyramidal traveling front to higher dimensional spaces.