Nonplanar Traveling Fronts Of The Belousov-Zhabotinskii Reaction System And A Class Of Nonlocal Dispersal Equations With The Bistable Nonlinearity

In the past twenty years,the theory of nonplanar traveling wave solutions of parabol-ic equations has developed rapidly.This is because nonplanar waves are widely found in natural sciences,such as chemical waves in chemical reactions,interfacial phe-nomena in physics,and bio-waves in living systems and so on,thus studies of the existence,uniqueness and stability on nonplanar waves have important theoretical and practical significance.Traveling wave solutions are a special kind of solutions to the reaction diffusion equations,which keep their shapes and velocity during the propagation process.Therefore,they can describe the oscillation phenomenon and finite velocity propagation phenomenon in nature very well.A nonplanar traveling wave solution is a traveling wave solution in high-dimensional space,and its level sets are no longer parallel hyperplanes,but more complex shapes such as V shapes,pyramidal shapes,conical shapes or other asymmetric convex geometric shapes.Thus,compared to one-dimensional traveling waves solutions whose theory are rela-tively complete,theoretical studies on the nonplanar ones are still lacked and more challenging.In this dissertation,we mainly study the nonplanar traveling wave so-lutions of a class of nonlocal dispersal equations with the bistable nonlinearity and the V-shaped traveling waves of a class of Belousov-Zhabotiskii chemical reaction diffusion system.First,we study the nonplanar traveling fronts(V-shaped traveling fronts)of the Belousov-Zhabotinskii(BZ for short)reaction system in the two dimensional space.By constructing proper super and sub solutions,then applying the comparison prin-ciple and the monotone iteration procedure,we establish the existence result of the V-shaped traveling fronts.Next,we study the global asymptotic stability of the V-shaped traveling fronts in the two dimensional space.Under the condition that the initial perturbation is not less than zero and decays to zero at infinity,by constructing a series of super and subsolutions and then applying the comparison principle,we prove the stability result;while when the initial perturbation is not lager than zero and decays to zero at infinity,the prove is more complicated.First,we introduce another kind of super and subsolutions as well as the corresponding comparison principle.Then,we construct a series of mild subsolutions,with the help of which as well as the corresponding comparison principle we prove the stability result.Combining the above two cases,we prove the global stability result under the condition that the the initial perturbation decays to zero at infinity.On the other hand,we study the existence as well as the qualitative properties of nonplanar traveling fronts of a class of nonlocal dispersal equations with the bistable nonlinearity.First,by constructing proper super and subsolutions and applying the comparison principle,we obtain the existence result of pyramidal traveling fronts in the three dimensional space in the weak sense(i.e.,in the integral sense),then we obtain the the existence result of pyramidal traveling fronts in the classical sense by the bootstrap argument.With the aid of relation of the super and subsolutions as well as their geometric shape,we further obtain the estimation of the global mean speed of the pyramidal traveling fronts,precisely,the global mean speed is equal to the speed of the planar traveling fronts.Based on the existence of pyramidal travel-ing fronts and with the help of the comparison principle,we construct a monotone function sequence of the pyramidal traveling fronts and then obtain the existence of conical traveling fronts by taking the limit of the function sequence.In addition,by means of the properties of the pyramidal traveling fronts,we obtain a series of properties of the conical traveling fronts.Parallel to the existence result of the pyramidal traveling fronts,it is easy to get the existence result as well as the global mean speed of the two dimensional V-shaped traveling fronts.Then and finally,we study the asymptotic stability of the V-shaped traveling fronts of the bistable nonlocal dispersal equations.Under the condition that the initial perturbation decays exponentially to zero,we prove the asymptotic stability of the V-shaped traveling fronts in the exponential weighted energy space by the weighted energy method,and the convergence rate is also obtained.