Multidimensional Stability Of Planar Waves In The Buffered Mono-stable System

Theory of reaction-diffusion equations is an important content of modernmathematics.In recent years, many scholars study dynamic character in thereaction-diffusion equations with buffered mono-stable and buffered bistable,andobtain the buffered mono-stable system and buffered bistable system,and bufferinghas a great influence on traveling wave of the system. As a special solution ofreaction-diffusion equations, traveling wave has widely used in subjects andtraveling wave can describe a lot of nature of the equations,so it is of greatsignificance on the study of traveling wave. But so far,the study result of travelingwave of the stability is limited in one dimensional space, but due to many areas ofresearch is carried out in multidimension space.so the stability of traveling wave inmultidimensional space makes more sense.So in this paper studiesmultidimentional stability of planar waves in the buffered mono-stable systems.First of all,we discuss the asymptotic behavior of traveling wave fronts.Secondly.wediscusse the relationship between the planar wave and the perturbation solution ofinitial value under certain conditions.1. Give the definition of planar wave.For buffered mono-stable systems, usestandard asymptotic theory and get the asymptotic property of the traveling wavefronts.2. According to the asymptotic property of the traveling wave fronts,useupper and lower solution to construct a set of supersolution and subsolution.Perturbation of initial value converges to planar wave by comparison principle, thatis,when|x|+|y|â†'∞, planar wave is asymptotic stability under the initialdsturbance.3. Reconstruct a set of supersolution and subsolution. according to the nature of the solution of the Cauchy problem,obtain the convergence speed of the planarwave. And this is the best convergence rate under certain conditions.4. In the case of different initial values,use the previous structure onsupersolution and subsolution, obtain the existence of a solution that vibrationpermanently between two planar wave,the solution can be seen the counter-exampleof asymptotic stability of the plane wave solutions.