A Class Of Traveling Wave Solutions Of The Viscous Balance Law Side The Existence And Stability

This paper is concerned with,the existence and stability of travelling wave solutions for the viscous balance lawwhich is an extension of viscous conservation law where a reaction term g(u) is added.l)the existence of travelling wave solutionsBy geometric singular perturbation method,we investigate the existence of travelling waves (A2) connecting a saddle point and a sink point and the existence of viscous shock waves C connecting two adjacent or disadjacent saddle points.By giving a detailed analysis of the fast and slow manifolds and verifying the transversality of the intersection of singular stable and unstable manifolds of the reduced problem along the singular heteroclinic orbit,we obtain the existence of travelling waves (A2) in the case of a convex flow function / and that of viscous shock waves C under the assumption that f" is bounded. Our results also show that the travellings waves (A2) and viscous shock waves C of viscous problem (I)are close to the discontinuous waves (A2) and shock waves C of the corresponding invicid problem respectively.The above results we obtained make up for the insufficiency in the proof of [11] and we extend the result of [11] on the existence of viscous shock waves to more general case.In addition,by phase plane method we obtain the existence of travelling fronts (Al) connecting two adjacent equilibria under the assumptions that f" is bounded and that of viscous shock waves C connecting two adjacent saddle points with / a convex function.2)the stability of travelling wave solutionsBy analytic semigroup theory and a detailed spectral analysis,the asymptotic stability of the travelling waves we obtained above is proved.For the travelling waves (Al) and (A2),we consider the distribution of spectra for the linearized operator in some weighted spaces. By choosing some appropriateexponential weight functions we prove that the essential spectra and the eigenvalues (except the simple zero eigenvalue) have negative real parts,thus we get the locally asymptotically exponential stability of travelling waves (Al) and (A2) in some weighted spaces.For viscous shock waves C,by the spectral analysis we prove that in L2(R) space the essential spectra and the eigenvalues (except the simple zero eigenvalue) of the linearized operator have negative real parts,thus we show that the viscous shock waves C is locally asymptotically exponentially stable in L2(R) space.