Global Classical Solutions And The Stability Of Traveling Wave Solutions Of The Initial-boundary Value Problem For The Equation Of Physical Models

In this paper, we study the important Born-Infeld type equations in applied physics.We get the global existence of classical solutions and the existence and stability of travelingwave solutions of the mixed initial-boundary value problem for some kinds of Born-Infeldtype equation under the appropriate assumptions.In the physics and other areas, most problem can be reduced to a mathematical prob-lem. Recently, Born-Infeld theory has received much attention mainly due to the fact thatthe Born-Infeld type Lagrangians naturally appear in string theory and relativity theory.This also triggers the revival of interests in the original Born-Infeld electromagnetism andthe exploration of Born-Infeld gauge theory. The equations of relativistic string moving andtime-like extremal surface in Minkowski space are Born-Infeld type equations.The mixed initial-boundary value problem of vibrating string equation plays an impor-tant role in string theory and particle physics. Therefore, it is particularly important to thestudy of its related problems. We get the global classical solutions for the initial-boundaryvalue problem of vibrating string equation under the appropriate assumptions on the initialand boundary data. As an important and non-trivial model in Born-Infeld type equations,extremal surface in Minkowski space is also an significant topic in mathematics. It alsoplays an important role in fluids mechanics, electromagnetism as well as in the theory ofblack hole. Using the related methods of quasilinear hyperbolic systems, we consider thestability of traveling wave solutions of the mixed initial boundary value problem for theequation of time-like extremal surface in Minkowski space R1+(1+n).