Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension. It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.
The global existence of solutions to the equations of one-dimensional compressible flow with density-dependent viscosity is proved. Specifically,the assumptions on initial data are that the modulo constant is stated at x=∞ ＋and x=-∞ ,which may be different ,the density and velocity are in L^z ,and the density is bounded above and below away from zero. The results also show that even under these conditions, neither vacuum states nor concentration states can be formed in finite time.
This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in H3 and Hs for semilinear wave equations with polynomial nonlinearity in u and (?)u. This ill-posed result concerns the focusing type equations with nonlinearity on u and (?)tu.
The authors study a resonant Klein-Gordon system with convenient nonlinearities in two space dimensions, prove that such a system has global solutions for small, smooth,compactly supported Cauchy data, and find that the asymptotic profile of the solution is quite different from that of the free solution.
