The Global Existence And Nonexistence Of The Initial Value Problem For The Rosenau Equation

We investigate the global existence, uniqueness and the finite time blow-up of the Cauchy problem of the n-dimensional (n ? 1) Rosenau equation utt By using the Fourier method, the Duhamel's principle, we get the linear estimate of the solution. The existence and uniqueness of local strong solutions are obtained by means of the contraction mapping principle. Moreover, we respectively provide the sufficient conditions of global solutions and finite time blow-up of solutions under some different conditions of the initial energy, through the method of the potential well. When the initial energy E(0) is more than the depth of the potential well d,there will be a global existence theorem and a finite time blow-up theorem, then when E(0) less than d, a global existence theorem will be presented.