On the Cauchy Problem for a Nonlinearly Dispersive Wave Equation
نویسنده
چکیده
We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.
منابع مشابه
The Cauchy Problem and the Stability of Solitary Waves of a Hyperelastic Dispersive Equation
We prove that the Cauchy problem for a certain sixth order hyperelastic dispersive equation is globally well-posed in a natural space. We also show that there exist solitary wave solutions u(x, y, t) = φc(x − ct, y) that come from an associated variational problem. Such solitary waves are nonlinearly stable in the sense that if a solution is initially close to the set of such solitary waves, it...
متن کاملInstability of Solitary Waves for a Nonlinearly Dispersive Equation
Solitary-wave solutions of a nonlinearly dispersive evolution equation are considered. It is shown that these waves are unstable in a certain parameter range.
متن کاملPeriodic Wave Shock solutions of Burgers equations
In this paper we investigate the exact peroidic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the cauchy problem of the in...
متن کاملBlow up in the Cauchy problem for a nonlinearly damped wave equation
In this paper we consider the Cauchy problem for the nonlinearly damped wave equation with nonlinear source utt −∆u + aut|ut|m−2 = bu|u|p−2, p > m. We prove that given any time T > 0, there exist always initial data with sufficiently negative initial energy, for which the solution blows up in time ≤ T. This result improves an earlier one by Todorova [11].
متن کاملStability of Solitary Waves for a Nonlinearly Dispersive Equation
Solitary-wave solutions of a nonlinearly dispersive equation are considered. It is found that solitary waves are peaked or smooth waves, depending on the wave speed. The stability of the smooth solitary waves also depends on the wave speed. Orbital stability is proved for some wave speeds, while instability is proved for others.
متن کامل