Critical Ostrovskiy-type equation on a half-line. (English) Zbl 1240.35243

The author studies the global existence and asymptotic behaviour of solutions to \[ u_t +\lambda | u| ^{\sigma }u + \int _0^{+\infty } {u_{yy}(y,t)\over \sqrt {| x-y| }}\,dy = 0, \, x >0,\, t>0, \] satisfying \( u(x,0) = u_0(x)\) for \(x>0\) and \(u_x(0,t) = 0\) for \(t>0\), where \(\lambda \) is a complex number and \(\sigma >0\). Starting with the linear problem (\(\lambda =0\)), the existence of local solutions to the original problem is derived by applying the contraction mapping principle. Then, for sufficiently small initial data \(u_0\), the existence of a unique global solution is derived along with the type of its asymptotic behaviour in time.


35K55 Nonlinear parabolic equations
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs