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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.

MSC:

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