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Existence and uniqueness for a nonlinear dispersive equation. (English) Zbl 1029.35203

Summary: We study the existence and uniqueness properties of solutions of some nonlinear dispersive evolution equations. We consider the equation \[ {\partial u\over\partial t}+{\partial \over\partial x} \bigl[f(u)\bigr] =\varepsilon {\partial\over \partial x}\left[g \left( {\partial u\over \partial x}\right) \right]- \delta{\partial^3 u\over\partial x^3} \qquad u(x,0)= \varphi(x) \] with \(x\in\mathbb{R}\), \(T\) an arbitrary positive time and \(t\in [0,T]\). The flux \(f=f(u)\) and the (degenerate) viscosity \(g=g(\lambda)\) are given smooth functions satisfying certain assumptions. We present a result that permits to obtain gain of regularity for equation (1), motivated by the results obtained by W. Craig, T. Kappeler and W. Strauss [Ann. Inst. Henri Poincaré, Anal. Nonlinéaire 9, 147-186 (1992; Zbl 0764.35021)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
47J35 Nonlinear evolution equations
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0764.35021
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