On the Study of Global Solutions for a Nonlinear Equation

الملخص EN

The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value v 0 ( x ) satisfies a sign condition and v 0 ( x ) ∈ H s ( R ) with s > 3 / 2 .

If the initial value v 0 ( x ) ∈ H s ( R ) ( 1 ≤ s ≤ 3 / 2 ) and the mean function of ( 1 - ∂ x 2 ) v 0 ( x ) satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space v ( t , x ) ∈ L 2 ( [ 0 , + ∞ ) , H s ( R ) ) in the sense of distribution and v x ∈ L ∞ ( [ 0 , + ∞ ) × R ) .