On the Study of Global Solutions for a Nonlinear Equation

Abstract

The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value $v_{\mathrm{0}}(x)$ satisfies a sign condition and $v_{\mathrm{0}}(x)\in H^s(R)$ with $s>\mathrm{3}/\mathrm{2}$. If the initial value $v_{\mathrm{0}}(x)\in H^s(R)(\mathrm{1}\le s\le \mathrm{3}/\mathrm{2})$ and the mean function of $(\mathrm{1}-{\partial }_x^{\mathrm{2}})v_{\mathrm{0}}(x)$ satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space $v(t,x)\in L^{\mathrm{2}}([\mathrm{0},+\mathrm{\infty }),H^s(R))$ in the sense of distribution and $v_x\in L^{\mathrm{\infty }}([\mathrm{0},+\mathrm{\infty })\times R)$.