The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

Abstract

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space $C([\mathrm{0},T);H^s(R))\hspace{0.17em}\cap C^{\mathrm{1}}([\mathrm{0},T);H^{s-\mathrm{1}}(R))$ provided that the initial value lies in $H^s(R)$ with $s>\mathrm{3}/\mathrm{2}$. Meanwhile, we prove the existence of global weak solutions in $L^{\mathrm{\infty }}([\mathrm{0},\mathrm{\infty });L^{\mathrm{2}}(R))$ for the equation.