The Local Strong and Weak Solutions for a Generalized Pseudoparabolic Equation

Abstract

The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space $C\left(\right[0,T);H^s(R\left)\right){\bigcap }_{}{C}^1\left(\right[0,T);H^{s-1}(R\left)\right)$ with $s>3/2$, while the existence of local weak solutions is proved in the space $H^s\left(R\right)$ with $1\le s\le 3/2$. Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space $C\left(\right[0,\infty );H^s(R\left)\right){\bigcap }_{}{C}^1\left(\right[0,\infty );H^{s-1}(R\left)\right)$ with $s\ge 2$.