Two Properties of Weak Solutions for a Nonlinear Pseudoparabolic Equation with Variable Exponent

The asymptotic behavior and monotonicity support of weak solutions to the initial-boundary value problem for a class of nonlinear pseudoparabolic equation with variable exponent are considered. The energy equality of weak solutions is obtained by using convexity of functional. By this and Poincaré’s and H?lder’s inequalities the asymptotic behavior of weak solutions to the nonlinear pseudoparabolic equation with variable exponent is discussed. The comparison principle is obtained by using of Steklov mean property of weak solutions. By this comparison principle, the monotonicity support of weak solutions is proved in 1-dimension.