Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

Abstract

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation v t − α v x x t − β v x x + γ v x + f ( v ) x = φ ( v x ) x + g ( v ) − α g ( v ) x x admits a unique global generalized solution in C 1 ( [ 0 , ∞ ) ; H s ( R ) ) , a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation v t − α v x x t − β v x x = g ( v ) − α g ( v ) x x has a unique global generalized solution in C 1 ( [ 0 , ∞ ) ; W m , p ( R ) ∩ L ∞ ( R ) ) , a unique global classical solution and asymptotic behavior of the solution.