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

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equationvt−αvxxt−βvxx+γvx+f(v)x=φ(vx)x+g(v)−αg(v)xxvt−αvxxt−βvxx+γvx+f(v)x=φ(vx)x+g(v)−αg(v)xx admits a unique global generalized solution in C1([0,∞);Hs(R))C1([0,∞);Hs(R)), a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equationvt−αvxxt−βvxx=g(v)−αg(v)xxvt−αvxxt−βvxx=g(v)−αg(v)xx has a unique global generalized solution in C1([0,∞);Wm,p(R)∩L∞(R))C1([0,∞);Wm,p(R)∩L∞(R)), a unique global classical solution and asymptotic behavior of the solution.