Asymptotic behavior and blow-up of solutions for the viscous diffusion equation

We study the initial–boundary value problem of the viscous diffusion equations which include GBBM equations, Sobolov–Galpern equations and some standard nonlinear diffusion equations as special cases. By using the integral estimate method and the eigenfunction method we prove that when the nonlinear terms satisfy some conditions the solutions of the problem decay to zero according to the exponent of tt. And when the nonlinear terms of the equation satisfy some other conditions the solutions blow up in finite time. Then the known results are improved and generalized.