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In this paper, we prove that the Cauchy problem for a non-linear wave equation (1)has a unique global generalized solutions in and a unique global classical solution in . We also prove that the Cauchy problem for the equation (1) admits a unique global generalized solution in C2([0, ∞); Wm, p(R) ∩ L∞(R))(m ≥ 0 is an integer,1 ≤ p ≤ ∞) and a unique global classical solution in . [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for the following damped generalized IMBq equation, utt-uxx-uxxtt+ν2uxxt=f(u)xx,x∈R,t>0, admits a unique global generalized solution in C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(1≤p≤∞,m≥0) and a unique global classical solution in C3([0,∞);Wm,p(R)∩L∞(R)∩L2(R))(m>2+1p). Moreover, the blow up of the solution for the Cauchy problem of damped generalized IMBq equation is studied. We also prove that the Cauchy problem of the above-mentioned equation has a unique global generalized solution in C2([0,∞);Hs(R))(s>12) and a unique global classical solution in C2([0,∞);Hs(R))(s>52), and discuss the blow up of the solution. [ABSTRACT FROM AUTHOR]

Abstract: The existence and uniqueness of the local generalized solution and the local classical solution to the Cauchy problem for three-dimensional damped nonlinear hyperbolic equation are proved. This paper arrives at some sufficient conditions for blow-up of the solution in finite time and an example. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation admits a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation has a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions. [Copyright &y& Elsevier]

In this paper, the global existence of small amplitude solution for the Cauchy problem of the multidimensional generalized IMBq equation is proved. Moreover, we obtain a nonlinear scattering result of the Cauchy problem of the IMBq equation for small initial data. [Copyright &y& Elsevier]

In this paper, the existence and the uniqueness of the global strong solution and the global classical solution for the Cauchy problem of the multidimensional generalized IMBq equation are proved. The nonexistence of the global solution for the Cauchy problem of the generalized IMBq equation is discussed. [Copyright &y& Elsevier]