Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity

Abstract

In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

“…for sufficiently large n. Since W is an open set in W m+1 0 (Ω), from u 0 (x) ∈ W and (13), we get u n (0) ∈ W , for sufficiently large n. Next we prove that u n (t) ∈ W ,for sufficiently large n. In fact, if it is false, then there exists a t 0 > 0 such that u n (t 0 ) ∈ ∂W , i.e., I(u n (t 0 )) = 0 and u n (t 0 ) = 0, which means u n (t 0 ) ∈ N . Hence we have J(u n (t 0 ) ≥ d, which contradicts (16).…”

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

“…Ruying Xue considered problem (1) with the nonlinearity f (u) = u k+1 , k = 1, 2, • • • , and proved the global existence for the small initial data and k > 4. The problem (1) with much more general nonlinear case f (u) = ±|u| p and |u| p−1 u was then considered in [15], and also by the potential well theory [4], [9], [8], [22], in the sub-critical initial energy case, i.e. E(0) ≤ d, the global and non-global solutions were classified in term of initial data.…”

A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

“…Later, in the case of infinite dimension in reference [25], Lian et al modified the potential well method and combined with Sobolev inequality to obtain the global existence of the solution and the blow up result under the condition of different initial energy(E(0) < d, E(0) = d, E(0) > d). When u ln |u| k in problem (3) becomes |u| p ln |u|, the problem is also considered by Lian et al [26], and they establish the global existence and finite time blow up of solutions at three different energy levels.…”