Abstract
In this paper, we obtain the precisely sharp criteria of blow-up and global existence for the cubic nonlinear beam equation in the H^{2} energy-critical and H^{2} sub-critical cases, respectively.
Highlights
In this paper, we consider the Cauchy problem of the nonlinear beam equation ⎧ ⎨ ∂2 ∂t2 u +2u + mu – |u|2u = 0, t ≥ 0, x ∈ Rd, ⎩u(0, x) = u0, ∂ ∂t u(0, x) = u1, (1.1)
Critical case: m = 1, d = 4, the sharp criteria and limiting profile of blow-up solutions were investigated by Zheng and Leng in [22]
This motivates us to further study the blow-up solutions of Eq (1.1) in the following sense: Under what conditions will the waves become unstable to collapse? Under what conditions will the waves be stable for all time? In other words, what are the sharp criteria of blow-up and global existence?
Summary
The asymptotic behavior and scattering properties of global solutions were widely studied in [3, 14, 17]. Critical case: m = 1, d = 4, the sharp criteria and limiting profile of blow-up solutions were investigated by Zheng and Leng in [22]. This motivates us to further study the blow-up solutions of Eq (1.1) in the following sense: Under what conditions will the waves become unstable to collapse (blow-up)? Under what conditions will the waves be stable for all time (global existence)? What are the sharp criteria of blow-up and global existence? This motivates us to further study the blow-up solutions of Eq (1.1) in the following sense: Under what conditions will the waves become unstable to collapse (blow-up)? Under what conditions will the waves be stable for all time (global existence)? In other words, what are the sharp criteria of blow-up and global existence?
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