Abstract
This article deals with the study of the higher-order Kirchhoff-type equation with delay term in a bounded domain with initial boundary conditions, where firstly, we prove the global existence result of the solution. Then, we discuss the decay of solutions by using Nakao’s technique and denote polynomially and exponentially. Furthermore, the blow-up result is established for negative initial energy under appropriate conditions.
Highlights
In this paper, we establish the higher-order Kirchhoff-type equation with delay term as follows: 8 >>>>>>>>>>>>< ð utt + Am/2u2 q dx Am u +μ1 jμt ðx, tÞjr−1 Ω+μ2jμtðx, t − τÞjr−1utðx, t − τÞ = jujp−1u, μt ðx, t Þ
Time delays often appear in many various problems, such as thermal, economic phenomena, biological, chemical, and physical
Motivated by the above works, we deal with the existence, decay, and blow-up results for the higher-order Kirchhoff type equation (1) with delay term and source term
Summary
We establish the higher-order Kirchhoff-type equation with delay term as follows:. The author obtained that the solution exists globally if p ≤ r, while if p > max fr, 2qg He established the blow-up result for E ð0Þ < 0. Messaoudi [24] studied the following equation utt + Δ2u + jutjr2ut = jujp2u ð8Þ and obtained an existence result for the equation (8) and proved that the solution continues to exists globally if r ≥ p; if r < p and the initial energy is negative, the solution blows up in finite time. Motivated by the above works, we deal with the existence, decay, and blow-up results for the higher-order Kirchhoff type equation (1) with delay term and source term.
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