Critical exponent for semi‐linear structurally damped wave equation of derivative type

Abstract

The main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: u tt − Δu + (−Δ) ∕2 u t = |u t | p , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), with > 0, n ≥ 1, ∈ (0, 2], and p > 1. In particular, we would like to prove the nonexistence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

Nonexistence Results for Semi-Linear Moore-Gibson-Thompson Equation With Non Local Operator

“…(x) + v 2 (x)) φ R (x)dx ≤ 0 and IR n (u 1 (x) + u 2 (x)) φ R (x)dx ≤ 0,which contradicts the assumption(11). For the critical case δ 2 = 0, from (25) we can see that J 1 ≤ C. Using Beppo Levi's theorem on monotone convergence, one obtains∞ 0 IR n |u t (x, t)| q dxdt = lim R→∞ R α 0 IR n |u t (x, t)| q ϕ R (x, t)dxdt = lim R→∞ J 1 ≤ C.…”

“…We assume that (u, v) = (u(x, t), v(x, t)), is a local solution to (2). In order to prove the lifespan estimate, we replace the initial data (0, u 1 , u 2 ), (0, v 1 , v 2 ) by (0, εf 1 , εf 2 ), (0, εg 1 , εg 2 ) with a small constant ε > 0, where (11). Repeating the steps in the above proofs, we arrive at the following estimate:…”

Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations

“…has been studied for a long time, where damping term satisfies g( [7,8,14,15,16,19]). Utilizing rescaled test function technique and iteration method, Ming et al [19] illustrate blow-up dynamic and lifespan estimate of solution to quasilinear wave equation with scattering damping µ (1+t) β u t (β > 1) and divergence form nonlinearity in the sub-critical and critical cases.…”

“…Utilizing rescaled test function technique and iteration method, Ming et al [19] illustrate blow-up dynamic and lifespan estimate of solution to quasilinear wave equation with scattering damping µ (1+t) β u t (β > 1) and divergence form nonlinearity in the sub-critical and critical cases. Dao et al [7,8] Recently, the study of weakly coupled system of semilinear wave equations…”

“…Inspired by the works in [3,4,7,8,9,16,20,24], our interest is to establish blow-up results of solutions to problems (1.1) and (1.2). It is worth to mention that Dao et al [7,8] establish formation of singularity of solution to the semilinear structurally damped wave equation with |u| p and |u t | p , respectively.…”