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- W3126433730 abstract "We consider the Cauchy problem on (mathbf {R_{0}^+} times mathbf {R^n}) for the semilinear damped wave equation $$displaystyle u_{tt}(t,x) - a^2(t) Delta u(t,x) + b(t) u_t(t,x)= |u(t,x)|{ }^p $$with decreasing in time coefficients, the propagation speed a(t) = (1 + t)−l, l ∈ (0, 1), the scale-invariant dissipation b(t) = β(1 + t)−1, β > 0, and a power nonlinearity of order p > 1. The solution u0 of the corresponding linear Cauchy problem will be represented in the explicit form using Fourier multipliers operators with multipliers expressed in terms of special functions. Our main goal is to prove a global in time existence result when initial data belongs to the space Hm(Rn) × Hm−1(Rn), m ≥ 1. We are focused in finding the critical exponent pc(n, l) such that if 1 < p < pc(n, l) there exist small data for which u blow-up in finite time. We also prove that if p ≥ pc(n, l) the global solution has the same long time behavior as u0. In order to estimate u we use Duhamel’s principle to represent u and then we apply L2 − L2 estimates of u0." @default.
- W3126433730 created "2021-02-15" @default.
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- W3126433730 date "2020-11-02" @default.
- W3126433730 modified "2023-10-17" @default.
- W3126433730 title "Critical Exponent for a Class of Semilinear Damped Wave Equations with Decaying in Time Propagation Speed" @default.
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- W3126433730 doi "https://doi.org/10.1007/978-3-030-61346-4_11" @default.
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