Global existence for the wave equation with nonlinear boundary damping and source terms

By Enzo Vitillaro. 2002

149 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ad7869e19f65fdd24ba78c6034666ab768f8f262

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In Computational Theory and Mathematics

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Citations in the corpus

Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping (2008, background): “With damping and source of polynomial structures: h(u) = |u|k−1u, g(u) = |u|q−1u, in [39] local existence of weak solutions was obtained provided that 1 ≤ k < 3 (up to the supercritical case) and with the following relationship between the exponents of the damping and the source: k < 4q/(q + 1).”
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction (2007, methodology): “Reference [34] studies homogeneous wave equation (f = 0) with both semilinear boundary conditions and the damping of a polynomial structure.”
On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source (2009, background): “See, for instance the followingworks: [1,5,7–12,14–16,18–20,25–32,34–36] and a long list of references therein.”
Blow‐up of solutions for a viscoelastic wave equation with variable exponents (2019, unknown): “Using Young inequality, the Sobolev embedding, and (17), we have ∫Ω|q(v1) − q(v2)||ut(s)|dx = ∫Ω|q′(ξ)||v||ut(s)|dx ≤ 2∫Ω|ut(s)|2dx + 1 2δ∫Ω|q′(ξ)|2|v|2dx ≤ δ 2∫Ω|ut(s)|2dx + b(2)(p2 − 1)2 2δ ∫Ω|ξ|2(p(x)−2)|v|2dx ≤ δ 2∫Ω|ut(s)|2dx + b(2)(p2 − 1)2 2δ ( ∫Ω|v| 2n n−2 dx ) n−2 n [( ∫Ω|ξ|n(p1−2)dx ) 2 n ”
On a wave equation with supercritical interior and boundary sources and damping terms (2011, methodology): “Full analysis of local and global existence for problem (1.1) for a supercritical source on the boundary (2 k ≤ 3), but without nonlinearities in the interior (g0 = f = 0), can be found in the work of Vitillaro [38].”
Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions (2008, methodology): “In this work, we cannot use “directly” the existence result of Georgiev and Todorova [15] nor the results of Vitillaro [33, 34] because of the presence of the strong linear damping −∆ut and the dynamic boundary conditions on Γ1.”
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping (2008, methodology): “This completes the picture for Hadamard wellposedness of weak solutions to the problem considered in (1), hence also in [31, 14].”