Existence and asymptotic stability for a semilinear damped wave equation with dynamic boundary conditions involving variable nonlinearity
DOI:
https://doi.org/10.24193/subbmath.2025.1.06Keywords:
Wave equation, Kelvin-Voigt damping, boundary damping, Faedo-Galerkin approximation, exponential growth, variable-exponent nonlinearities, viscoelastic equation, global existence, nonlinear dissipation, energy estimatesAbstract
We study the solvability of a class of quasilinear elliptic equations with (p(x),k(x))(p(x),k(x))-growth structure and with nonlinear boundary conditions in the context of Kelvin-Voigt damping with arbitrary data. We approach our problem in a suitable functional classes by considering the so-called Lebesgue and Sobolev spaces with variable exponents. In the first step we establish existence and uniqueness results of solutions for the considered model if the data are regular enough. Our main idea is essentially based on using fixed point theory and Faedo-Galerkin approaches and includes some new techniques. Second, we assume the data is large enough and show that the energy grows exponentially.
Mathematics Subject Classification (2010): 35L05, 35L70, 35G30.
Received 13 September 2023; Accepted 17 November 2023.
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