What is it about?
In this work, we consider a semi-linear generalized wave equation with nonlocal boundary conditions and variable exponent nonlinearities. We prove the local existence solution and uniform decay rates if the initial datum possesses suitable assumptions. Our results are generalizations of the corresponding results in the constant exponent cases. The main difficulty of this problem is related to the presence of quasilinear terms; then our objective in this paper is to apply the compactness method in the framework of the Lebesgue and Sobolev spaces with variable exponents to get the existence of the local solution.
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Why is it important?
1. Our work helps in studying many mathematical nonlinear models of hyperbolic, parabolic, and elliptic equations with nontrivial boundary conditions. There are only a few works regarding problems with nontrivial boundary conditions. 2. Our work, It knowing us how to deal with the trace embedding on the boundary of the domain in Rn 3. The importance of this research is to give the existence result by the study of the generalized problem with nonstandard p(.) growth, then in the framework of the Lebesgue and Sobolev spaces with variable exponents, for this aim, I think the readership of the journal would be interested in it
Perspectives
Some perspectives are treated in the next works such as; 1) The general decay rate, 2) Existence and uniqueness with damping and time-varying delay, 3) Stability and instability result in the boundary or internal feedbacks.
abita Rahmoune
University Amar Telidji de Laghouat
Read the Original
This page is a summary of: Existence and asymptotic stability for the semilinear wave equation with variable-exponent nonlinearities, Journal of Mathematical Physics, December 2019, American Institute of Physics,
DOI: 10.1063/1.5089879.
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