Unique solvability of initial-boundary-value problem for second order equations of Kirchhoff type with variable exponents of nonlinearity
Анотація
The paper devoted to investigation strong solutions of the initial-boundary-value problem for some equations of the Kirchhoff type with avariable exponent of nonlinearity. As we know the equations of the Kirchhoff type of the second order with variable exponents of the nonlinearity are not studied yet. Problems for nonlinear partial differential equations with variable exponents of the nonlinearity are investigated in the generalized Lebesgue and Sobolev spaces.
In present paper we investigate strong solutions of the initial-boundary value problem for these equations. This article continues the research which starts in the paper Adv. Math. Sci. App., 23 (2013), 509--528, where we investigated the equation with strong damping. We found the sufficient conditions of the existence and uniqueness of the strong solution to given problem. The proof is based on the Faedo-Galerkin method, the derivation of a priori energy estimates, and the use the embedding results for Lebesgue spaces with variable exponent of nonlinearity. The obtained results can be applied to further studies of global solvability and long-time behavior of solutions for our problem.
Посилання
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