Hyperbolic Variational Inequality of the Third Order with Variable Exponent of Nonlinearity

Abstract

In Sobolev spaces with variable exponent, we consider the problem for a semilinear hyperbolic variational inequality of the third order. We establish conditions for the existence of a solution u of this problem such that u ∈ L ^∞((0, T); V _1,0(Ω)), u _t  ∈ L ^∞((0, T); V _1,0(Ω)) ∩ L ^p(x)(Q _T ), and u _tt  ∈ L ^∞((0, T); L ²(Ω)), where V _1,0(Ω) ⊂ H ¹(Ω).