hyperbolic problems on generalized sobolev spaces

What is it about?

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation utt = Lu + f(x, t) in ΩT = Ω × (0, T), where L is a nonlinear operator and ϕ(x, t, ⋅ ), f(x, t) and the exponents of the nonlinearities p(x, t) and μ(x, t) are given functions. Keywords: Dirichlet problem, hyperbolic problem, global existence, nonstandard growth conditions

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Why is it important?

Our aim in this article is to show the existence of globally weak solutions to some hyperbolic problem in framework of sobolev spaces with variable exponet with the presence of a non-coercive lower-order term in divergential form − div(ϕ(x, t, ut) which satisfies only a growth condition with respect to ut, and induce a lack of coercivity, and therefore a lack of estimate of energy. To overcome this difficulties we use the generalized Gagliardo–Nirenberg inequality

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