Abstract
In this paper, we consider the following p-Kirchhoff type hyperboc equation with variable exponents Equation We prove that a global existence of the solution with positive initial energy, the stability based of Komorniks inequality.
Highlights
We prove the global existence of the solution with positive initial energy, the stability established based on Komornik’s inequality
They proved a finite-time blow-up result for the solution with negative initial energy as well as for certain solutions with positive initial energy; in the cas where m (x) = 2 and under suitable conditions on the exponents, they established a blow-up result for solutions with arbitrary positive initial energy
Acknowledgement The authors wish to thank deeply the anonymous referee for useful remarks and careful reading of the proofs presented in this paper
Summary
We begin this section with some notations and definitions. Denote by . p , the Lp (Ω) norm of a Lebesgue function u ∈ Lp (Ω). We define the variable-exponent Sobolev space W 1,q(.) (Ω) as follows:. W 1,q(.) (Ω) := v ∈ Lq(.) (Ω) such that ∇v exists and |∇v| ∈ Lq(.) (Ω) This is a Banach space with respect to the norm v W 1,q(.)(Ω) = v q(.) + ∇v q(.). For the existence of the local solution of problem (1.1), we refer the reader to [13]. Their result is given in the following theorem: Theorem 2.1 Suppose that r, m ∈ C Ω with n−1 2 ≤ r1 ≤ r (x) ≤ r2 < 2 n − 2 , if n ≥ 3, r (x) ≥ 2, if n = 1, 2, and. For any (u0, u1) ∈ W01,p (Ω) × L2 (Ω) , problem (1.1) has a unique weak local solution u ∈ L∞ (0, T ) , W01,p (Ω) , ut ∈ L∞ (0, T ) , L2 (Ω) ∩ Lm(.) (Ω × (0, T )) , utt ∈ L2 (0, T ) , W −1,p′ (Ω)
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