Blow-Up of Certain Solutions to Nonlinear Wave Equations in the Kirchhoff-Type Equation with Variable Exponents and Positive Initial Energy

Abstract

This paper is concerned with the blow-up of certain solutions with positive initial energy to the following quasilinear wave equation: u t t − M N u t Δ p · u + g u t = f u . This work generalizes the blow-up result of solutions with negative initial energy.

BLOW UP FOR A VISCOELASTIC WAVE EQUATION WITH SPACE-TIME POTENTIAL IN Rn P. A. OGBIYELE AND P. O. ARAWOMO

“…Our technique follows the one in [22]. however we employ Lemma 4 in obtaining the blow up estimate of solution to (1).…”

“…The weighted functions λ(t) and β(t) are used here to compensate for the lack of L 2 → L p injection arising as a result of the unboundedness of the domain for 0 ≤ 2 < p. Theorem 2. Let u(t, x) be a solution of the problem (1) with compact support in the ball B(R) and suppose that the assumptions (l 1 ), (l 21 ), (l 3 ) and (l 41 ) are satisfied. In addition, assume that f (•, u) satisfies…”

“…∫ ∞ 0 g(s)ds ≤ q−2 q−1 , for positive constants ρ 0 and q with q ∈ (2, p). Then no weak solution of (1) with compact support and satisfying E(0) < 0 and ∫ B(R) u 0 u 1 dx > 0 can exist on the whole of [0, ∞).…”

“…Theorem 3. Let u(t, x) be a solution of the problem (1) and suppose that the assumptions (l 1 ), (l 21 ), (l 3 ) and (l 51 ) are satisfied. In addition, assume that f (u) satisfies…”

“…when p > m and obtained blow up results for positive initial energy solution. For a review on recent results on global existence, energy decay and blow up of solutions to nonlinear wave equations in bounded domains and their extensions to variable exponents, see [1,4,16,17] and for other relevant results on blow-up and global existence for nonlinear wave equations in bounded smooth domains, the reader is referred to [2,6,7,11,13,18,19,21,23,26].…”