Finite Time Blow-Up for Wave Equations with Strong Damping in an Exterior Domain
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Finite Time Blow-Up for Wave Equations with Strong Damping in an Exterior Domain Ahmad Z. Fino Abstract. We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power-type nonlinearity |u|p . We will establish blow-up results under some conditions on the initial data and the exponent p, using the method of test function with an appropriate harmonic functions. We also study the existence of mild solution and its relation with the weak formulation. Mathematics Subject Classification. Primary 35B44, 35L71; Secondary 35B33, 34K10. Keywords. Semilinear wave equation, blow-up, exterior domain, strong damping.
1. Introduction This paper concerns the initial boundary value problem of the strongly damped wave equation in an exterior domain. Let Ω ⊂ Rn be an exterior domain whose obstacle O ⊂ Rn is bounded with smooth compact boundary ∂Ω. We consider the initial boundary value problem ⎧ utt − Δu − Δut = |u|p , t > 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎨ x ∈ Ω, u(0, x) = u0 (x), ut (0, x) = u1 (x), (1.1) ⎪ ⎪ ⎪ ⎪ ⎩ u = 0, t > 0, x ∈ ∂Ω, where the unknown function u is real-valued, n ≥ 1, and p > 1. Throughout this paper, we assume that (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω).
(1.2)
Without loss of generality, we assume that 0 ∈ O ⊂⊂ B(R), where B(R) := {x ∈ Rn : |x| < R} is a ball of radius R centred at the origin. Moreover, we assume that 0123456789().: V,-vol
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⎧ ⎪ 1 < p < ∞, ⎨ ⎪ ⎩1 < p ≤
n , n−2
MJOM
for n = 1, 2, for n ≥ 3.
(1.3)
For the simplicity of notations, · q and · H01 (1 ≤ q ≤ ∞) stand for the usual Lq (Ω)-norm and H01 (Ω)-norm, respectively. For any confusion, we remember that uH01 = u2 + ∇u2 . For the Cauchy problem for the semilinear wave equation with weak damping utt − Δu + ut = |u|p ,
t > 0, x ∈ Rn ,
(1.4)
Todorova and Yordanov [23] proved the global existence of solution for sufficiently small data in the energy space if p > 1 + 2/n and the blowing up of solution when p < 1+2/n. Moreover, Zhang [25] succeeded in completing this study by proving that the critical case p = 1 + 2/n belongs to the blow-up existence of solution. The number pc (n) = 1 + 2/n is known (due to Fujita [9]) as the critical exponent of corresponding heat equation −Δu + ut = |u|p since it divides (1, ∞) into two subintervals so that the following take place: If p ∈ (1, pc (n)], then solutions with non-negative (in some sense) initial values blow-up in finite time, while if p ∈ (pc (n), ∞), then solutions with small (and sufficiently regular ) initial values exist for all time. For problem (1.4) in exterior domain utt − Δu + ut = |u|p ,
t > 0, x ∈ Ω.
Ikehata [12,13] proved that the solution exists globally when n = 2 for p > pc (n), and n = 3, 4, 5 for 1 + 4/(n + 2) < p ≤ 1 + 2/(n − 2) under the assumption of the compactness of the initial data. Ono [22] obtained the same result of global existence of solution without compactness of the initial data by applying the result of Dan–Shibata [3] and cut-off method. Ogawa and Takeda [21] proved the non-existence of non-negative
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