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Existence and Blow up Time Estimate for a Negative Initial Energy Solution of a Nonlinear Cauchy Problem P.A. Ogbiyele1

· P.O. Arawomo1

Received: 26 November 2019 / Accepted: 17 June 2020 © Springer Nature B.V. 2020

Abstract In this paper, we consider nonlinear wave equations with dissipation having the form   utt − div |∇u|γ −2 ∇u + b(t, x)|ut |m−2 ut = g(x, u) for (t, x) ∈ [0, ∞) × Rn . We obtain existence and blow up results under suitable assumptions on the positive function b(t, x) and the nonlinear function g(x, u). The existence result was obtained using the Galerkin approach while the blow up result was obtained via the perturbed energy method. Our result improves on the perturbed energy technique for unbounded domains. Keywords Nonlinear wave equation · Global existence · Blow up · Finite speed of propagation Mathematics Subject Classification (2010) 35A01 · 35B45 · 35L15 · 35L70

1 Introduction In this paper, we consider existence and blow up of solution to a nonlinear wave equation    utt − div |∇u|γ −2 ∇u + b(t, x)|ut |m−2 ut = g(x, u) t ∈ [0, ∞), x ∈ Rn (1) u(0, x) = u0 (x), ut (0, x) = u1 (x) x ∈ Rn with space–time dependent dissipation. u = u(t, x) is an unknown real valued function on [0, ∞) × Rn and the initial data u0 , u1 is assumed to have compact support in a ball B(R)

B P.A. Ogbiyele

[email protected] P.O. Arawomo [email protected]

1

Department of Mathematics, University of Ibadan, Ibadan, 200284, Nigeria

P.A. Ogbiyele, P.O. Arawomo

of radius R about the origin, where R satisfies the condition supp{u0 (x), u1 (x)} ⊂ {|x| ≤ R} and such that the solution satisfy the finite speed of propagation property supp u(t) ∈ B(R + t)

t ∈ [0, ∞)

Under these circumstances, the expectation is that the spread of the support could hinder finite-time blow-up of solution that is seen in the case of bounded domains, except for the case where the damping is absent or linear as blow-up can indeed occur. In the case of bounded smooth domains Ω ⊂ Rn , there is an extensive literature on global existence and blow up of solutions of non-linear wave equations having negative initial energy and of the form

 ⎧ ⎨ utt − ut − div |∇u|γ ∇u + |∇ut |r ∇ut + |ut |m ut = |u|p u x ∈ Ω, t > 0 (2) ⎩ u(x, 0) = u0 , ut (x, 0) = u1 , x ∈ Ω u(x, t)|∂Ω = 0, t > 0 that is, when the dissipation being considered, arises from an internal nonlinear damping term. see [1, 16, 18, 19] to mention but a few Yang in [17], obtained blow up of solutions to (2) under the condition p > max{γ , m} and where the blow up time depends on |Ω|. In [10] Messaoudi and Said-Houari studied a class of nonlinear wave equations having the form (2) and obtained blow up result for p > max{γ , m} and γ > r, where the blow up result holds regardless of the size of Ω. Thus extending the result of Yang [17]. Liu and Wang [9] considered a class of wave equations of the form (2) and established blow up results for certain solutions with non-positive initial energy as well as positive initial energy. This further improves the results of Yang [17]