Global Existence and Global Nonexistence of Solutions of the Cauchy Problem for a Nonlinearly Damped Wave Equation

Levine, Howard A.; Park, Sang Ro; Serrin, James

doi:10.1006/jmaa.1998.6126

Cited by 138 publications

(92 citation statements)

References 7 publications

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“…Precisely, they showed that solutions with negative energy exist globally 'in time' if m ≥ p and blow up in finite time if p > m and initial energy is 'sufficiently' negative. This result was later generalized to an abstract setting and to unbounded domains by Levine et al [47], Levine and Serrin [48], Levine and Park [49], and Messaoudi [55,57]. In all these papers, the authors showed that no solution with negative or sufficiently negative energy can be extended on [0, ∞), if the nonlinearity dominates the damping effect ( p > m).…”

Section: Blowup In the Case Of Constant Exponentsmentioning

confidence: 91%

On wave equation: review and recent results

Messaoudi

Talahmeh

2017

Arab. J. Math.

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The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. Mathematics Subject Classification

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Section: Blowup In the Case Of Constant Exponentsmentioning

confidence: 91%

On wave equation: review and recent results

Messaoudi

Talahmeh

2017

Arab. J. Math.

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“…More recently and independently, in [11], [23], [24] the following result was established for the Cauchy problem (1.10)-(1.11): Remark 1. As a matter of fact, in [23], [24] when µ = 0, the following result was proved: If µ 2 = q 2 (x) is a locally bounded, measurable function on R n which satisfies…”

Section: Solutions With Positive Initial Energy 795mentioning

confidence: 92%

Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

Levine

Todorova

2000

Proc. Amer. Math. Soc.

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Abstract. In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the formHere a, b > 0. We prove that if p > m ≥ 1, then for any λ > 0 there are choices of initial data from the energy space with initial energy E(0) = λ 2 , such that the solution blows up in finite time. If we replace b|u| p−1 u by b|u| p−1 u − q(x) 2 u, where q(x) is a sufficiently slowly decreasing function, an analogous result holds.

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“…Similarly, authors in [17,24,28,29,11,12,21,22,5,15,25] discussed the case when the solution blows up in finite time. The closest equation to the one we investigate here has been studied by Zhijian [26], who considered:…”

Section: Introductionmentioning

confidence: 99%