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Nonexistence of Global Solutions to a Class of Nonlinear Differential Inequalities and Application to Hyperbolic–Elliptic Problems
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N. Alaa
Published/Copyright:
June 9, 2010
Abstract
We consider the problem
utt + δut + εaΔu + ϕ(∫Ω|∇u|2dx)Δu ≥ f(x, t),
posed in Ω × (0, +∞). Here 
is a an open smooth bounded domain and ϕ is like ϕ(s) = bsγ, γ > 0, a > 0 and ε = ±1. We prove, in certain conditions on f and ϕ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type
w″ + δw′ ≥ αw + βwp.
It is also shown that a global positive solution, when it exists, must decay at least exponentially.
Key words and phrases.: Nonlinear differential inequalities; hyperbolic and elliptic problems; blow-up; asymptotic behavior of solutions
Received: 2001-05-10
Revised: 2003-01-20
Published Online: 2010-06-09
Published in Print: 2003-June
© Heldermann Verlag
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Keywords for this article
Nonlinear differential inequalities;
hyperbolic and elliptic problems;
blow-up;
asymptotic behavior of solutions
Articles in the same Issue
- Skew Products of Ideals
- Crowded and Selective Ultrafilters under the Covering Property Axiom
- Solution of the Stieltjes Truncated Moment Problem
- Minimax Solutions of the Dual Hamilton-Jacobi Equation
- Nonexistence of Global Solutions to a Class of Nonlinear Differential Inequalities and Application to Hyperbolic–Elliptic Problems
- Functions of Two Variables Whose Vertical Sections Are Equiderivatives
- On Maximal Element Problem and Quasi-Variational Inequality Problem in L.C. Metric Spaces
- On a Certain Generalization of the Krasnosel'skii Theorem
