10,196 research outputs found

    SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SUPERCRITICAL SOURCES

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    We consider the local and global well-posedness of the coupled nonlinear wave equations utt – Δu + g1(ut) = f1(u, v) vtt – Δv + g2(vt) = f2(u, v); in a bounded domain Ω subset of the real numbers (Rn) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents representing strong sources, while g1(ut) and g2(vt) act as damping. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this dissertation are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Moreover, we prove a blow up result for weak solutions with nonnegative initial energy. Finally, we establish important generalization of classical results by H. Brézis in 1972 on convex integrals on Sobolev spaces. These results allowed us to overcome a major technical difficulty that faced us in the proof of the local existence of weak solutions

    Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

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    We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite time blow-up and as well as global existence of solutions of the problem.Comment: 11 pages. Minor changes and added reference

    Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

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    In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, utt−Luxx=B(−∣u∣p−1u)xx, (p>1)u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1), where the nonlocality enters through two pseudo-differential operators LL and BB. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory, Methods & Application

    A Positivity Criterion for the Wave Equation and Global Existence of Large Solutions

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    In dimensions one to three, the fundamental solution to the free wave equation is positive. Therefore, there exists a simple positivity criterion for solutions. We use this to obtain large global solutions to two well-studied energy-supercritical semilinear wave equations, as well as some new results in the subcritical and critical cases.Comment: 26 pages; added some references, fixed some typo
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