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The Digital Collections digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 10 Dec 2018 14:23:24 GMT2018-12-10T14:23:24ZDecay of Solutions of a Degenerate Hyperbolic Equation
https://digital.library.txstate.edu/handle/10877/7800
Decay of Solutions of a Degenerate Hyperbolic Equation
Dix, Julio G.
This article studies the asymptotic behavior of solutions to the damped, non-linear wave equation
ü + yů - m(||∇u||2)∆u = ƒ(x,t), for x ∈ Ω, t ≥ 0
u(x,0) = g(x), ů(x,0) = h(x), for x ∈ Ω
u(x,t) = 0, for x ∈ ∂Ω t ≥ 0;
where Ω is a bounded domain in Rn, with smooth boundary ∂Ω; y is a positive constant; m is a non-negative, bounded, and continuous function; ů denotes the derivative of u with respect to time; and as usual
∆u = n∑i=1 ∂2u/∂x2i, ||∇u||2 = n∑i=1 ƒΩ |∂u/∂xi|2 dx.
This equation appears in mathematical physics as the Carrier or Kirchoff equation, when modeling planar vibrations. For a background and physical properties of this model, we refer the reader to [3], [4], [8], [12], and their references.
Fri, 28 Aug 1998 00:00:00 GMThttps://digital.library.txstate.edu/handle/10877/78001998-08-28T00:00:00ZOn a Mixed Problem for a Linear Coupled System with Variable Coefficients
https://digital.library.txstate.edu/handle/10877/7799
On a Mixed Problem for a Linear Coupled System with Variable Coefficients
Clark, H. R.; San Gil Jutuca, L. P.; Milla Miranda, M.
We prove existence, uniqueness and exponential decay of solutions to the mixed problem
u" (x,t) - μ(t)Δu(x,t) + Σn i=1 ∂θ/∂xi (x,t) = 0,
θ' (x,t) - Δθ(x,t) + Σn i=1 ∂u'/∂xi (x,t) = 0,
with a suitable boundary damping, and a positive real-valued function μ.
Fri, 13 Feb 1998 00:00:00 GMThttps://digital.library.txstate.edu/handle/10877/77991998-02-13T00:00:00ZEigenvalue Comparisons for Differential Equations on a Measure Chain
https://digital.library.txstate.edu/handle/10877/7798
Eigenvalue Comparisons for Differential Equations on a Measure Chain
Chyan, Chuan Jen; Davis, John M.; Henderson, Johnny; Yin, William K. C.
The theory of u0-positive operators with respect to a cone in a Banach space is applied to eigenvalue problems associated with the second order Δ-differential equation (often referred to as a differential equation on a measure chain) given by
yΔΔ (t) + λp(t)y(σ(t)) = 0, t ∈ [0,1]
satisfying the boundary conditions y(0) = 0 = y(σ2(1)). The existence of a smallest positive eigenvalue is proven and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type.
Sat, 19 Dec 1998 00:00:00 GMThttps://digital.library.txstate.edu/handle/10877/77981998-12-19T00:00:00ZExistence of Axisymmetric Weak Solutions of the 3-D Euler Equations for Near-Vortex-Sheet Initial Data
https://digital.library.txstate.edu/handle/10877/7797
Existence of Axisymmetric Weak Solutions of the 3-D Euler Equations for Near-Vortex-Sheet Initial Data
Chae, Dongho; Imanuvilov, Oleg Yu
We study the initial value problem for the 3-D Euler equation when the fluid is inviscid and incompressible, and flows with axisymmetry and without swirl. On the initial vorticity ω0, we assumed that ω0/r belongs to L(log L(R3))ɑ with ɑ > 1/2, where r is the distance to an axis of symmetry. To prove the existence of weak global solutions, we prove first a new a priori estimate for the solution.
Thu, 15 Oct 1998 00:00:00 GMThttps://digital.library.txstate.edu/handle/10877/77971998-10-15T00:00:00Z