Results 231 to 240 of about 855 (241)

Certain nonlocal problem for a degenerate hyperbolic equation

Mathematical Notes, 1992
The equation \((*)\) \(| y|^ m u_{xx}- u_{yy}=0\), \(m>0\) is considered in a domain bounded by characteristics of \((*)\). Values of integrals of \(u\) along characteristics are given. The author proves uniqueness and existence of the solution to the problem mentioned above.
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On a variational inequality for a degenerate quasilinear hyperbolic equation

Applied Mathematics and Computation, 2003
The author studies a unilateral problem for the nonhomogeneous degenerate Kirchhoff equation with a blowing up term. Using the penalty method and Galerkin's approximation, existence and uniqueness results are obtained.
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Boundary problems for a class of degenerate hyperbolic equations

Siberian Mathematical Journal, 1984
The author considers the equation \(xu_{xx}-yu_{yy}+(\alpha -m)u_ x- (\beta -n)u_ y=0\), provided ...
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Energy estimates for a class of degenerate hyperbolic equations

Mathematische Annalen, 2009
Considered is the Cauchy problem for a linear degenerate inhomogeneous hyperbolic equation in one dimension. The leading coefficient which is allowed to vanish is decomposed in \(a(x ,t)= q(x ,t)K(x ,t)\), where \(K(x ,t)\), is a nonnegative smooth function and \(q(x ,t)\) is a positive smooth function. In case all coefficients are smooth functions on \
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Continuous Observability of Hyperbolic Equations under Degenerate Sensors

2017
In this chapter we intend to extend our studies of the observability properties of partial differential equations to a large class of linear second-order partial differential equations (PDEs) of hyperbolic type in n spatial dimensions. We will show that both the L ∞ (0, T; R n+1)- and C([0, T[; R n+1)-continuous observability properties are possible ...
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On local solutions of some mildly degenerate hyperbolic equations

Nonlinear Analysis: Theory, Methods & Applications, 1993
The paper is concerned with the Cauchy problem for the abstract Kirchhoff-type equation \[ u''- M(\langle Au,u \rangle) Au=0, \quad u(0)= u_ 0, \quad u'(0)=u_ 1, \tag{\(*\)} \] where \(M: \mathbb{R}^ +\to \mathbb{R}^ +\) is a non-negative \(C^ 1\)-function and \(A: V\to V'\) a selfadjoint, linear isomorphism acting in a Hilbert triplet \(V\subseteq H ...
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Local solution for a degenerate hyperbolic equation with memory

Nonlinear Analysis: Theory, Methods & Applications, 1994
In this paper, the local solvability in suitable Sobolev-type spaces for the nonlocal equation \[ u'' - m \bigl( (\Lambda u,u) \bigr) \cdot \Lambda u + \dot a* \biggl( m \bigl( (\Lambda u,u) \bigr) \cdot \Lambda u \biggr) = 0, \quad u(0) = u_0,\;u'(0) = u_1, \tag{*} \] is studied. Here \(m(r)\) is a nonnegative function (possibly vanishing somewhere), \
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The Dirichlet Problem for a Degenerate Hyperbolic Equation in a Rectangle

Differential Equations, 2001
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Singular perturbation hyperbolic–parabolic for degenerate nonlinear equations of Kirchhoff type

Nonlinear Analysis: Theory, Methods & Applications, 2001
The Cauchy problem for the operator equation \(\varepsilon u''+ \) \(\delta u'+\) \(m(|A^{1/2}u|^2)\) \(Au=0\) is studied. The convergence \(u^{\varepsilon}\to u\), as \(\varepsilon\to 0\), is proved to a solution of the limit problem with \(\varepsilon =0\). The function \(m\) is allowed to be locally Lipschitz continuous.
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Cauchy problem for degenerate hyperbolic equations

Communications on Pure and Applied Mathematics, 1980
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