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  • Pseudoparabolic Partial Differential Equations

    Creator:
    Showalter, R. E. and Ting, T. W.
    Resource Type:
    Article
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    differential operator. The initial and boundary conditions for this equation are the same as those posed for

  • Well-Posed Problems for a Partial Differential Equation of Order 2m + 1

    Creator:
    Showalter, R. E.
    Abstract:
    We are concerned here with well-posed problems for the partial differential equation uₜ(x, t) + yMuₜ(x, t) + Lu(x, t) = f(x, t) containing the elliptic differential operator M of order 2m and the differential operator L of order ≤2m. Hilbert space methods are used to formulate and solve an...
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    Article
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    conditions on the operator M are given for the existence and uniqueness of a weak solution to the generalized

  • Existence and Representation Theorems for a Semilinear Sobolev Equation in Banach Space

    Creator:
    Showalter, R. E.
    Abstract:
    An existence theory is developed for a semilinear evolution equation in Banach space which is modeled on boundary value problems for partial differential equations of Sobolev type. The operators are assumed to be measurable and to satisfy coercive estimates which are not necessarily uniform in their time dependence, and to...
    Resource Type:
    Article
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    Existence and representation theorems for a semilinear sobolev

  • A Priori Error Estimates for Approximation of Parabolic Boundary Value Problems

    Creator:
    Showalter, R. E.
    Abstract:
    The L²-error estimates are established for the continuous time Faedo-Galerkin approximation to solutions of a linear parabolic initial boundary value problem that has elliptic part of order 2m. Properties of analytic semigroups are used to obtain these estimates directly from the L²-estimates for the corresponding steady state elliptic problem under...
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    Article
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    and approximation. Let Vand H be Hilbert spaces for which V is a dense subspace ofH and the injection

  • Nonlinear Degenerate Evolution Equations and Partial Differential Equations of Mixed Type

    Creator:
    Showalter, R. E.
    Abstract:
    The Cauchy problem for the evolution equation Mu’(t) + N(t,u(t)) = 0 is studied, where M and N(t,•) are, respectively, possibly degenerate and nonlinear monotone operators from a vector space to its dual. Sufficient conditions for existence and for uniqueness of solutions are obtained by reducing the problem to an...
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    Article
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    2 and the preceding argument applied to //+ ff(t), we obtain a sufficient condition for uniqueness

  • Regularization and Approximation of Second Order Evolution Equations

    Creator:
    Showalter, R. E.
    Abstract:
    We give a nonstandard method of integrating the equation Bu" + Cu’ + Au = f in Hilbert space by reducing it to a first order system in which the differentiated term corresponds to energy. Semigroup theory gives existence for hyperbolic and for parabolic cases. When C = εA, ε...
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    Article
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    term corresponds to energy. Semigroup theory gives existence for hyperbolic and for parabolic cases

  • Implicit Degenerate Evolution Equations and Applications

    Creator:
    Di Benedetto, Emmanuele and Showalter, R. E.
    Abstract:
    The initial-value problem is studied for evolution equations in Hilbert space of the general form d/dt A(u) + B(u) ϶ f, where and are maximal monotone operators. Existence of a solution is proved when A is a subgradient and either is strongly monotone or B is coercive; existence is established...
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    Article
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    J. MATH. ANAL. Vol. 12, No. 5, September 1981 1981 Society for Industrial and Applied Mathematics

  • Diffusion in Fissured Media

    Creator:
    Böhm, Michael and Showalter, R. E.
    Abstract:
    See article for Abstract.
    Resource Type:
    Article
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    , May 1985 (C) 1985 Society for Industrial and Applied Mathematics O07 DIFFUSION IN FISSUREDMEDIA

  • A Nonlinear Pseudoparabolic Diffusion Equation

    Creator:
    Böhm, Michael and Showalter, R. E.
    Abstract:
    Diffusion in a fissured medium with absorption or partial saturation effects leads to a pseudoparabolic equation nonlinear in both the enthalpy and the permeability. The corresponding initial-boundary value problem is shown to have a solution in various Sobolev-Besov spaces, and sufficient conditions are given for the problem to be well-posed.
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    Article
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    . ANAL. Vol. 16, No. 5, September 1985 (C) 1985 Society for Industrial and Applied Mathematics 006 A

  • Diffusion of Fluid in a Fissured Medium with Microstructure

    Creator:
    Showalter, R. E. and Walkington, N. J.
    Abstract:
    A system of quasilinear degenerate parabolic equations arising in the modeling of diffusion in a fissured medium is studied. There is one such equation in the local cell coordinates at each point of the medium, and these are coupled through a similar equation in the global coordinates. It is shown...
    Resource Type:
    Article
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    SIAM J. MATH. ANAL. Vol. 22, No. 6, pp. 1702-1722, November 1991 ()1991 Society for Industrial and