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  • Pseudoparabolic Partial Differential Equations

    Creator:
    Showalter, R. E. and Ting, T. W.
    Resource Type:
    Article
    Full Text:
    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...
    Resource Type:
    Article
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    term corresponds to energy. Semigroup theory gives existence for hyperbolic and for parabolic cases