Philip W. Schaefer

Lower bounds for blow-up time in parabolic problems under Neumann conditions

We consider an initial boundary value problem for the semilinear heat equation under homogeneous Neumann boundary conditions in which the solution may blow up in finite time. A lower bound for the blow-up time is determined by means of a differential inequality argument when blow up occurs. Under alternative conditions on the nonlinearity, some additional bounds for blow-up time are also determined.

Spatial decay estimates in time-dependent Stokes flow

This paper considers the time-dependent slow flow of an incompressible viscous fluid in a semi-infinite cylindrical pipe of smooth cross section. An exponential decay estimate in terms of the distance from the finite end of the pipe is obtained from a second-order differential inequality for a weighted energy integral defined on the solutions of the system. The decay constant depends only on the geometry and the first positive eigenvalues for the fixed and free membrane problems for the cross sectional geometry. The paper also indicates how to bound the total weighted energy.

Some Phragmén-Lindelöf type results for the biharmonic equation

We derive estimates for weighted and unweighted energy expressions for the solution of a biharmonic boundary value problem in the plane by means of a second order differential inequality. We consider three kinds of unbounded domains with boundary conditions of Dirichlet type. For each domain we develop exponential estimates that either grow or decay. In the case of decay we also present a method for obtaining explicit bounds for the total energy.ZusammenfassungWir leiten Abschätzungen für gewichtete und ungewichtete Energieausdrücke für die Lösung einer biharmonischen Randwertaufgabe in der Ebene mittels einer zweite Ordnung Differentialungleichung ab. Wir betrachten drei Arten von unbeschränkten Gebieten mit Randbedingungen des Dirichletschen Typs. Für jedes Gebiet entwickeln wir exponential Abschätzungen mit entweder positivem oder negativem Wachstum. Im Falle des negativem Wachstums geben wir auch eine Methode an, um explizite Abschätzungen für die totale Energie zu erhalten.

Energy bounds for some nonstandard problems in partial differential equations

Abstract We consider problems of the form d 2 u dt 2 +Au=F, αu(0)+u(T)=g, β du dt (0)+ du dt (T)=h, for t∈(0,T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.

On a nonstandard problem for heat conduction in a cylinder

Decay bounds are derived for the solution of a heat conduction problem in a semi-infinite cylinder when the lateral surface is held at zero temperature, a nonzero temperature is prescribed on the finite base, and the temperature at time T is prescribed to be a constant multiple of the temperature at initial time. Both energy and pointwise decay bounds are computed for a range of values of the constant multiple. Such problems were originally introduced as a means of stabilizing the backward-in-time problem for the heat equation.

Existence theorems for some nonlinear fourth order elliptic boundary value problems

The general outline of our method is to that of of the second author second order elliptic parabolic equations [S, 71. of our effort is in deriving estimates for linear equations which lead to for the FrCchet derivatives of the relevant nonlinear to which Section 2 devoted, appear to be sharp in to explained below, and feel are independent interest. are stated in and the existence theorems developed in 4. a concluding section, a comparison is made between our results and on the theory of and pseudomonotone operators. There is substantial overlap in classes of to which the results and those obtained by The latter generally allows much greater in the nonlinearity (without priori estimate being available) whereas our results allow for a certain amount of ‘nonmonotonicity’ in the nonlinear terms. Furthermore, here we obtain a constructive algorithm (a variation of Newton’s method) for the solution, and this is a large part of the motivation for this work. In fact, the second author has recently utilized the second order results in [5] to obtain numerical approximations by the finite element method in [2]. We anticipate that the present work can be similarly utilized.

On the development of functionals which satisfy a maximum principle

We present a systematic method for constructing functions, defined on the solutions of a higher order differential equationor weakly coupled system of differential equations, which satisfy a maximum principle. The construction is related to the construction of Lyapunov functions and to the theory of invariant sets. One can obtain various consequences from the resulting principle, such as a priori estimates, uniqueness of solutions, and comparison theorems

Solution, gradient, and Laplacian bounds in some nonlinear fourth order elliptic equations

In order to obtain bounds of the type in the title, a suitable function is defined on the solutions to a certain class of semilinear fourth order elliptic partial differential equations. The subharmonicity of this function under appropriate conditions on the coefficients and nonlinear terms leads one immediately to the desired bounds.

Uniqueness in some higher order elliptic boundary value problems

SummaryThe uniqueness of the solution to two boundary value problems for the linear equation Δ3u −a Δ2u +b Δu −cu =F and to two boundary value problems for the quasilinear differential equation Δ2u +ρw(u) =f are proved. The proofs follow as a consequence of maximum principles for a functional which is defined on solutions to the differential equation.ZusammenfassungDie Eindeutigkeit der Lösung zweier Randwertaufgaben für die lineare Gleichung Δ3u −a Δ2u +b Δu −cu =F und zweier Randwertaufgaben für die quasilineare Differentialgleichung Δ2u +ρw(u) =f wird bewiesen. Der Beweis folgt aus einem Maximumprinzip für ein Funktional, das für die Lösungen der Gleichung definiert ist.