Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.     

Boundary value problems for a second-order differential equation with selfadjoint operator coefficients

Translated from Sibirskii Matematicheskii, Vol. 37, No. 6, pp. 1397–1406, November–December, 1996.     

Solutions of the main boundary value problems for a loaded second-order parabolic equation with constant coefficients

We give well-posed statements of the main initial–boundary value problems in a rectangular domain and in a half-strip for a second-order parabolic equation that contains partial Riemann–Liouville fractional derivatives with respect to one of the two independent variables. We construct Green functions and representations of solutions of these problems. We prove existence and uniqueness theorems for the first boundary value problem and the problem in the half-strip with the boundary condition of the first kind.     

Boundary value problems for strongly degenerate parabolic equations

Solutions of strongly degenerate parabolic partial differential equations are known to develop infinite spatial derivatives in finite time from smooth initial conditions over the real line. However, when Dirichlet or Neumann boundary conditions are prescribed on a finite interval, a smooth classical solution may exist for all Eq., with derivatives vanishing as t tends to infinity. With some simple extra conditions relating two nonlinear coefficients in the degenerate equation, classical solvability is proved in general.     

Boundary value problems with mixed lateral conditions for parabolic operators

Summary We study a parabolic problem in a cylinder with lateral conditions of mixed type. We get an existence, uniqueness and regularity result in dissymmetric function spaces nicely fitting the geometry of the problem.Partially supported by G.N.A.F.A. of C.N.R., Italy.Partially supported by I.N.F.N., Bologna, Italy.     

Boundary value problems for higher order parabolic equations

We consider a constant coefficient parabolic equation of order and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order lie in with respect to surface measure. In addition, a regularity result for the solution is obtained if the data has one more derivative. The boundary of the space domain is given by the graph of a Lipschitz function. This provides an extension of the methods of Pipher and Verchota on elliptic equations to parabolic equations.

     

Cauchy problem for a parabolic higher-order equation with pulse action

We prove the existence and establish some estimates of a solution of the Cauchy problem for a parabolic pulse-action equation of higher order in t. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 17–24, January–March, 2008.     

An inverse problem for a higher-order parabolic equation

We prove existence and uniqueness theorems for the inverse problem of finding the right-hand side of a higher-order parabolic equation with two independent variables and an additional condition in the form of integral overdetermination. The results obtained are used to study the passage to the limit in a sequence of such inverse problems with weakly convergent coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 680–691, November, 1998.     

Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane

For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (k _j ?1), j = 1,..., l, where 1 ≤ k ₁ < ··· < k _l, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for k _j = j + 1. We obtain a sufficient condition for the Fredholm property of which problem and derive an index formula.     

Boundary value problems for nonuniformly elliptic equations with measurable coefficients

Summary Let A be a symmetric N × N real-matrix-valued function on a connected region in Rⁿ, with A positive definite a.e. and A, A⁻¹ locally integrable. Let b and c be locally integrable, non-negative, real-valued functions on Ω, with c positive a.e. Put a(u, v) = = ((A∇u, ∇v) + buv) dx. We consider in X the weak boundary value problem a(u, v) = = fvcdx, all v ε X; where X is a suitable Hilbert space contained in H _loc ^1,1 (Ω). Criteria are given in order that the Green's operator for this problem have an integral representation and bounded eigenfunctions; in addition, criteria for compactness are given. Entrata in Redazione il 21 giugno 1975. Research was partially supported by the National Science Foundation under Grant GP-28377A2.     

Schauder estimates for higher-order parabolic systems with time irregular coefficients

We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and a half space. We also provide an existence result for the divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables.     

Stabilization of a solution to the Cauchy problem for a nondivergence parabolic equation with growing lower order coefficients

We obtain sharp sufficient conditions on the growth of lower order coefficients of a second-order parabolic equation under which a solution to the Cauchy problem stabilizes to zero uniformly in x on every compact set K ∈ ℝ ^N in some classes of growing initial functions.     

Boundary value problem for an elliptic equation with rapidly oscillating coefficients in a rectangle

An elliptic equation in a rectangle with coefficients depending on a fast variable and with its period being a small parameter is considered. An asymptotic expansion of the solution up to an arbitrary degree of the small parameter is constructed and substantiated by applying the two-scale expansion method.     

Boundary value problems for the one-dimensional Willmore equation

The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested in smooth graph solutions, since for suitable boundary data, we expect the stable solutions to be among these. In the first part, classical symmetric solutions for symmetric boundary data are studied and closed expressions are deduced. In the Navier case, one has existence of precisely two solutions for boundary data below a suitable threshold, precisely one solution on the threshold and no solution beyond the threshold. This effect reflects that we have a bending point in the corresponding bifurcation diagram and is not due to that we restrict ourselves to graphs. Under Dirichlet boundary conditions we always have existence of precisely one symmetric solution. In the second part, we consider boundary value problems with nonsymmetric data. Solutions are constructed by rotating and rescaling suitable parts of the graph of an explicit symmetric solution. One basic observation for the symmetric case can already be found in Euler’s work. It is one goal of the present paper to make Euler’s observation more accessible and to develop it under the point of view of boundary value problems. Moreover, general existence results are proved.