Problem For Nonlinear Parabolic Equation Research Articles

Exact local estimates for the supports of solutions in problems for non-linear parabolic equations

The phenomenon of instantaneous shrinking of the support in the Cauchy problem for non-linear parabolic equations with a positive initial function that is infinitesimal as is considered. Exact local estimates for the boundary of the support of the solutions are proved. For example, the exact asymptotic formula holds for the solution of the equation , , , where and .

Sequential approximation to nonlinear inverse heat conduction problems

Sequential approximations to the solution of nonlinear Inverse Heat Conduction Problems in one spatial dimension are analyzed. The problem is illposed and is also known as a noncharacteristic Cauchy problem for nonlinear parabolic equations. The nonlinearity caused by a temperature dependent thermal diffusivity is linearized by freezing the temperature for certain steps in the sequential procedure known as the Beck method. In the analysis, a relation for the approximating heat fluxes showing the explicit dependence on the previous ones is proved. Moreover, an error analysis is established providing estimates for the errors in the surface heat fluxes and in the surface temperatures when the temperature data are perturbed.

The symmetry of positive solutions of periodic-parabolic problems

In this paper, we generalized a result of Gidas, Ni and Nirenberg (1979) on the forced symmetry of positive solutions of nonlinear elliptic problems to the periodic (in time) problem for nonlinear parabolic equations with Dirichlet boundary conditions.

Uniqueness for the cauchy problem for nonlinear parabolic equations and inequalities

Let L be a linear second order parabolic partial differential operator, and suppose u is a solution to the differential inequality on satisfying u(x,0) = 0 for all . We show that the only solution for which is the trivial one provided the function h satisfies

On self-adjoint operators generated by nonhomogeneous elliptic problems with discontinuous boundary conditions and conjugation conditions

We construct and study self-adjoint operators corresponding to problems mentioned in the title. We describe a correlation between domains of definition of fractional powers of these operators and Sobolev spaces. We state new results on the solvability of several problems for nonlinear parabolic equations which have not yet been studied.

Periodic-Neumann boundary value problem for nonlinear parabolic equations and application to an elliptic equation

In this paper we study the periodic-Neumann boundary value problem for a class of nonlinear parabolic equations. We prove a new uniqueness result and study the structure of the set of solutions when there exist more than one solution. The ideas are applie

Boundary value problem for nonlinear parabolic equations of infinite order in Sobolev-Orlicz spaces

Boundary value problem for nonlinear parabolic equations of infinite order in Sobolev-Orlicz spaces

Method of straight lines in application to some two-dimensional nonlinear parabolic equations

We consider the first mixed problem for nonlinear parabolic equation. Assuming that the exact solution of the problem is u(t,x,y) ∈ C4,−0(Q), Q={(x,y)∈Ω, 0⩽t⩽T} we construct a scheme of the method of straight lines of accuracy 0(h2) for the cases when ώ is a rectangle or a trapezoid.

A class of inverse problems for nonlinear parabolic equations

A class of inverse problems for nonlinear parabolic equations

Criticality conditions and comparison therems for difference solutions of non-linear parabolic equations

Conditions for special pointwise estimates of difference solutions, i.e., criticality conditions, to exist, are established, and are used to prove comparison theorems for the solutions of implicit difference schemes for various non-linear parabolic equations, i.e., the conditions are found, under which one solution is majorized by another at all nodes of the space-time mesh. The results reveal that the solutions of implicit schemes display important properties of “monotonicity”, inherent in differential problems for non-linear parabolic equations.

Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions

Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions

The method of lines in nonlinear boundary value problems for nonlinear parabolic equations

The method of lines in nonlinear boundary value problems for nonlinear parabolic equations

VARIATIONAL METHODS IN NONSTATIONARY PROBLEMS (PARABOLIC CASE)

In this paper we present a method for constructing a generalized solution of boundary problems for nonlinear parabolic equations and systems. We assume that the stationary (elliptic) part of these equations is generated by a convex functional. A feature of the scheme we propose is that the construction of the generalized solution can be carried out by using the functional itself without considering its variation.