Well posedness of a nonlinear mixed problem for a parabolic equation with integral condition (original) (raw)

On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

Axioms, 2021

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. Then by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.

Mixed problem with integral conditions for a certain class of hyperbolic equations

Journal of Applied Mathematics, 2001

We study a mixed problem with purely integral conditions for a class of two-dimensional second-order hyperbolic equations. We prove the existence, uniqueness, and the continuous dependence upon the data of a generalized solution. We use a functional analysis method based on a priori estimate and on the density of the range of the operator generated by the considered problem.

Mixed Problem with an Integral Two-Space-Variables Condition for a Class of Hyperbolic Equations

International Journal of Analysis, 2013

This paper is devoted to the proof of the existence and uniqueness of the classical solution of mixed problems which combine Neumann condition and integral two-space-variables condition for a class of hyperbolic equations. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered.

On the solvability of a class of nonlinear singular parabolic equation with integral boundary condition

Applied Mathematics and Computation, 2020

In this paper, the existence and uniqueness of a weak solution for nonlinear singular parabolic equation with integral boundary conditions is proved. First, the associated linear problem is solved. After writing the linear problem on its operatorial form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the weak solution of the nonlinear problem.

A Strong Solution of an Evolution Problem with Integral Conditions

2002

The paper is devoted to proving the existence and uniqueness of a strong solution of a mixed problem with integral boundary conditions for a certain singular parabolic equation. A functional analysis method is used. The proof is based on an energy inequality and on the density of the range of the operator generated by the studied problem.

Solvability and Blow-up of Solutions of Semi Linear Parabolic Problem with Integral Condition of Second Type

Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES, 2020

In this paper, semi-linear parabolic equation with integral boundary condition of second type is investigated. The existence, uniqueness and Blow-up of weak solutions in finite time are established. The proof is proceeds in two steps; using the variable separation method for the solvability of the linear cas and applying an iterative process and a priori estimate, we prove the existence, uniqueness of the weak solution of the semilinear problem. Finally, we study a blow-up of solution in finite time for a super-linear problem by using eigen functions method introduced by Kaplan.

Mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator

Journal of Applied Mathematics and Stochastic Analysis, 2002

In this paper, we prove the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. The proof uses a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the considered problem.

Solution to a Semilinear Pseudoparabolic Problem with Integral Conditions

2006

In this article, we use the Rothe time-discretization method to prove the well-posedness of a mixed problem with integral conditions for a third order semilinear pseudoparabolic equation. Also we establish the convergence of the method and an error estimate for a semi-discrete approximation.

A mixed problem with only integral boundary conditions for a hyperbolic equation

International Journal of Mathematics and Mathematical Sciences, 2004

We investigate an initial boundary value problem for a second-order hyperbolic equation with only integral conditions. We show the existence, uniqueness, and continuous dependence of a strongly generalized solution. The proof is based on an energy inequality established in a nonclassical function space, and on the density of the range of the operator associated to the abstract formulation of the studied problem by introducing special smoothing operators.