Solvability and dynamics study of a weak solution for a nonlinear parabolic problem with the bessel operator

Zainouba Chebana,Taki-Eddine Oussaeif

doi:10.55059/ijm.2022.1.1/14

Abstract

In this paper, we present a class of parabolic equation with nonlinear where we show two part of this study the theoretical part we prove the existence and uniqueness of the solution by energy inequality method. Then we present finite time blow up by using the energy method.

Highlights

Partial differential equations play an important role in modeling many disciplines including engineering, physics, chemistry economics, and biology, and their study has occupied mathematicians since the eighteenth century with the work of Euler, D’Alambert, Lagrange, and Laplace...; over the last forty years many modern physical, mechanical, biological and technological phenomena, and problems have been shaped by partial differential equations (PDEs), many physical phenomena are modeled by nonclassical parabolic problems associated with boundary conditions
As the first question to be asked in the theoretical study is to know if, for a nonlinear evolution equation with initial conditions and boundary conditions, there exists at least one local solution and if it is unique in the case considered, these problems have been solved for a large class of nonlinear evolution equations by a series of useful
The dynamic questions in the field of partial differential equations problems like the blowup and decay studies are considered among the most important issues because they fall under the framework of controlling the studied phenomenon

Summary

Introduction

Partial differential equations play an important role in modeling many disciplines including engineering, physics, chemistry economics, and biology, and their study has occupied mathematicians since the eighteenth century with the work of Euler, D’Alambert, Lagrange, and Laplace...; over the last forty years many modern physical, mechanical, biological and technological phenomena, and problems have been shaped by partial differential equations (PDEs), many physical phenomena are modeled by nonclassical parabolic problems associated with boundary conditions. As the first question to be asked in the theoretical study is to know if, for a nonlinear evolution equation with initial conditions and boundary conditions, there exists at least one local solution and if it is unique in the case considered, these problems have been solved for a large class of nonlinear evolution equations by a series of useful. There are many works focused on the blow-up property of various parabolic equations (or systems) with homogenous Newmann boundary conditions

Formulation of the nonlinear problem

Position of the associated linear problem

The operator

Existence and uniqueness of a weak solution of the nonlinear problem