New results on IBVP for Class of Nonlinear Parabolic Equations

Svetlin G Georgi̇ev,Khaled Zenni̇r

doi:10.31197/atnaa.417824

Abstract

In this article we propose a new approach for investigation the local existence of classical solutions of IBVP for a class of nonlinear parabolic equations.

Highlights

In this article we investigate the IBVP ut − u = f (t, x, u, Du), t ≥ 0, x1 ≥ 0, (x2, . . . , xn) ∈ Rn−1, u(0, x) = u0(x), x1 ≥ 0, (x2, . . . , xn) ∈ Rn−1, u(t, 0, x2, . . . , xn) = v(t, x2, . . . , xn), t ≥ 0, (x2, . . . , xn) ∈ Rn−1, (1.1) where n ≥ 2, u = n i=1 uxixi Du =(ux1, ux2, . . . , uxn ), u : Rn+1 −→R is unknown function, fR × Rn × R × Rn −→ R, u0 : Rn −→ R and v : Rn −→ R are given functions which satisfy the following conditions
In this article we propose a new approach for investigation the local existence of classical solutions of IBVP for a class of nonlinear parabolic equations
In this article we propose new idea which tell us that the local existence of classical solutions of the IBVP is connected with the integral representation of the solutions, it is not connected with the dimension n and if the domain is bounded or not

Summary

Introduction

The main question which we consider here is local existence of classical solutions to the problem (1.1). In this article we propose new idea which tell us that the local existence of classical solutions of the IBVP is connected with the integral representation of the solutions, it is not connected with the dimension n and if the domain is bounded or not. At this moment the problem for existence of classical solutions for the problem (1.1) for arbitrary dimension n ≥ 2 was opened. There exist positive constant m such that there exists a solution u ∈ C1([0, m], C2(Rn1+)) to the problem (1.1).

Here x c xi c

The above lemma motivate us to define the operator xs xs

We define the sets