Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in .


Introduction
In this paper, we study the follwoing parabolic problem: where is a ball in is the outward normal and is smooth, nonzero, nonnegative, radially symmetric, satisfiing the following condition: Many real problems in the fields of fluid dynamics, population, heat propagation, and others are modeled using the forms of parabolic partial differntional equations assosiated with different types of initial-boudary conditions [1][2][3]. In some cases, the solutions of parabolic problems cannot be continued globally in time. This is called the blow-up phenomenon which, in time-dependent problems, has been studied over the past years by many authoes [3][4][5][6][7]. One of these problems is the problem of the heat equation defined in a ball with a nonlinear Neumann boundary condition, on which has been introduced in previous articles [7][8][9][10][11].

ISSN: 0067-2904
Rasheed Iraqi Journal of Science, 2020, Special Issue, pp: 200-203 In an earlier study [10], it was proved that if is integrable at infinity for positive values, and is nondecreasing, then with any positive initial function the blow-up occurs in a finite time. In addition, if is a convex and increasing function in then the blow-up can only occur on the boundary.
In another study [8], a speial case was considerd, where showing that for any the blow-up occurs for and it can only occur on the boundary. Futhermore, in other investigations [9,11], it was proved that the upper (lower) blow-up rate estimates are as follows: Another special case, where was considered in another work [7], showing that all positive solutions blow up in finite times, and the blow-up can only occur on the boundary. Moreover, the upper (lower) blow-up rate estimates are as follows: A similer result has been obtained [3] for another case: 1. In this paper, we derive that upper blow-up rate estimate for problem (1), showing that: The rest of this paper is organized as follows: In section two, we discuss the local existance, blowup, and blow-up set with stating the main properties of classical solutions for problem (1). In section three, we derive the upper blow-up rate estimate. Section four is devoted to state some conclusions.

Basic properties and Blow-up Set
It is well known that with any smooth initial function, problem (1) has a local unique classical solution [12]. Moreover, since is positive, increasing , convex, and is integrable at infinity for , then each solution of problem (1) blows up in a finite time and the blow-up can only occur on the boundary [10] . The next lemma, proved in an earlier study [3], presents some solution properties of problem (1). For simplicity, we denote Lemma 2.1. Let be a solution to problem (1). Then 1. is positive and radial on , and is nonnegative in 2.
is positive in and if in then in

Upper Blow-up Rate Estimate
The next theorem is concerned with deraving the upper blow-up rate estimate for problem (1). Theorem 3.2 Let be a blow-up solution to (1), such that . Then there is a positive constant such that (2) Proof.
As in another work [5], we define the function:

Conclusions
This paper is devoted to derive the upper blow-up rate estimate for problem (1). The results show that the upper blow-up rate estimate formula does not depend on , which means that the power function, appeared in the boundary condition, does not make any effect on the blow-up profile to problem (1). Therefore, the influence of the power function may only appear on the blow-up time.