• 沒有找到結果。

In this paper, we study the following initial value problem:

N/A
N/A
Protected

Academic year: 2021

Share "In this paper, we study the following initial value problem:"

Copied!
2
0
0

加載中.... (立即查看全文)

全文

(1)

1 Introduction

In this paper, we study the following initial value problem:

( g ′′ − βyg

m1

−1 g + αmg

m1

= 0, y > 0,

g (0) = mg 1+

mk

(0), (1)

where k < 0, m < 0, and α, β are given by

α := 1

1 − m − 2k , β := −k

1 − m − 2k = −kα.

This problem arises in the study of the quenching behavior of the problem

 

 

u t = (u m−1 u x ) x , x ∈ (0, 1), t > 0, (u m−1 u x )(0, t) = u m+k (0, t), t > 0,

(u m−1 u x )(1, t) = 0, t > 0, u(x, 0) = u 0 (x), x ∈ [0, 1].

(2)

We say that a solution quenches, if its minimum reaches zero in finite time. Quenching phe- nomena (when, where, and how the solution quenches) have been studied by many authors, since the first work of Kawarada [10] in 1975. For more references, we refer the reader to the survey papers by Levine [11, 12] and Chan [1]. See also the references [3, 4, 5, 6, 7, 9].

We assume that u 0 is a C 1 function in [0, 1] such that

u 0 ≥ δ > 0, u 0 ≥ 0 in [0, 1], u 0 (0) = u 1+k 0 (0), u 0 (1) = 0. (3) Note that from (3) and the maximum principle, it follows that u x > 0 as long as u >

0. We shall prove in §2 that there is T ∈ (0, ∞) such that u > 0 in [0, 1] × [0, T ) and lim inf t↑T

u(0, t) = 0.

We are concerned with the existence of the self-similar positive solutions of (2) in the form

u(x, t) = (T − t) α ϕ(x/(T − t) β ). (4)

Set y = x/(T − t) β . It follows that u satisfies (2) if and only if ϕ satisfies

m−1 ϕ ) − βyϕ + αϕ = 0, 0 < y < (T − t) −β , (ϕ m−1 ϕ )(0) = ϕ m+k (0),

m−1 ϕ )((T − t) −β ) = 0.

(5)

Let g = ϕ m . Then ϕ satisfies (5) if and only if g satisfies

g ′′ − βyg

m1

−1 g + αmg

m1

= 0, 0 < y < (T − t) −β , g (0) = mg 1+

mk

(0),

g ((T − t) −β ) = 0.

Since we are interested in the behavior of u as t ↑ T , we end up with the problem (1) and we shall prove the existence of the globally monotone decreasing solution of (1).

1

(2)

The case k = 0 has been studied by Ferreira-de Pablo-Quir´os-Rossi in [2]. In [2], they proved that the quenching always occurs for any positive solution. They also study the quenching set, quenching rate, and beyond quenching. In particular, they found the so- called super-fast quenching for certain cases. Indeed, this is due to the lack of positive self-similar solutions. Therefore, it is very important to study the self-similar solutions. We shall apply the shooting method to study the initial value problem (1).

The organization of this paper is as follows. In §2, we shall first prove that quenching always occurs and then give some preliminary results related to the initial value problem (1).

Then we prove in §3 the existence of self-similar solutions by a shooting method. Finally, we study the asymptotic behavior of any globally monotone decreasing solution of (1) in §4.

2

參考文獻

相關文件

In Case 1, we first deflate the zero eigenvalues to infinity and then apply the JD method to the deflated system to locate a small group of positive eigenvalues (15-20

In Sections 3 and 6 (Theorems 3.1 and 6.1), we prove the following non-vanishing results without assuming the condition (3) in Conjecture 1.1, and the proof presented for the

Since we use the Fourier transform in time to reduce our inverse source problem to identification of the initial data in the time-dependent Maxwell equations by data on the

Given a shift κ, if we want to compute the eigenvalue λ of A which is closest to κ, then we need to compute the eigenvalue δ of (11) such that |δ| is the smallest value of all of

In this paper, we have shown that how to construct complementarity functions for the circular cone complementarity problem, and have proposed four classes of merit func- tions for

In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases.. In addition,

We give some numerical results to illustrate that the first pass of Algorithm RRLU(r) fails but the second pass succeeds in revealing the nearly rank

In the following we prove some important inequalities of vector norms and matrix norms... We define backward and forward errors in