We study the following nonlinear parabolic equation:

Forward and backward self-similar solutions for a nonlinear parabolic equation

Chi-Jen Wang June 18, 2004

1 Introduction

We study the following nonlinear parabolic equation:

u t = u ^σ (∆u + u ^p ), x ∈ R ⁿ , t > 0, (1.1) where σ ∈ R, p > 1, and n ≥ 1. The equation (1.1) has been extensively studied for the past years, see, for example, [1], [3], [8] for σ = 0; [6] for σ < 0; [2] for σ ∈ (0, 1);

[5], [4] for σ = 1. In all of these works, much attentions have been paid to the blow-up behaviours of solutions in order to understand the mechanism of thermal runway in combustion problem.

We are interested in the global and non-global existence of positive solutions of (1.1) for σ > 1. In particular, we study the existence of forward and backward self-similar positive solutions of (1.1) in the forms

U (x, t) = (t + 1) ^−α ϕ( |x|

(t + 1) ^β ), (1.2)

V (x, t) = (T − t) ^−α ϕ( | x |

(T − t) ^β ), (1.3)

where T > 0 is given and the similarity exponents are necessarily given by

α = 1

p + σ − 1 , β = p − 1 2 α.

We set ξ= |x|/(t + 1) ^β . It follows that U satisfies (1.1) if and only if ϕ satisfies the following equation

ϕ ⁰⁰ + n − 1

ξ ϕ ⁰ + ϕ ^p + αϕ ^1−σ + βξϕ ^−σ ϕ ⁰ = 0, ξ > 0, (1.4)

1

2

and ϕ ⁰ (0) = 0. Similarly, set ξ = | x | /(T − t) ^β . It follows that V satisfies (1.1) if and only if ϕ satisfies the equation

ϕ ⁰⁰ + n − 1

ξ ϕ ⁰ + ϕ ^p − αϕ ^1−σ − βξϕ ^−σ ϕ ⁰ = 0, ξ > 0, (1.5) and ϕ ⁰ (0) = 0.

In Section 2, following [5], we shall prove that the solution of (1.4) with initial condition

ϕ ⁰ (0) = 0, ϕ(0) = η, (1.6)

exists globally for any η > 0. Also, we prove that ϕ(ξ) and ϕ ⁰ (ξ) tend to 0 as ξ → ∞.

Next, we study the equation (1.5) with initial condition (1.6) in Sections 3 and 4 for σ ∈ (1, 2). In Section 3, we study the one-dimensional case. The multiple-dimensional case is treated in Section 4. We show that, for n = 1, (1.5)-(1.6) has at least N −1 distinct positive global solutions such that ϕ(ξ) → 0 as ξ → ∞, where −N is the largest integer which is less or equal to −(p + σ − 1)/(p − 1) = −1/(2β). Set p ^c (n) = (n + 2)/(n − 2) for n ≥ 3 and p ^c (2) = ∞. In parallel to Section 3 of [4], we can also derive that, for n ≥ 2 and 1 < p < p c (n), (1.5)-(1.6) has at least N 1 distinct positive global solutions such that ϕ(0) > κ and ϕ(ξ) → 0 as ξ → ∞, where N 1 is the largest integral which is less than or equal to N/2 and κ = α ^α .

In Section 5, we study the asymptotic behaviors of bounded global solutions of (1.4) and (1.5) as ξ → ∞. We shall show that for any bounded global solution ϕ(ξ) of (1.4) or (1.5) the limit

ξ→∞ lim {ξ ^α/β ϕ(ξ) } = A exists and A > 0. It follows that

t→T lim

V (x, t) = A |x| ^−α/β , x 6= 0 for any nonconstant bounded global solution of (1.5).

From these results there always exists a symmetric positive monotone self-similar solution V of (1.1) in the form (1.3) such that V blows up only at single point x = 0 at T for a given finite time T . Note that N = 2 for p ≥ σ + 1 and N ≥ 3 for p ∈ (1, σ + 1). Also, there are some other self-similar single-point blow-up patterns with different oscillations, if p ∈ (1, σ + 1).

It is also interesting to remark that for any solution ϕ of (1.4), (1.6) the corresponding forward self-similar solution U defined by (1.2) exists globally. Note that the initial data u 0 (x) := U (x, 0) = ϕ( |x|) satisfies

|x|→∞ lim |x| ^α/β u 0 (x) = A.