Blow-up for a porous medium equation with a localized source

Abstract

In this paper we investigate the blow-up properties of the positive solutions to a localized porous medium equation v_τ=Δv^m+av^p₁v^q₁(x₀,τ) subject to homogeneous Dirichlet condition and positive initial datum v₀(x). Under appropriate hypotheses, we establish the local existence and obtain that in the case of p₁+q₁<m or p₁+q₁=m and a is sufficiently small, there exists a global solution of the above problem; in the case of p₁+q₁>m, the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum. Moreover, for the special case p₁=0, q₁>m and a is large, under an additional hypothesis on the initial datum, we can also obtain the asymptotic behavior of the blow-up solution.

Introduction

In this paper we investigate the blow-up properties of the positive solution to the following localized porous medium equation

$$\text{v(x,τ)=0,}\text{x∈}\text{∂}\text{Ω,}\text{τ>0,}$$$$\text{v(x,0)=v}{}_0\text{(x),}\text{x∈Ω,}$$

where m>1, a>0, p₁⩾0 and q₁>0 are constants, $\text{Ω}$ is a bounded domain in R^N with smooth boundary $\text{∂}\text{Ω}$, and $\text{x}{}_0\text{∈Ω}$ is a fixed point.

Problem (1.1) models a variety of physical phenomena, which arises, for example, in the study of the flow of a fluid through a porous medium with an internal localized source or in the study of population dynamics (see [4], [7]).

Porous medium equations and the equations of porous medium type with or without local source have been studied by a large number of authors since 70s (see [1], [9], [10], [11], [12]).

Recently, porous medium equation with non-local source has attracted more and more interest (see [2], [6]). But to our best knowledge, there are few literature on problem (1.1). Many works have been devoted to the case m=1 (see [3], [5], [14], [15]). In this case, Eq. (1.1) is a semilinear heat equation. The case p₁=0, q₁>1 was studied by Cannon and Yin [3] and Chadam et al. [5]. Paper [3] studied its local solvability and [5] investigated its blow-up property. The cases a=1, p₁=0, q₁⩾1 and a=1, p₁, q₁⩾1 were studied by Souplet [14], [15]. Paper [14] showed that the positive solution blows up in finite time if the initial value v₀(x) is large by using the method of self-similar subsolution. In the case a=1, p₁=0, q₁>1, paper [15] demonstrated that the solution u(x,t) blows up globally and the limit

converges uniformly on the compact subsets of $\text{Ω}$, where T^′ denotes the blow-up time.

To get the blow-up properties for problem (1.1), we need some transformations to (1.1) first. Let v^m=u, τ=t/m in (1.1), then it becomes

$$\text{u}{}_{\mathrm{t}}\text{=u}{}^{\mathrm{r}}\text{(}\text{Δ}\text{u+au}{}^{\mathrm{p}}\text{u}{}^{\mathrm{q}}\text{(x}{}_0\text{,t)),}\text{x∈Ω,}\text{t>0,}$$$$\text{u(x,t)=0,}\text{x∈}\text{∂}\text{Ω,}\text{t>0,}$$$$\text{u(x,0)=u}{}_0\text{(x),}\text{x∈Ω,}$$

where 0<r=(m−1)/m<1, p=p₁/m⩾0, q=q₁/m>0, and u₀(x)=v₀^m(x). In the sequel, we will focus on problem (1.2).

Throughout this paper, we assume that

$$\text{u}{}_0\text{∈C}{}^1\text{(}\text{Ω}\text{),}\text{u}{}_0\text{(x)>0}\text{in}\text{Ω;}\text{u}{}_0\text{(x)=0,}\text{∂}\text{u}{}_0\text{∂}\text{ν}\text{<0}\text{on}\text{∂}\text{Ω,}$$

where ν is the unit outward normal vector on $\text{∂}\text{Ω}$.

Under hypothesis (1.3), by a slight modification of the method developed by Souplet in [14], [15], we can show that problem (1.2) admits a positive classical solution u(x,t) and obtain: (i) In case of p+q<1 or p+q=1 and a is sufficiently small, there exists a global positive solution. (ii) In case of p+q>1, the solution u(x,t) of problem (1.2) blows up in finite time if the initial value u₀(x) is large enough or there exists a global positive solution u(x,t) to problem (1.2) provided u₀(x) is sufficiently small. Furthermore, for the special case p=0, q>1 and a is large, under an additional hypothesis (H) on the initial datum u₀ which will be given in Section 4, the blow-up set of the blow-up solution u(x,t) of (1.2) is the whole domain $\text{Ω}$, and the limit

converges uniformly on the compact subsets of $\text{Ω}$, where $\text{T}{}^{\mathrm{\ast }}$ is the blow-up time of u(x,t). And these generalize the results of [14], [15].

This paper is organized as follows. In Section 2, we establish the local existence of the classical positive solution of problem (1.2). Results regarding to global existence and blow-up in finite time for problem (1.2) are presented in Section 3. In Section 4, we show the asymptotic behavior of the blow-up solution.