Global existence and blow-up for degenerate and singular parabolic system with localized sources

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Abstract

This paper deals with the blow-up properties of the solution to degenerate and singular parabolic system with localized sources and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally and blows up in finite time are obtained. Furthermore, under certain conditions, it is proved that the blow-up set of the solution is the whole domain.

Introduction

In this paper, we consider the following degenerate and singular nonlinear reaction–diffusion equations with localized sources:xq1ut-(xr1ux)x=vp1(x0,t),(x,t)(0,a)×(0,T),xq2vt-(xr2vx)x=up2(x0,t),(x,t)(0,a)×(0,T),u(0,t)=u(a,t)=v(0,t)=v(a,t)=0,t(0,T),u(x,0)=u0(x),v(x,0)=v0(x),x[0,a],where T > 0, a > 0, r1, r2  [0, 1), ∣q1 + r1  0, ∣q2 + r2  0 and p1 > 1, p2 > 1, x0  (0, a) is a fixed point, and u0(x),v0(x)C2+α(D¯) for some α  (0, 1) are nonnegative nontrivial functions and satisfy the compatibility condition.

Let D = (0, a) and Ωt = D × (0, t), D¯ and Ω¯t are their closures respectively. Since ∣q1 + r1  0, ∣q2 + r2  0, the coefficients of ut, ux, uxx and vt, vx, vxx may tend to 0 or ∞ as x tends to 0, we can regard the equations as degenerate and singular.

Floater [1] and Chan et al. [2] investigated the blow-up properties of the following degenerate parabolic problem:xqut-uxx=up,(x,t)(0,a)×(0,T),u(0,t)=u(a,t)=0,t(0,T),u(x,0)=u0(x),x[0,a],where q > 0 and p > 1. Under certain conditions on the initial datum u0(x), Floater [1] proved that the solution u(x, t) of (1.2) blows up at the boundary x = 0 for the case 1 < p  q + 1. This contrasts with one of the results in [3], which showed for the case q = 0, the blow-up set of solution u(x, t) of (1.2) is a proper compact subset of D. For the case p > q + 1, under certain conditions, in [2] Chan et al. proved that x = 0 is not a blow-up point and the blow-up set is a proper compact subset of D.

In [4], Chen et al. consider the following degenerate nonlinear reaction–diffusion equation with nonlocal source:xqut-(xγux)x=0aupdx,(x,t)(0,a)×(0,T),u(0,t)=u(a,t)=0,t(0,T),u(x,0)=u0(x),x[0,a],they established the local existence and uniqueness of classical solution. Under appropriate hypotheses, they also got some sufficient conditions for the global existence and blow-up of positive solution. Furthermore, under certain conditions, it is proved that the blow-up set of the solution is the whole domain.

In [5], Zhou et al. discussed the following degenerate and singular nonlinear reaction–diffusion equations with nonlocal source:xq1ut-(xr1ux)x=0avp1dx,(x,t)(0,a)×(0,T),xq2vt-(xr2vx)x=0aup2dx,(x,t)(0,a)×(0,T),u(0,t)=u(a,t)=v(0,t)=v(a,t)=0,t(0,T),u(x,0)=u0(x),v(x,0)=v0(x),x[0,a],they established the local existence and uniqueness of classical solution. Under appropriate hypotheses, they obtained some sufficient conditions for the global existence and blow-up of positive solution.

In [6], Chan et al. considered the following degenerate semilinear parabolic equation:xqut-uxx=f(u(x0,t)),(x,t)(0,a)×(0,T),u(0,t)=u(a,t)=0,t(0,T),u(x,0)=u0(x),x[0,a],where q  0, x0  (0, a). They established the local existence and uniqueness of classical solution and got some sufficient conditions for the blow-up of positive solution. Furthermore, under certain conditions, it is proved that the blow-up set of the solution is the whole domain.

In this paper, we want to know the effect of the singularity, degeneracy and localized reaction on the behavior of the solution of (1.1) and show that the blow-up set of the solution of (1.1) is the whole domain. Our main results are stated as follows.

Theorem 1.1

Let (u(x, t), v(x, t)) be the solution of (1.1), if there exist a1, a2 > 0, such that u0(x)  a1ψ(x), v0(x)  a2φ(x), then (u(x, t), v(x, t)) exists globally, where ψ(x)=a2-r12-r1(xa)1-r1(1-xa) and φ(x)=a2-r22-r2(xa)1-r2(1-xa).

Theorem 1.2

If u0(x), v0(x) are sufficiently large in a neighborhood of x0, then the solution (u(x, t), v(x, t)) of problem (1.1) blows up in a finite time.

Theorem 1.3

When q1 > 0, r1 = 0 or q2 > 0, r2 = 0, if the solution of problem (1.1) blows up in a finite time, then the blow-up set is D¯.

Theorem 1.4

When q1 = 0, 0  r1 < 1, or q2 = 0, 0  r2 < 1, if there exists M > 0 such that (xr1u0x(x))xM or (xr2v0x(x))xM in (0, a), and the solution of problem (1.1) blows up in a finite time, then the blow-up set is D¯.

Remark 1

When q1 = q2, r1 = r2 and p1 = p2, we also can establish the local existence of the nonnegative solution and get the finite time blow-up results for the following single equation xqut  (xrux)x = up(x0, t).

This paper is organized as follows. In next section, we show the local existence and uniqueness of the solution of (1.1). In Section 3, some criteria for the solution (u(x, t), v(x, t)) to exists globally and blows up in finite time are given. In the last section, we discuss the blow-up set.

Section snippets

Local existence

In order to prove the existence of the solution to (1.1), we start with the following comparison principle.

Lemma 2.1

Let b1(x, t) and b2(x, t) be continuous nonnegative functions defined on Ω¯T, and (u(x,t),v(x,t))(C(Ω¯T)C2,1(ΩT))2 satisfiesxq1ut-(xr1ux)xb1(x,t)v(x0,t),(x,t)ΩT,xq2vt-(xr2vx)xb2(x,t)u(x0,t),(x,t)ΩT,u(0,t)0,u(a,t)0,v(0,t)0,v(a,t)0,t(0,T],u(x,0)0,v(x,0)0,xD¯.Then u(x, t)  0, v(x, t)  0 on Ω¯T.

Proof

The proof is similar to the proof of Lemma 2.1 in [5], so we omit it. 

Obviously, (u, v) = (0, 

Global existence and blow-up

In this section, we give global existence and blow-up result of the solution to (1.1) and prove Theorem 1.1, Theorem 1.2.

Proof of Theorem 1.1

Leta1=a2p1a2-r22-r2x0a1-r21-x0ap1,a2=a1p2a2-r12-r1x0a1-r11-x0ap2,

and u¯=a1ψ(x), v¯=a2φ(x), where ψ(x) and φ(x) are given by Theorem 1.1, then we havexq1u¯t(x,t)-(xr1u¯x(x,t))x=-(xr1a1ψ(x))=a1=a2p1a2-r22-r2x0a1-r21-x0ap1=v¯p1(x0,t),(x,t)(0,a)×(0,T),xq2v¯t(x,t)-(xr2v¯x(x,t))x=u¯p2(x0,t),(x,t)(0,a)×(0,T),u¯(0,t)=u¯(a,t)=v¯(0,t)=v¯(a,t)=0,t(0,T),u¯(x,0)=a1ψ(x)u0(x),v¯(x,0)=a

Blow-up set

In this section, we discuss the blow-up set in two special cases.

Case 1. q1 > 0, r1 = 0 or q2 > 0, r2 = 0.

Chan et al. [10], [6] proved that there exists Green’s function G(x, ξ, t  τ) associated with the operator L=xq1t-2x2 with the first boundary condition, and obtained the following lemmas.

Lemma 4.1

(a) For t > τ, G(x, ξ, t  τ) is continuous for (x, t, ξ, τ)  ([0, a] × (0, T]) × ((0, a] × [0, T)).

(b) For each fixed (ξ, τ)  (0, a] × [0, T), Gt(x, ξ, t  τ)  C([0, a] × (τ, T]).

(c) In {(x, t, ξ, τ): x and ξ are in (0, a), T  t > τ  0}, G(x, ξ, t  τ) is

Acknowledgement

This work is supported in part by NNSF of China (10771226) and in part by Natural Science Foundation Project of CQ CSTC (2007BB0124).

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