Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition

Abstract

This paper deals with a nonlinear p -Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatis x=1 are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Xiliu Li 1 and Chunlai Mu 1 and Qingna Zhang 1 and Shouming Zhou 1

Recommended by Kanishka Perera

1, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 2 December 2011; Accepted 10 January 2012

1. Introduction

In this paper, we consider the following problem: [figure omitted; refer to PDF] where ψ(u) is a monotone increasing function with ψ(0)=0 , p>1 , g(u)>0 , ^{g[variant prime]} (u)<0 for u>0 , and _{lim u[arrow right]⁰⁺} g(u)=+∞ . The initial value _u0 (x) is positive and satisfying some compatibility conditions.

If ψ(u)=^u1/m with m>0 , (1.1) becomes the well-known non-Newtonian filtration equation, which is used to describe the non-stationary flow in a porous medium of fluids with a power dependence of the tangential stress on the velocity of the displacement under polytropic conditions (see [1, 2]).

Many papers have been devoted to the study of critical exponents of non-Newtonian filtration equation, see [3-5]. There are many results on the quenching phenomenon, see, for instance [6-12]. By the quenching phenomenon we mean that the solution approaches a constant but its derivative with respect to time variable t tends to infinity as (x,t) tends to some point in the spatial-time space. The study of the quenching phenomenon began with the work of Kawarada through the famous initial boundary problem for the reaction-diffusion equation: _ut =_uxx +1/(1-u) (see [13]).

As an example of the type of results, we wish to obtain, let us recall results for a closely related problem [figure omitted; refer to PDF] where q>0 . In [14], it was shown that u quenches in finite time for all _u0 , and the only quenching point is x=1 . Furthermore, the behavior of u near quenching was described there. It is easily seen that (1.2) is a special case of (1.1).

If p=2 , then (1.1) reduces to the following equation: [figure omitted; refer to PDF] Deng and Xu proved in [15] the finite time quenching for the solution and established results on quenching set and rate for (1.3). If ψ(u)=u , then (1.1) reduces to the following equation, see [11], [figure omitted; refer to PDF] and they obtained that the bounds for the quenching rate, and the quenching occurs only at x=1 .

In this paper, we extend the equation _ut =_{(^{|ux |p-2}ux )x} , see [11], to a more general form _(ψ(u))t =_{(^{|ux |p-2}ux )x} . We prove that quenching occurs only at x=1 . We determine the bounds for the quenching rate, and present an example which shows the applicability of our results.

The main results are stated as follows.

Theorem 1.1.

Suppose that the initial data satisfies ^{u0[variant prime]} (x)...4;0 and ^{u0[variant prime][variant prime]} (x)...4;0 for 0...4;x...4;1 , and one of the following conditions holds:

(i) ^{ψ[variant prime][variant prime]} (u)>0 for u>0 ,

(ii) ^{ψ[variant prime][variant prime]} (u)<0 for u>0 , _{limsup u[arrow right]⁰⁺} (g[variant prime](u)/ψ[variant prime](u))<0 .

Then every solution of (1.1) quenches in finite time, and the only quenching point is x=1 .

Next, we deal with the quenching rate. Before we establish upper bounds for the quenching rate, we introduce the following hypothesis:

( _H1 ): ^{ψ[variant prime][variant prime]} (u)g(u)...5;2(p-1)^{ψ[variant prime]} (u)^{g[variant prime]} (u) , ^{ψ[variant prime]} (u)[(p-2)^{g[variant prime] 2} (u)+g(u)^{g[variant prime][variant prime]} (u)]...5;^{ψ[variant prime][variant prime]} (u)g(u)^{g[variant prime]} (u) .

Theorem 1.2.

Suppose that the conditions of Theorem 1.1 and the hypothesis (_H1 ) hold. Then there exists a positive constant _C1 such that [figure omitted; refer to PDF]

Next, we will give the lower bound on the quenching rate, the derivation of which is in the spirit of [15]. We need the following additional hypotheses: there exists a constant σ(-∞<σ...4;_σ0 =min {1,2-1/(p-1)}) such that

( _H2 ): ^{(g(p-1)(σ-1) (u)g[variant prime](u)/ψ[variant prime](u))[variant prime][variant prime]} <0 ,

( _H3 ): (5p-2σp-2σ-6)(^g(p-1)(σ-1) (u)g[variant prime](u))[variant prime]ψ[variant prime](u)...5;(p-1)(3-σ)^g(p-1)(σ-1) (u)^{g[variant prime]} (u)^{ψ[variant prime][variant prime]} (u) ,

( _H4 ): (^{g(p-1)(σ-1)-1} (u)g[variant prime](u)/ψ[variant prime](u))[variant prime]<0 .

Theorem 1.3.

Suppose that the hypotheses of Theorem 1.1 hold. Furthermore, suppose that the hypotheses (_H2 ) - (_H4 ) hold. Then there exists a positive constant _C2 such that [figure omitted; refer to PDF] Furthermore, if (_H1 ) holds, then the quenching rates are [figure omitted; refer to PDF]

Next, as an application of the main results of this paper, we study the following concrete example: [figure omitted; refer to PDF] where q>0 , p>1 , and m>0 . We will verify that (1.8) satisfies the hypotheses (_H1 ) - (_H4 ) , and we give the following theorem.

Theorem 1.4.

Suppose that ^{u0[variant prime]} (x)...4;0 and ^{u0[variant prime][variant prime]} (x)...4;0 for 0...4;x...4;1 . Then the solution of (1.8) satisfies [figure omitted; refer to PDF] where _C3 and _C4 are positive constants.

The plan of this paper is as follows. In Section 2, we prove that quenching occurs only at x=1 , that is the proof of Theorem 1.1. In Section 3, we derive the estimates for the quenching rate, that is the proof of Theorems 1.2 and 1.3. In Section 4, we present results for certain ψ(u) and g(u) , that is the proof of the Theorem 1.4.

2. Quenching on the Boundary

In this section, we prove finite time quenching. We rewrite problem (1.1) into the following form: [figure omitted; refer to PDF] where a(u)=1/(ψ[variant prime](u)) . Clearly, ψ[variant prime](u)...0;0 for u>0 .

Lemma 2.1.

Assume the solution u of problem (2.1) exists in (0,_T0 ) for some _T0 >0 , and ^{u0[variant prime]} (x)...4;0 , ^{u0[variant prime][variant prime]} (x)...4;0 for 0...4;x...4;1 . Then _ux (x,t)<0 and _ut (x,t)<0 in (0,1]×(0,_T0 ) .

Proof.

Let v(x,t)=_ux (x,t) . Then v(x,t) satisfies [figure omitted; refer to PDF] The maximum principle leads to v(x,t)<0 , and thus _ux (x,t)<0 in (0,1]×(0,_T0 ) . Then it is easy to see that the problem (2.2) is nondegenerate in (0,1]×(0,_T0 ) . So _ux (x,t) is a classical solution of (2.2). Similarly, letting w(x,t)=_ut (x,t) , we have [figure omitted; refer to PDF] Making use of the maximum principle, we obtain _ut (x,t)<0 in (0,1]×(0,_T0 ) . Hence, the solutions of problem (2.1) u∈^C2,1 ((0,1]×(0,_T0 )) with _ux (x,t)<0 and _ut (x,t)<0 in (0,1]×(0,_T0 ) .

The Proof of Theorem 1.1

By the maximum principle, we know that 0<u(·,t)...4;M for all t in the existence interval, where M=_{max 0...4;x...4;1 u0} (x) . Define F(t)=^∫01 ψ(u(x,t))dx . Then F(t) satisfies [figure omitted; refer to PDF] Thus F(t)...4;F(0)-^gp-1 (M)t , which means that F(_t0 )=0 for some _t0 >0 . From the fact that ψ(u)>0 for u>0 and _ux (x,t)<0 for 0<x...4;1 , we find that there exists a T (0<t...4;_T0 ) such that _{lim t[arrow right]^T-} u(1,t)=0 . By virtue of the singular nonlinearity in the boundary condition, u must quench at x=1 . In what follows, we only need to prove that quenching cannot occur in ((1/2),1)×(η,T) for some η(0<η<T) . Consider two cases.

Case 1.

^{ψ[variant prime][variant prime]} (u)>0 for u>0 . Let h(x,t)=^{|_ux |p-2}_ux +[straight epsilon](x-(1/4))^gp-1 (M) in ((1/4),1)×(η,T) , where [straight epsilon] is a positive constant. Then h(x,t) satisfies [figure omitted; refer to PDF] for (x,t)∈((1/4),1)×(η,T) , since (a[variant prime](u))/(a(u))=-(ψ[variant prime][variant prime](u))/(ψ[variant prime](u))...4;0 . On the parabolic boundary, h((1/4),t)=|_ux^|p-2_ux ((1/4),t)<0 for η...4;t<T ; if [straight epsilon] is sufficiently small, h(1,t)...4;^gp-1 (M)((3[straight epsilon]/4)-1)<0 for η...4;t<T , and h(x,η)...4;-|_ux ((1/4),η)^|p-1 +(3[straight epsilon]/4)^gp-1 (M)<0 for (1/4)...4;x...4;1 . Thus by the maximum principle, we have h(x,t)...4;0 in ((1/4),1)×(η,T) , which leads to [figure omitted; refer to PDF] So we have [figure omitted; refer to PDF] Integrating (2.7) from x to 1, we obtain [figure omitted; refer to PDF] It then follows that u(x,t)>0 if x<1 .

Case 2.

^{ψ[variant prime][variant prime]} (u)<0 for u>0 . Let k(x,t)=_ut -[straight epsilon](x-(1/2))_ux in ((1/2),1)×(η,T) . Then k(x,t) satisfies [figure omitted; refer to PDF] for (x,t)∈((1/2),1)×(η,T) . On the boundary, k((1/2),t)=_ut ((1/2),t)<0 for η...4;t<T . Since _{lim t[arrow right]^T-} u(1,t)=0 and lim _{sup u[arrow right]⁰⁺} ((g[variant prime](u))/(ψ[variant prime](u)))<0 , if η is close to T and [straight epsilon] is small enough, ^{g[variant prime]} (u(1,t))+([straight epsilon]^{ψ(u(1,t))[variant prime]} /2(p-1)^g(p-2) u(1,t))+2...4;0 for η...4;t<T . Thus [figure omitted; refer to PDF] It is easily seen that k(x,η)...4;0 for (1/2)...4;x...4;1 . Hence, the maximum principle yields that k(x,t)...4;0 in [(1/2),1]×[η,T) . In particular, [figure omitted; refer to PDF] Integrating (2.11) from t to T , we obtain [figure omitted; refer to PDF] Define G(u)=^∫0u (1/g(s))ds . Since ^{G[variant prime]} (u)=1/g(u)>0 for u>0 , the inverse ^G-1 exists. In view of (2.12), we can see [figure omitted; refer to PDF] Let H(x,t)=u(x,t)-_c1 (1-^x2 )-_c2^G-1 (([straight epsilon]/2)(T-t)) , where _c1 and _c2 are positive constants. Since g[variant prime](u)<0 , _ux (x,t)<0 , and by (2.13), we have that in (0,1)×(η,T) [figure omitted; refer to PDF] provided [straight epsilon]_c2 g(M)^{ψ[variant prime]} (M)...5;4_c1 (p-1)|_ux^|p-2 , which is true since 0...4;|_ux (x,t)|...4;g(u(1,t))=g(_M0 ) , where 0<_M0 ...4;M . On the other hand, _Hx (0,t)=0 ; H(1,t)=u(1,t)-_c2^G-1 (([straight epsilon]/2)(T-t))...5;0 if _c2 ...4;1 and H(x,η)...5;0 if _c1 and _c2 are small enough. Thus by the maximum principle, we find that H(x,t)...5;0 in (0,1)×(η,T) , which implies that u(x,t)...5;_c1 (1-^x2 )>0 if x<1 .

3. Bounds for Quenching Rate

In this section, we establish bounds on the quenching rate. We first present the upper bound.

The Proof of Theorem 1.2

We define a function Φ(x,t)=^{|_ux |p-2}_ux +^{[straight phi]p-1} (x)^gp-1 (u(x,t)) in (0,1)×(η,T) , where [straight phi](x) is given as follows: [figure omitted; refer to PDF] with some _x0 <1 and l...5;max {3,(1/(p-1))} is chosen so large that [straight phi](x)...4;-(^{u0[variant prime]} (x)/g(_u0 (x)) for _x0 <x...4;1 . It is easy to see that Φ(0,t)=Φ(1,t)=0 , and Φ(x,0)...4;0 . On the other hand, in (0,1)×(η,T) , Φ satisfies [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By (_H1 ) and the definition of [straight phi](x) , it follows that [figure omitted; refer to PDF] Thus, the maximum principle yields Φ(x,t)...4;0 , that is [figure omitted; refer to PDF] Moreover, by the definition of the limit, we see that _Φx (1,t)...5;0 since Φ(x,t)...4;0 . In fact, [figure omitted; refer to PDF] which means [figure omitted; refer to PDF] Integrating (3.7) from t to T , we get [figure omitted; refer to PDF]

We then give the lower bound.

The Proof of Theorem 1.3

Let d(u)=a(u)^g(p-1)(σ-1) (u)^{g[variant prime]} (u) . Notice that the hypotheses (_H2 ) - (_H4 ) are equivalent to

( _H...2 ): ^{d[variant prime][variant prime]} (u)...4;0 ,

( _H...3 ): ^{d[variant prime]} (u)(5p-2σp+2σ-6)...5;(d(u)/a(u))^{a[variant prime]} (u)(2p-σp+σ-3) ,

( _H...4 ): d(u)g[variant prime](u)>d[variant prime](u)g(u) ,

respectively. Letting τ be close to T , we consider Ψ(x,t)=_ut -[straight epsilon]d(u)^{(-_ux )(p-1)(2-σ)} in (1-T+τ,1)×(τ,T) , where [straight epsilon] is a positive constant. Through a fairly complicated calculation, we find that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Since the hypotheses (_H...2 ) - (_H...3 ) hold, and d(u)<0 , a(u)>0 , we see that _R2 (x,t)<0 . Thus, we have [figure omitted; refer to PDF] for (x,t)∈(1-T+τ,1)×(τ,T) . On the parabolic boundary, since x=1 is the only quenching point, if [straight epsilon] is small enough, then both Ψ(1-T+τ,t) and Ψ(x,τ) are negative. At x=1 , in view of (_H...4 ) , we have [figure omitted; refer to PDF] provided [straight epsilon] is sufficiently small. Hence, by the maximum principle, we have Ψ(x,t)...4;0 on [1-T+τ,1]×[τ,T) . In particular, Ψ(1,t)...4;0 , that is, [figure omitted; refer to PDF] Integration of (3.13) over (t,T) , then leads to [figure omitted; refer to PDF]

4. Results for Certain Nonlinearities

In this section, we give the concrete quenching rate of solutions for (1.8).

The Proof of Theorem 1.4

We first present the upper bound. Consider two cases.

Case 1.

m+q(p-1)...5;2 . We only need to verify the hypothesis (_H1 ) . Since [figure omitted; refer to PDF] then we have [figure omitted; refer to PDF] Therefore, we get the upper bound as [figure omitted; refer to PDF]

Case 2.

m+q(p-1)<2 . We use a modification of an argument from [16]. For t∈[η,T) with some η such that u(1,η)<1 , set [figure omitted; refer to PDF] with [figure omitted; refer to PDF] where -(m+q+1)<γ...4;-(m+1)(q+1) .

A routine calculation shows [figure omitted; refer to PDF] where I(t)=γ^{∫1-ξ(t)1um} (x,t)dx+(q+1)^uq+1 (1,t)^um (1-ξ(t),t) . Since _ux ...4;0 and _uxx ...4;0 in [0,1]×[η,T) , we find [figure omitted; refer to PDF] for any x∈[1-ξ(t),1] and t∈[η,T) . By (4.4), (4.5) and (4.7), we have [figure omitted; refer to PDF] or equivalently, [figure omitted; refer to PDF] We now claim that I(t)...4;0 on [η,T) . In fact [figure omitted; refer to PDF] where 1-ξ(t)<ζ(t)<1 . When 0<m<1 , γ...4;-(((m+2)(q+1))/2) , because _ux ,_uxx ...4;0 and ^∫1-ξ(t)1 (1-ξ(t),t)) dx=(1/2)^u2q+2 (1,t) , we have [figure omitted; refer to PDF] when 1...4;m<2 , γ...4;-(q+1)(m+1) , we have [figure omitted; refer to PDF] In conclusion, when 0<m<2 , γ...4;-(q+1)(m+1) , we have [figure omitted; refer to PDF] From (4.6), (4.9), and (4.13), it then follows that [figure omitted; refer to PDF] Integrating the above inequality from t to T , we obtain [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] which in conjunction with (4.9) yields the desired upper bound.

We then give the lower bound. We examine the validity of hypotheses (_H2 ) - (_H4 ) .

Firstly, for (_H2 ) , we find [figure omitted; refer to PDF] provided σ<_σ1 =-((m+1)/(q(p-1)))+((p-2)/(p-1)) or σ>_σ2 =-(m/(q(p-1)))+((p-2)/(p-1)) .

Secondly, for (_H3 ) , we have [figure omitted; refer to PDF] provided _σ3 ...4;σ...4;_σ4 with [figure omitted; refer to PDF] where a=-2q^(p-1)2 , b=(p-1)(7pq-10q-m-1) , and c=-5^p2 q+16pq-12q+3pm+2p-3m-3 .

Thirdly, for (_H4 ) , we obtain [figure omitted; refer to PDF] provided σ<_σ5 =1-(q/(m(p-1))) . Since _σ2 >_σ5 , we choose a σ such that _σ3 <σ<min {_σ1 ,_σ4 ,_σ5 } and hypotheses (_H2 ) - (_H4 ) hold. Thus the proof is completed.

Acknowledgments

This work is supported in part by NSF of PR China (11071266) and in part by Natural Science Foundation Project od CQ CSTC (2010BB9218).

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